\chapter{A Voyage of Discovery}
\pagenumbering{arabic}
The role of the ocean\index{ocean} on weather and climate is often discussed in the news. Who has not heard of El Ni\~{n}o and changing weather patterns, the Atlantic hurricane season and storm surges? Yet, what exactly is the role of the ocean? And, why do we care?
\section[Physics of the ocean]{Why study the Physics of the ocean?}
The answer depends on our interests, which devolve from our use of the ocean. Three
broad themes are important:
\begin{enumerate}
\vitem We get food from the ocean. Hence we may be interested
in processes which influence the sea just as farmers are interested in the weather and
climate. The ocean not only has weather such as temperature changes and currents,
but the oceanic weather fertilizes the sea. The atmospheric weather seldom
fertilizes fields except for the small amount of nitrogen fixed by lightning.
\vspace{-0.5ex}
\vitem We use the ocean. We build structures on the shore or just offshore. We use
the ocean for transport. We obtain oil and gas below the ocean. And, we use the
ocean for recreation, swimming, boating, fishing, surfing, and diving. Hence we are
interested in processes that influence these activities, especially waves, winds,
currents, and temperature.
\vspace{-0.5ex}
\vitem The ocean influence the atmospheric weather and climate. The ocean
influence the distribution of rainfall, droughts, floods, regional climate, and the
development of storms, hurricanes, and typhoons. Hence we are interested in
air-sea interactions, especially the fluxes of heat and water across the sea surface,
the transport of heat by the ocean, and the influence of the ocean on climate and
weather patterns.
\end{enumerate}
These themes influence our selection of topics to study. The topics then
determine what we measure, how the measurements are made, and the geographic areas of
interest. Some processes are local, such as the breaking of waves on a beach, some
are regional, such as the influence of the North Pacific on Alaskan weather, and some
are global, such as the influence of the ocean on changing climate and global
warming.
If indeed, these reasons for the study of the ocean are important, lets
begin a voyage of discovery. Any voyage needs a destination. What is ours?
\section{Goals}
At the most basic level, I hope you, the students who are reading this text, will become aware of some of the major conceptual schemes (or theories) that form the foundation of physical oceanography\index{physical oceanography!goals of}, how they were arrived at, and why they are widely accepted, how oceanographers achieve order out of a random ocean, and the role of experiment in oceanography (to paraphrase Shamos, 1995: p. 89).
More particularly, I expect you will be able to describe physical processes influencing the ocean and coastal regions: the interaction of the ocean with the atmosphere, and the distribution of oceanic winds, currents, heat fluxes\index{heat flux}, and water masses. The text emphasizes ideas rather than mathematical techniques. I will try to answer such questions as:
\begin{enumerate}
\item What is the basis of our understanding of physics of the ocean?
\vspace{-0.5ex}
\begin{enumerate}
\item What are the physical properties of sea water?
\vspace{-0.5ex}
\item What are the important thermodynamic and dynamic processes influencing the
ocean?
\vspace{-0.5ex}
\item What equations describe the processes and how were they derived?
\vspace{-0.5ex}
\item What approximations were used in the derivation?
\vspace{-0.5ex}
\item Do the equations have useful solutions?
\vspace{-0.5ex}
\item How well do the solutions describe the process? That is, what is the
experimental basis for the theories?
\vspace{-0.5ex}
\item Which processes are poorly understood? Which are well understood?
\end{enumerate}
\item What are the sources of information about physical variables?
\vspace{-0.5ex}
\begin{enumerate}
\vspace{-0.5ex}
\item What instruments are used for measuring each variable?
\vspace{-0.5ex}
\item What are their accuracy and limitations?
\vspace{-0.5ex}
\item What historic data exist?
\vspace{-0.5ex}
\item What platforms are used? Satellites, ships, drifters\index{drifters}, moorings?
\end{enumerate}
\vspace{-0.5ex}
\item What processes are important? Some important process we will study include:
\begin{enumerate}
\vspace{-0.5ex}
\item Heat storage and transport \index{transport!heat}in the ocean.
\vspace{-0.5ex}
\item The exchange of heat with the atmosphere and the role of the ocean in climate.
\vspace{-0.5ex}
\item Wind and thermal forcing of the surface mixed layer\index{mixed layer!external
forcing of}.
\vspace{-0.5ex}
\item The wind-driven circulation including the Ekman circulation, Ekman pumping\index{Ekman
pumping} of the deeper circulation, and upwelling\index{upwelling!due to Ekman pumping}.
\vspace{-0.5ex}
\item The dynamics of ocean currents, including geostrophic\index{geostrophic currents}
currents and the role of vorticity.
\vspace{-0.5ex}
\item The formation of water types\index{water!type} and masses.
\vspace{-0.5ex}
\item The deep circulation of the ocean.
\vspace{-0.5ex}
\item Equatorial dynamics, El Ni\~{n}o, and the role of the ocean in weather.
\vspace{-0.5ex}
\item Numerical models of the circulation.
\vspace{-0.5ex}
\item Waves in the ocean including surface waves, inertial
oscillations\index{inertial!oscillation}, tides, and tsunamis\index{tsunami}.
\vspace{-0.5ex}
\item Waves in shallow water, coastal processes, and tide predictions.
\end{enumerate}
\vspace{-0.5ex}
\item What are a few of the major currents and water masses in the ocean, and what governs their
distribution?
\end{enumerate}
\section{Organization}
Before beginning a voyage, we usually try to learn about the places we will
visit. We look at maps and we consult travel guides. In this book, our guide will be
the papers and books published by oceanographers. We begin with a brief overview of
what is known about the ocean. We then proceed to a description of the ocean basins,
for the shape of the seas influences the physical processes in the water. Next, we
study the external forces, wind and heat, acting on the ocean, and the ocean's
response. As we proceed, I bring in theory and observations as necessary.
By the time we reach chapter 7, we will need to understand the equations describing
dynamic response of the ocean. So we consider the equations of motion, the
influence of earth's rotation, and viscosity. This leads to a study of wind-driven
ocean currents, the geostrophic approximation\index{geostrophic approximation}, and the
usefulness of conservation of vorticity.
Toward the end, we consider some particular examples: the deep circulation, the
equatorial ocean and El Ni\~{n}o, and the circulation of particular areas of the
ocean. Next we look at the role of numerical models in describing the ocean.
At the end, we study coastal processes, waves, tides, wave and tidal
forecasting, tsunamis\index{tsunami}, and storm surges.
\section{The Big Picture}
The ocean is one part of the earth system. It mediates processes in the atmosphere by the transfers of mass, momentum, and energy through the sea surface. It receives water and dissolved substances from the land. And, it lays down sediments that eventually become rocks on land. Hence an understanding of the ocean is important for understanding the earth as a system, especially for understanding important problems such as global change or global warming. At a lower level, physical oceanography and meteorology are merging. The ocean provides the feedback leading to slow changes in the atmosphere.
As we study the ocean, I hope you will notice that we use theory,
observations, and numerical models to describe ocean dynamics.
\index{physical oceanography!big picture} \textit{None is sufficient by itself}.
\begin{enumerate}
\vitem Ocean processes\index{ocean!processes in} are nonlinear and
turbulent. Yet we don't really understand the theory
of non-linear, turbulent flow in complex basins. Theories used to describe the
ocean are much simplified approximations to reality. \vitem
Observations\index{observations} are sparse in time and space.
They provide a rough description of the time-averaged flow, but
many processes in many regions are poorly observed. \item
Numerical models\index{numerical models} include
much-more-realistic theoretical ideas, they can help interpolate
oceanic observations in time and space, and they are used to
forecast climate change, currents, and waves. Nonetheless, the
numerical equations are approximations to the continuous analytic
equations that describe fluid flow, they contain no information
about flow between grid points, and they cannot yet be used to
describe fully the turbulent flow seen in the ocean.
\end{enumerate}
By combining theory and observations in numerical models we avoid some of the
difficulties associated with each approach used separately (figure 1.1). Continued
refinements of the combined approach are leading to ever-more-precise descriptions of
the ocean. The ultimate goal is to know the ocean well enough to predict the future
changes in the environment, including climate change or the response of fisheries to
over fishing.
\begin{figure}[h!]
\makebox[121mm] [c]{\includegraphics{bigpicture}}
\footnotesize
Figure 1.1 Data, numerical models, and \rule{0mm}{4ex}theory are
all necessary to understand the ocean. Eventually, an
understanding of the ocean-atmosphere-land system will lead to
predictions of future states of the system.
\label{fig:bigpicture}
\vspace{-1ex}
\end{figure}
The combination of theory, observations, and computer
models\index{numerical models} is relatively new. Four decades of
exponential growth in computing power has made available desktop
computers capable of simulating important physical processes and
oceanic dynamics.
\begin{quote} \small
All of us who are involved in the sciences know that the computer has become
an essential tool for research \dots scientific computation has reached the point where
it is on a par with laboratory experiment and mathematical theory as a tool for research
in science and engineering---Langer (1999).
\end{quote}
The combination of theory, observations, and computer models also
implies a new way of doing oceanography\index{oceanography!new
methods of}. In the past, an oceanographer would devise a theory,
collect data to test the theory, and publish the results. Now, the
tasks have become so specialized that few can do it all. Few excel
in theory, collecting data, and numerical simulations. Instead,
the work is done more and more by teams of scientists and
engineers.
\section{Further Reading}
If you know little about the ocean and oceanography, I suggest you begin by
reading MacLeish's (1989) book \textit{The Gulf Stream\index{Gulf Stream}: Encounters With the
Blue God}, especially his Chapter 4 on ``Reading the ocean.'' In my
opinion, it is the best overall, non-technical, description of how oceanographers
came to understand the ocean.
You may also benefit from reading pertinent chapters from any introductory oceanographic textbook. Those by Gross, Pinet, or Segar are especially useful. The three texts produced by the Open University provide a slightly more advanced treatment.
\begin{description}
\item[Gross,] M. Grant and Elizabeth Gross (1996) \textit{Oceanography---A View of Earth.} 7th edition. Prentice Hall.
\vspace{-0.8ex}
\item[MacLeish,] William (1989) \textit{The Gulf Stream: Encounters With the Blue God.} Houghton Mifflin Company.
\vspace{-0.8ex}
\item[Pinet,] Paul R. (2006) \textit{Invitation to Oceanography.} 4nd edition. Jones and Bartlett Publishers.
\vspace{-0.8ex}
\item[Open University] (2001) \textit{Ocean Circulation.} 2nd edition. Pergamon Press.
\vspace{-0.8ex}
\item[Open University] (1995) \textit{Seawater: Its Composition, Properties and Behavior.} 2nd edition. Pergamon Press.
\vspace{-0.8ex}
\item[Open University] (1989) \textit{Waves, Tides and Shallow-Water Processes.} Pergamon Press.
\vspace{-0.8ex}
\item[Segar,] Douglas A. (2007) \textit{Introduction to Ocean Sciences.} 2nd edition. W. W. Norton.
\end{description}
\chapter{The Historical Setting}
Our knowledge of oceanic currents, winds, waves, and tides goes back thousands of
years. Polynesian navigators traded over long distances in the Pacific as
early as 4000 \textsc{bc} (Service, 1996). Pytheas explored the Atlantic
from Italy to Norway in 325 \textsc{bc}. Arabic traders used their knowledge of
the reversing winds and currents in the Indian Ocean to establish trade routes
to China in the Middle Ages and later to Zanzibar on the African coast. And, the
connection between tides and the sun\index{sun} and moon\index{moon} was described in the Samaveda of
the Indian Vedic period extending from 2000 to 1400 \textsc{bc} (Pugh, 1987).
Those oceanographers who tend to accept as true only that which has been
measured by instruments, have much to learn from those who earned their living
on the ocean.
Modern European knowledge of the ocean began with voyages of discovery by
Bartholomew Dias (1487--1488), Christopher Columbus (1492--1494), Vasco da Gama
(1497--1499), Ferdinand Magellan (1519--1522), and many others. They laid the
foundation for global trade routes stretching from Spain to the Philippines in
the early 16th century. The routes were based on a good working knowledge of
trade winds, the westerlies, and western boundary currents in the Atlantic and
Pacific (Couper, 1983: 192--193).
The early European explorers were soon followed by scientific voyages of
discovery led by (among many others) James Cook (1728--1779) on the
\textit{Endeavour},
\textit{Resolution}, and \textit{Adventure}, Charles Darwin (1809--1882) on the
\textit{Beagle}, Sir James Clark Ross and Sir John Ross who surveyed the
Arctic and Antarctic regions from the \textit{Victory}, the \textit{Isabella},
and the \textit{Erebus}, and Edward Forbes (1815--1854) who studied the vertical
distribution of life in the ocean. Others collected oceanic observations and
produced useful charts, including Edmond Halley who charted the trade winds and
monsoons and Benjamin Franklin who charted the Gulf Stream\index{Gulf Stream!mapped by
Benjamin Franklin}.
Slow ships of the 19th and 20th centuries gave way to satellites, drifters, and autonomous instruments toward the end of the 20th century. Satellites now observe the ocean, air, and land. Thousands of drifters observe the upper two kilometers of the ocean. Data from these systems, when fed into numerical models allows the study of earth as a system.
For the first time, we can study how biological, chemical, and physical systems
interact to influence our environment.
\section{Definitions}
The long history of the study of the ocean has led to the development of
various, specialized disciplines each with its own interests and
vocabulary. The more important disciplines include:
\textit{Oceanography} is \index{oceanography|textbf}the study of the ocean, with emphasis on
its character as an environment. The goal is to obtain a description sufficiently quantitative
to be used for predicting the future with some certainty.
\textit{Geophysics} is \index{geophysics|textbf}the study of the physics of the earth.
\textit{Physical Oceanography} is \index{physical oceanography|textbf}the study of
physical properties and dynamics of the ocean. The primary interests are the interaction of
the ocean with the atmosphere, the oceanic heat budget, water mass formation, currents, and
coastal dynamics. Physical Oceanography is considered by many to be a subdiscipline of
geophysics.
\textit{Geophysical Fluid Dynamics} is \index{geophysical fluid dynamics|textbf}the study of
the dynamics of fluid motion on scales influenced by the rotation of the earth. Meteorology
and oceanography use geophysical fluid dynamics to calculate planetary flow fields.
\textit{Hydrography} is \index{hydrography|textbf}the preparation of nautical charts, including charts of ocean
depths, currents, internal density field of the ocean, and tides.
\textit{Earth-system Science} is \index{earth-system science|textbf}the study of earth as a single system comprising many interacting subsystems including the ocean, atmosphere, cryosphere, and biosphere, and changes in these systems due to human activity.
\begin{figure}[t!]
\includegraphics{Fig2-1}
\centering
\footnotesize
Figure 2.1 Example from the era\rule{0pt}{3ex} of deep-sea exploration: Track of
H.M.S. \textit{Challenger}\\ during the British Challenger Expedition
1872--1876. After Wust (1964).
\label{fig:Fig2-1}
\vspace{-3ex}
\end{figure}
\begin{figure}[t!]
\makebox[121mm] [c] {\includegraphics{Fig2-2}}
\centering
\footnotesize
Figure 2.2 Example of a survey from the era of national\rule{0pt}{3ex} systematic
surveys. Track of the R/V \textit{Meteor} during the German Meteor Expedition. Redrawn from
Wust (1964).
\label{fig:Fig2-2}
\vspace{-3ex}
\end{figure}
\section{Eras of Oceanographic Exploration}
The exploration \index{oceanography!eras of exploration|(}of the
sea can be divided, somewhat arbitrarily, into various eras (Wust,
1964). I have extended his divisions through the end of the 20th
century.
\begin{enumerate}
\vitem Era of Surface Oceanography: Earliest times to 1873. The era is
characterized by systematic collection of mariners' observations of winds,
currents, waves, temperature, and other phenomena observable from the
deck of sailing ships. Notable examples include Halley's charts of the trade
winds, Franklin's map of the Gulf Stream\index{Gulf Stream!mapped by
Benjamin Franklin}, and Matthew Fontaine Maury's
\textit{Physical Geography of the Sea}.
\vitem Era of Deep-Sea Exploration: 1873--1914.
Characterized by a few, wide-ranging oceanographic expeditions to survey surface
and subsurface conditions, especially near colonial claims. The major example is the
\textit{Challenger} Expedition (figure 2.1), but also the \textit{Gazelle} and
\textit{Fram} Expeditions.
\vitem Era of National Systematic Surveys: 1925--1940.
Characterized by detailed surveys of colonial areas. Examples include
\textit{Meteor} surveys of the Atlantic (figure 2.2), and the \textit{Discovery}
Expeditions.
\vitem Era of New Methods: 1947--1956.
Characterized by long surveys using new instruments (figure 2.3). Examples
include seismic surveys of the Atlantic by \textit{Vema} leading to Heezen's
maps of the sea floor.
\begin{figure}[t!]
\makebox[121 mm] [c] {\includegraphics{Fig2-3}}
\centering
\footnotesize
Figure 2.3 Example from the era of new \rule{0pt}{4ex}methods. The cruises of
the R/V \textit{Atlantis} out of Woods Hole Oceanographic Institution. After Wust
(1964).
\label{fig:Fig2-3}
\vspace{-3ex}
\end{figure}
\vitem Era of International Cooperation: 1957--1978. Characterized by
multinational surveys of ocean and studies of oceanic processes. Examples
include the Atlantic Polar Front Program, the \textsc{norpac} cruises, the
International Geophysical Year cruises, and the International Decade of Ocean
Exploration (figure 2.4). Multiship studies of oceanic processes include
\textsc{mode}, \textsc{polymode}, \textsc{norpax}, and \textsc{jasin}
experiments.
\begin{figure}[t!]
\includegraphics{Fig2-4}
\centering
\footnotesize
Figure 2.4 Example from the era of international cooperation
\rule{0pt}{3ex}. Sections measured by the International
Geophysical Year Atlantic Program 1957-1959. After Wust (1964).
\label{fig:Fig2-4}
\vspace{-3ex}
\end{figure}
\vitem Era of Satellites: 1978--1995. Characterized by global
surveys of oceanic processes from space. Examples include Seasat,
\textsc{noaa} 6--10, \textsc{nimbus}--7, Geosat\index{Geosat},
Topex/\-Poseidon\index{Topex/Poseidon}, and \textsc{ers}--1 \& 2\index{ERS satellites}. \vitem
Era of Earth System Science: 1995-- Characterized by global studies of the interaction of
biological, chemical, and physical processes in the ocean and atmosphere and on land using \textit{in situ} \index{in situ|textbf} (which means from measurements made in the water) and space data in numerical models. Oceanic examples include the World Ocean Circulation
Experiment (\textsc{woce})\index{World Ocean Circulation Experiment} (figure 2.5) and Topex/Poseidon (figure 2.6), the Joint Global Ocean Flux Study \index{oceanography!eras of exploration|)} (\textsc{jgofs}), the Global Ocean Data Assimilation Experiment (\textsc {godae}), and the SeaWiFS, Aqua, and Terra satellites.
\end{enumerate}
\begin{figure}[t!]
\includegraphics{wocesurvey}
\centering
\footnotesize
Figure 2.5 World Ocean\index{World Ocean Circulation Experiment} Circulation
Experiment:\rule{0pt}{4ex} Tracks of research ships making a one-time global survey of the
ocean of the world. From World Ocean Circulation Experiment.
\label{fig:wocesurvey}
\vspace{-3ex}
\end{figure}
\begin{figure}[b!]
\vspace{-1ex}
\makebox[121mm][c]{\includegraphics{Fig2-6}}
\centering
\footnotesize
Figure 2.6 Example \rule{0mm}{3ex}from the era of satellites.
Topex/Poseidon\index{Topex/Poseidon!ground tracks} tracks in the Pacific\\Ocean during a 10-day
repeat of the orbit. From Topex/Poseidon Project.
\label{fig:Fig2-6}
%\vspace{-3ex}
\end{figure}
\vspace{-1ex}
\section{Milestones in the Understanding of the Ocean}
What have all these programs and expeditions taught us about the ocean?
Let's look at some milestones in our ever increasing understanding of the ocean
beginning with the first scientific investigations of the 17th century.
Initially progress was slow. First came very simple observations of far reaching
importance by scientists who probably did not consider themselves
oceanographers, if the term even existed. Later came more detailed descriptions
and oceanographic experiments by scientists who specialized in the study of the
ocean.
\vspace{-1.0ex}
\begin{description}
\item[1685] Edmond Halley, investigating \index{ocean!milestones
in understanding|(}the oceanic wind systems and currents,
published ``An Historical Account of the Trade Winds, and
Monsoons, observable in the Seas between and near the Tropicks,
with an attempt to assign the Physical cause of the said Winds''
\textit{Philosophical Transactions}. \vspace{-1.0ex}
\item[1735] George Hadley published his theory for the trade winds
based on conservation of angular momentum in ``Concerning the
Cause of the General Trade-Winds'' \textit{Philosophical
Transactions}, 39: 58-62. \vspace{-1.0ex}
\item[1751] Henri Ellis
made the first deep soundings of temperature in the tropics,
finding cold water below a warm surface layer, indicating the
water came from the polar regions. \vspace{-1.0ex}
\item[1769]
Benjamin Franklin, as postmaster, made the first map of the Gulf
Stream\index{Gulf Stream!mapped by
Benjamin Franklin} using information from mail ships sailing between New England
and England collected by his cousin Timothy Folger (figure 2.7).
\begin{figure}[t!]
\makebox[121mm][c]{\includegraphics{Fig2-7}}
\centering
\footnotesize
Figure 2.7 The 1786 version of Franklin-Folger map \rule{0mm}{3ex}of the Gulf
Stream\index{Gulf Stream!Franklin-Folger map of}.
\label{fig:Fig2-7}
\vspace{-2ex}
\end{figure}
\vspace{-1.0ex}
\item[1775] Laplace's published his theory of
tides. \vspace{-1.0ex} \item[1800] Count Rumford proposed a meridional\index{circulation!meridional overturning} circulation of the ocean with water sinking near the poles and rising near the Equator. \vspace{-1.0ex}
\item[1847] Matthew Fontaine Maury published his first chart of winds and
currents based on ships logs. Maury established the practice of
international exchange of environmental data, trading logbooks for
maps and charts derived from the data. \vspace{-1.0ex}
\item[1872--1876] Challenger Expedition marks the beginning of
the systematic study of the biology, chemistry, and physics of the
ocean of the world. \vspace{-1.0ex}
\item[1885] Pillsbury made
direct measurements of the Florida Current using current meters
deployed from a ship moored in the stream. \vspace{-1.0ex}
\item[1903] Founding of the Marine Biological Laboratory of the University of
California. It later became the Scripps Institution of Oceanography.
\vspace{-1.0ex}
\item[1910--1913] Vilhelm Bjerknes published \textit{Dynamic
Meteorology and Hydrography} which laid the foundation of
geophysical fluid dynamics. In it he developed the idea of fronts,
the dynamic meter, geostrophic\index{geostrophic currents} flow, air-sea interaction, and
cyclones. \vspace{-1.0ex}
\item[1930] Founding of the Woods Hole Oceanographic Institution.
\vspace{-1.0ex}
\item[1942] Publication of \textit{The ocean} by Sverdrup, Johnson, and Fleming,
a comprehensive survey of oceanographic knowledge up to that time. \vspace{-1.0ex}
\item[Post WW 2] The need to detect submarines led the navies of the world to greatly expand their studies of the sea. This led to the founding of oceanography departments at state universities,
including Oregon State, Texas A\&M University, University of Miami, and
University of Rhode Island, and the founding of national ocean
laboratories such as the various Institutes of Oceanographic
Science. \vspace{-1.0ex}
\item[1947--1950] Sverdrup, Stommel, and Munk publish their theories of the
wind-driven circulation of the ocean. Together the three papers lay the foundation
for our understanding of the ocean's circulation. \vspace{-1.0ex}
\item[1949] Start of California Cooperative Fisheries
Investigation of the California Current. The most complete study ever undertaken
of a coastal current. \vspace{-1.0ex}
\item[1952] Cromwell and Montgomery rediscover the Equatorial
Undercurrent in the Pacific. \vspace{-1.0ex}
\item[1955] Bruce Hamon and Neil Brown develop the CTD\index{CTD} for measuring conductivity
and temperature as a function of depth in the ocean.
\vspace{-1.0ex}
\item[1958] Stommel publishes his theory for the
deep circulation of the ocean. \vspace{-1.0ex}
\item[1963] Sippican Corporation (Tim Francis, William Van Allen Clark, Graham
Campbell, and Sam Francis) invents the Expendable BathyThermograph
\textsc{xbt} now perhaps the most widely used oceanographic
instrument deployed from ships. \vspace{-1.0ex}
\item[1969] Kirk Bryan and Michael Cox
develop the first numerical model of the oceanic circulation.
\vspace{-1.0ex}
\item[1978] \textsc{nasa} launches the first oceanographic satellite, Seasat. The
project developed techniques used by generations of remotes sensing satellites.
\vspace{-1.0ex}
\item[1979--1981] Terry Joyce, Rob Pinkel, Lloyd Regier, F. Rowe
and J. W. Young develop techniques leading to the acoustic-doppler
current profiler for measuring ocean-surface currents from moving
ships, an instrument widely used in oceanography. \vspace{-1.0ex}
\item[1988] \textsc{nasa} Earth System Science Committee headed by Francis
Bretherton outlines how all earth systems are interconnected, thus breaking down the
barriers separating traditional sciences of astrophysics, ecology, geology,
meteorology, and oceanography.
\item[1991] Wally Broecker proposes that changes in the deep
circulation of the ocean modulate the ice ages, and that the deep
circulation in the Atlantic could collapse, plunging the northern
hemisphere into a new ice age.\index{ocean!milestones in
understanding|)}
\vspace{-1.0ex}
\item[1992] Russ Davis and Doug Webb invent the autonomous, pop-up drifter
that continuously measures currents at depths to 2 km.
\vspace{-1.0ex}
\item[1992] \textsc{nasa} and \textsc{cnes} develop and launch
Topex/Poseidon\index{Topex/Poseidon}, a satellite that maps ocean surface
currents, waves, and tides every ten days, revolutionizing our understanding of ocean dynamics and tides.
\item[1993] Topex/Poseidon science-team members publish first accurate global maps of the tides\index{tides}.
\end{description}
\vspace{-1.0ex}
More information on the history of physical
oceanography can be found in Appendix A of W.S. von Arx (1962): \textit{An
Introduction to Physical Oceanography}.
Data collected from the centuries of oceanic expeditions have been used
to describe the ocean. Most of the work went toward describing the steady
state of the ocean, its currents from top to bottom, and its interaction
with the atmosphere. The basic description was mostly complete by the early
1970s. Figure 2.8 shows an example from that time, the surface circulation
of the ocean. More recent work has sought to document the variability of
oceanic processes, to provide a description of the ocean sufficient to
predict annual and interannual variability, and to understand the role of the
ocean in global processes.
\begin{figure}[t!]
\makebox[121mm][c]{\includegraphics{Fig2-8}}
\centering
\footnotesize
Figure 2.8 The time-averaged, surface circulation \rule{0mm}{3ex}of the ocean during
northern hemisphere winter deduced from a century of oceanographic expeditions. After
Tolmazin (1985: 16).
\label{fig:Fig2-8}
\vspace{-3ex}
\end{figure}
\section{Evolution of some Theoretical Ideas}
A theoretical understanding of oceanic processes is based on classical physics
coupled with an evolving understanding of chaotic systems in mathematics and the
application to the theory of turbulence\index{turbulence!theory of}. The dates given below are
approximate.
\vspace{-1.0ex}
\begin{description}
\item[19th Century] Development of analytic hydrodynamics. Lamb's
\textit{Hydrodynamics} is the pinnacle of this work. Bjerknes develops
geostrophic\index{geostrophic currents} method widely used in meteorology and oceanography.
\vspace{-1.0ex}
\item[1925--40] Development of theories for turbulence based on aerodynamics and
mixing-length\index{mixing-length theory} ideas. Work of Prandtl and von Karman.
\vspace{-1.0ex}
\item[1940--1970] Refinement of theories for turbulence\index{turbulence!theory of} based on
statistical correlations and the idea of isotropic homogeneous turbulence. Books by
Batchelor (1967), Hinze (1975), and others.
\vspace{-1.0ex}
\item[1970--] Numerical investigations of turbulent geophysical fluid dynamics
based on high-speed digital computers.
\vspace{-1.0ex}
\item[1985--] Mechanics of chaotic processes. The application to hydrodynamics
is just beginning. Most motion in the atmosphere and ocean may be inherently
unpredictable.
\end{description}
\section{The Role of Observations in Oceanography}
\index{observations}The brief tour of theoretical ideas suggests
that observations are essential for understanding the ocean. The
theory describing a convecting, wind-forced, tur\-bulent fluid in
a rotating coordinate system has never been sufficiently well
known that important features of the oceanic circulation could be
predicted before they were observed. In almost all cases,
oceanographers resort to observations to understand oceanic
processes.
At first glance, we might think that the numerous expeditions mounted since
1873 would give a good description of the ocean. The results are indeed
impressive. Hundreds of expeditions have extended into all ocean. Yet, much
of the ocean is poorly explored.
By the year 2000, most areas of the ocean will have been sampled from
top to bottom only once. Some areas, such as the Atlantic, will have been sparsely sampled three times: during the International Geophysical Year in 1959, during the Geochemical Sections cruises in the early 1970s, and during the World Ocean Circulation Experiment\index{World Ocean Circulation Experiment} from 1991 to 1996. All areas will be vastly under sampled. This is the sampling problem\index{sampling error} (See box on next
page). Our samples of the ocean are insufficient to describe the ocean well enough to predict
its variability and its response to changing forcing. \textit{Lack of sufficient samples is the
largest source of error in our understanding of the ocean.}
The lack of observations has led to a very important and widespread conceptual error:
\begin{quote} \small
\textit{``The absence of evidence was taken as evidence of absence.''} The great difficulty of
observing the ocean meant that when a phenomenon was not observed, it was assumed it was not
present. The more one is able to observe the ocean, the more the complexity and subtlety that
appears---Wunsch (2002a).
\end{quote}
As a result, our understanding of the ocean is often too simple to be correct.
\begin{figure} [t!]
\fbox{\parbox{12cm}{
\centering
\vspace{-0.5 em}
\section*{Sampling Error}
\begin{minipage}{11.5cm}
\vspace{0.5 em} \hspace*{1 em}Sampling error \index{sampling error|textbf}is the largest source
of error in the geosciences. It is caused by a set of samples not representing the population
of the variable being measured. A population is the set of all possible measurements, and a
sample is the sampled subset of the population. We assume each measurement is perfectly
accurate.
\hspace*{1 em}To determine if your measurement has a sampling error, you must
first completely specify the problem you wish to study. This defines the
population. Then, you must determine if the samples represent the population.
Both steps are necessary.
\hspace*{1 em}Suppose your problem is to measure the annual-mean sea-surface
temperature of the ocean to determine if global warming is occurring. For this
problem, the population is the set of all possible measurements of surface
temperature, in all regions in all months. If the sample mean is to equal the
true mean, the samples must be uniformly distributed throughout the year and
over all the area of the ocean, and sufficiently dense to include all important
variability in time and space. This is impossible. Ships avoid stormy regions
such as high latitudes in winter, so ship samples tend not to represent the
population of surface temperatures. Satellites may not sample uniformly
throughout the daily cycle, and they may not observe temperature at high
latitudes in winter because of persistent clouds, although they tend to sample
uniformly in space and throughout the year in most regions. If daily variability
is small, the satellite samples will be more representative of the population
than the ship samples.
\hspace*{1 em}From the above, it should be clear that oceanic samples rarely
represent the population we wish to study. We always have sampling errors.
\hspace*{1 em}In defining sampling error, we must clearly distinguish between
instrument errors and sampling errors. Instrument errors are due to the
inaccuracy of the instrument. Sampling errors are due to a failure to make a
measurement. Consider the example above: the determination of mean sea-surface
temperature. If the measurements are made by thermometers on ships, each
measurement has a small error because thermometers are not perfect. This is an
instrument error. If the ships avoids high latitudes in winter, the absence of
measurements at high latitude in winter is a sampling error.
\hspace*{1 em}Meteorologists designing the Tropical Rainfall Mapping Mission
have been investigating the sampling error in measurements of rain. Their
results are general and may be applied to other variables. For a general
description of the problem see North \& Nakamoto (1989).
\vspace{0.7ex}
\end{minipage}
}}
\vspace{-4ex}
\end{figure}
\paragraph{Selecting Oceanic Data Sets}
\index{data sets}Much of the existing oceanic data have been
organized into large data sets. For example, satellite data are
processed and distributed by groups working with \textsc{nasa}.
Data from ships have been collected and organized by other groups.
Oceanographers now rely more and more on such collections of data
produced by others.
The use of data produced by others introduces problems: i) How accurate
are the data in the set? ii) What are the limitations of the data set? And, iii)
How does the set compare with other similar sets? Anyone who uses public or
private data sets is wise to obtain answers to such questions.
If you plan to use data from others, here are some guidelines.
\begin{enumerate}
\vitem \textit{Use well documented data sets}. \index{data sets!what makes good data?|(}Does the documentation
completely describe the sources of the original measurements, all steps used to process the data, and all criteria
used to exclude data? Does the data set include version numbers to identify changes to the set? \vitem \textit{Use
validated data}. \index{data!validated|textbf}Has accuracy\index{accuracy} of data been well
documented? Was accuracy determined by comparing with different measurements of the same
variable? Was validation global or regional? \vitem \textit{Use sets that have been used by
others and referenced in scientific papers}. Some data sets are widely used for good reason.
Those who produced the sets used them in their own published work and others trust the data.
\vitem
\textit{Conversely, don't use a data set just because it is handy}. Can you document
the source of the set? For example, many versions of the digital, 5-minute maps of
the sea floor are widely available. Some date back to the first sets produced by the
U.S. Defense Mapping Agency, others are from the \textsc{etopo-5} set. Don't rely on
a colleague's statement about the source. Find the documentation. If it is missing,
find another data set.\index{data sets!what makes good data?|)}
\end{enumerate}
\paragraph{Designing Oceanic Experiments}
\index{oceanic experiments}Observations are exceedingly important
for ocean\-ography, yet observations are expensive because ship
time and satellites are expensive. As a result, oceanographic
experiments must be carefully planned. While the design of
experiments may not fit well within an historical chapter, perhaps
the topic merits a few brief comments because it is seldom
mentioned in oceanographic textbooks, although it is prominently
described in texts for other scientific fields. The design of
experiments is particularly important because poorly planned
experiments lead to ambiguous results, they may measure the wrong
variables, or they may produce completely useless data.
The first and most important aspect of the design of any experiment is to
determine \textit{why} you wish to make a measurement before deciding how you will
make the measurement or what you will measure.
\begin{enumerate}
\vitem What is the purpose of the observations? Do you wish to test hypotheses or
describe processes?
\vitem What accuracy\index{accuracy} is required of the observation?
\vitem What resolution in time and space is required?
What is the duration of measurements?
\end{enumerate}
Consider, for example, how the purpose of the measurement changes how you might
measure salinity or temperature as a function of depth:
\begin{enumerate}
\vitem If the purpose is to describe water masses in an ocean basin, then measurements with
20--50 m vertical spacing and 50--300 km horizontal spacing, repeated once per 20--50 years in
deep water are required.
\vitem If the purpose is to describe vertical mixing\index{mixing!vertical} in the open equatorial Pacific, then 0.5--1.0 mm vertical spacing and 50--1000 km spacing between locations repeated once per
hour for many days may be required.
\end{enumerate}
\paragraph{Accuracy, Precision, and Linearity}
While we are on the topic of experiments, now is a good time to introduce three
concepts needed throughout the book when we discuss experiments: precision,
accuracy, and linearity of a measurement.
\textit{Accuracy} \index{accuracy|textbf}is the difference between the measured value and the true value.
\textit{Precision} \index{precision|textbf}is the difference among repeated measurements.
The distinction between accuracy and precision is usually illustrated by the
simple example of firing a rifle at a target. Accuracy is the average distance
from the center of the target to the hits on the target. Precision is the average
distance between the hits. Thus, ten rifle shots could be clustered within a
circle 10 cm in diameter with the center of the cluster located 20 cm from the
center of the target. The accuracy is then 20 cm, and the precision is roughly 5
cm.
\textit{Linearity} \index{linearity|textbf}requires that the output of an instrument be a linear function
of the input. Nonlinear devices rectify variability to a constant value. So a non-linear
response leads to wrong mean values. Non-linearity can be as important as accuracy.
For example, let
\begin{align}
Output &= Input + 0.1(Input)^2 \notag \\
Input &= a \sin \omega t \notag
\end{align}
then
\begin{align}
Output &= a \sin \omega t + 0.1\,(a \sin \omega t)^2 \notag \\
Output &= Input + \frac{0.1}{2} a^2 - \frac{0.1}{2} a^2 \cos 2\omega t
\notag
\end{align}
Note that the mean value of the input is zero, yet the output of this non-linear
instrument has a mean value of \(0.05 a^2\) plus an equally large term at
twice the input frequency. In general, if \textit{input} has frequencies
\(\omega_1\) and
\(\omega_2\), then \textit{output} of a non-linear instrument has frequencies
\(\omega_1
\pm
\omega_2\). Linearity of an instrument is especially important when the
instrument must measure the mean value of a turbulent variable. For example, we
require linear current meters when measuring currents near the sea surface where
wind and waves produce a large variability in the current.
\paragraph{Sensitivity to other variables of interest.}
Errors may be correlated with other variables of the problem. For example,
measurements of conductivity are sensitive to temperature. So, errors in
the measurement of temperature in salinometers leads to errors in the
measured values of conductivity or salinity.
\section{Important Concepts}
From the above, I hope you have learned:
\begin{enumerate}
\vitem
The ocean is not well known. What we know is based on data collected from
only a little more than a century of oceanographic expeditions supplemented with
satellite data collected since 1978.
\vitem
The basic description of the ocean is sufficient for describing the
time-averaged mean circulation of the ocean, and recent work is beginning to
describe the variability.
\vitem
Observations are essential for understanding the ocean. Few processes
have been predicted from theory before they were observed.
\vitem
Lack of observations has led to conceptual pictures of oceanic processes that are often too
simplified and often misleading.
\vitem
Oceanographers rely more and more on large data sets produced by others.
The sets have errors and limitations which you must understand before using
them.
\vitem
The planning of experiments is at least as important as conducting the
experiment.
\vitem
Sampling errors arise when the observations, the samples, are not
representative of the process being studied. Sampling errors are the
largest source of error in oceanography.
\vitem
Almost all our observations of the ocean now come from satellites, drifters, and autonomous instruments. Fewer and fewer observations come from ships at sea.
\end{enumerate}
\chapter{The Physical Setting}
\addtocounter{figure}{1}
Earth \index{earth!radii of}is an oblate ellipsoid, an ellipse rotated about its minor axis, with an
equatorial radius of $R_e = 6,378.1349$ km (West, 1982) slightly greater than the polar radius of
$R_p = 6,356.7497$ km. The small equatorial bulge is due to earth's rotation.
Distances on earth are measured in many different units, the most common are degrees of
latitude or longitude, meters, miles, and nautical miles.
\textit{Latitude}\index{latitude|textbf} is the angle between the local vertical and the
equatorial plane. A meridian is the intersection at earth's surface of a plane perpendicular
to the equatorial plane and passing through earth's axis of rotation.
\textit{Longitude}\index{longitude|textbf} is the angle between the standard meridian and any
other meridian, where the standard meridian is the one that passes through a point at the
Royal Observatory at Greenwich, England. Thus longitude is measured east or west of Greenwich.
A degree of latitude is not the same length as a degree of longitude except
at the equator. Latitude is measured along great circles with radius $R$, where
$R$ is the mean radius of earth. Longitude is measured along circles with
radius $R \cos \varphi$, where
$\varphi$ is latitude. Thus $1^{\circ}$ latitude $ = 111$ km, and $1^{\circ}$
longitude $= 111 \cos \varphi$ km.
Because distance in degrees of longitude is not constant, oceanographers measure
distance on maps using degrees of latitude.
Nautical miles and meters are connected historically to the size of earth. Gabriel Mouton proposed in 1670
a decimal system of measurement based on the length of an arc that is one minute of a great circle of
earth. This eventually became the nautical mile. Mouton's decimal system eventually became the metric
system based on a different unit of length, the meter, which was originally intended to be one
ten-millionth the distance from the Equator to the pole along the Paris meridian. Although the
tie between nautical miles, meters, and earth's radius was soon abandoned because it was not
practical, the approximations are very good. For example, earth's polar circumference
is approximately 40,008 km. Therefore one ten-millionth of a quadrant is 1.0002
m. Similarly, a nautical mile should be 1.8522 km, which is
very close to the official definition of the\index{nautical mile|textbf}\index{international
nautical mile|textbf} \textit{international nautical mile}: 1 nm $\equiv$ 1.8520 km.
\section{Ocean and Seas}
There is only one ocean. It is divided into three named parts by international agreement: the Atlantic, Pacific, and Indian ocean\index{ocean!defined} (International Hydrographic Bureau,
1953)\index{International Hydrographic Bureau}. Seas, which are part of the ocean, are
defined in several ways. I consider two.
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{atlantic}} \footnotesize
\centering Figure 3.1 The Atlantic Ocean viewed with an Eckert
VI\rule{0mm}{3ex} equal-area projection. Depths, in meters, are
from the \textsc{etopo} 30$'$ data set. The 200 m contour outlines
continental shelves.
\label{fig:atlantic}
\vspace{-4ex}
\end{figure}
\textbf{The Atlantic Ocean} \index{ocean!Atlantic Ocean}extends
northward from Antarctica and includes all of the Arctic Sea, the
European Mediterranean, and the American Mediterranean more
commonly known as the Caribbean sea (figure 3.1). The boundary
between the Atlantic and Indian Ocean is the meridian of Cape
Agulhas (20\degrees E). The boundary between the Atlantic and
Pacific is the line forming the shortest distance from Cape
Horn to the South Shetland Islands. In the north, the Arctic Sea
is part of the Atlantic Ocean, and the Bering Strait is the
boundary between the Atlantic and Pacific.
\textbf{The Pacific Ocean} \index{ocean!Pacific Ocean}extends
northward from Antarctica to the Bering Strait (figure 3.2). The
boundary between the Pacific and Indian Ocean follows the line
from the Malay Peninsula through Sumatra, Java, Timor, Australia
at Cape Londonderry, and Tasmania. From Tasmania to Antarctica it
is the meridian of South East Cape on Tasmania 147\degrees E.
\textbf{The Indian Ocean} \index{ocean!Indian Ocean}extends from
Antarctica to the continent of Asia including the Red Sea and
Persian Gulf (figure 3.3). Some authors use the name Southern
Ocean to describe the ocean surrounding Antarctica.
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{pacific}}
\centering
\footnotesize
Figure 3.2 The Pacific Ocean viewed with an Eckert VI\rule{0mm}{3ex} equal-area
projection. Depths, in meters, are from the \textsc{etopo} 30$'$ data set. The
200 m contour outlines continental shelves.
\label{fig:pacific}
\vspace{-4ex}
\end{figure}
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{indian}}
\footnotesize
\centering
Figure 3.3 The Indian Ocean viewed with an Eckert VI\rule{0pt}{3ex}
equal-area projection. Depths, in meters, are from the \textsc{etopo} 30$'$ data
set. The 200 m contour outlines continental shelves.
\label{fig:indian}
\vspace{-4ex}
\end{figure}
\textbf{Mediterranean Seas} \index{seas!Mediterranean}are mostly
surrounded by land. By this definition, the Arctic and Caribbean
Seas are both Mediterranean Seas, the Arctic Mediterranean and the
Caribbean Mediterranean.
\textbf{Marginal Seas} \index{seas!marginal}are defined by only an
indentation in the coast. The Arabian Sea and South China Sea are
marginal seas.
\section{Dimensions of the ocean}
\index{ocean!dimensions of}The ocean and seas cover 70.8\% of the surface of earth, which amounts to 361,254,000
km$^2$. The areas of the named parts vary considerably (table 3.1).
\begin{table} [b!]\centering \small
\vspace{-3ex}
\begin{tabular*}{65mm}{@{}l @{\extracolsep{\fill}} r@{}}
\multicolumn{2}{@{}l@{}}{\bfseries Table 3.1 Surface Area of the ocean} $^{\dag }$ \\
\hline
\rule{0ex}{2.5ex}Pacific Ocean & $181.34 \times 10^6 \hbox{ km}^2$ \\
Atlantic Ocean & $ 106.57 \times 10^6 \hbox{ km}^2$ \\
Indian Ocean & $74.12 \times 10^6 \hbox{ km}^2$ \\[0.5ex]
\hline
\multicolumn{2}{@{}l@{}} {\rule{0ex}{2.5ex}$^{\dag }$ From Menard and Smith (1966)}
\end{tabular*} \\[0.5ex]
% \vspace{-3ex}
\end{table}
Oceanic dimensions range from around 1500 km for the minimum width of the
Atlantic to more than 13,000 km for the north-south extent of the Atlantic and
the width of the Pacific. Typical depths are only 3--4 km. So horizontal
dimensions of ocean basins are 1,000 times greater than the vertical dimension. A
scale model of the Pacific, the size of an $8.5 \times 11$ in sheet of paper,
would have dimensions similar to the paper: a width of 10,000 km scales to 10 in,
and a depth of 3 km scales to 0.003 in, the typical thickness of a piece of
paper.
Because the ocean is so thin, cross-sectional plots of ocean basins must have a
greatly exaggerated vertical scale to be useful. Typical plots have a vertical scale
that is 200 times the horizontal scale (figure 3.4). This exaggeration distorts
our view of the ocean. The edges of the ocean basins, the continental slopes, are
not steep cliffs as shown in the figure at 41\degrees W and 12\degrees E. Rather,
they are gentle slopes dropping down 1 meter for every 20 meters in the
horizontal.
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{bathy}}
\footnotesize
Figure 3.4 Cross-section of the south Atlantic\rule{0pt}{3ex} along 25\degrees S
showing the continental shelf offshore of South America, a seamount near
35\degrees W, the mid-Atlantic Ridge near 14\degrees W, the Walvis Ridge
near 6\degrees E, and the narrow continental shelf off South Africa.
\textbf{Upper} Vertical exaggeration of 180:1. \textbf{Lower} Vertical
exaggeration of 30:1. If shown with true aspect ratio, the plot would be the
thickness of the line at the sea surface in the lower plot.
\label{fig:bathy}
\vspace{-4ex}
\end{figure}
The small ratio of depth to width of the ocean basins is very important for understanding ocean currents. Vertical velocities must be much smaller than horizontal velocities. Even over distances of a few hundred kilometers, the vertical velocity must be less than 1\% of the horizontal velocity. I will use this information later to simplify the equations of motion.
The relatively small vertical velocities have great influence on
turbulence\index{turbulence}. Three dimensional turbulence is fundamentally
different than two-dimensional turbulence\index{turbulence!two dimensional}. In two dimensions,
vortex lines must always be vertical, and there can be little vortex stretching. In three
dimensions, vortex stretching plays a fundamental role in turbulence.
\section{Sea-Floor Features}
Earth's rocky surface is divided into two types: oceanic, with a thin
dense crust about 10 km thick, and continental, with a thick light crust about
40 km thick. The deep, lighter continental crust floats higher on the denser
mantle than does the oceanic crust, and the mean height of the crust relative to
sea level has two distinct values: continents have a mean elevation of 1100 m, the
ocean has a mean depth of -3400 m (figure 3.5).
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{depth-r}}
\footnotesize
Figure 3.5 Histogram of height\rule{0pt}{3ex} of land and depth of the sea as percentage of area of earth in 100 m intervals, showing the clear distinction between continents and sea floor. The cumulative frequency curve is the integral of the histogram. The curves are calculated from the \textsc{etopo} 2 data set by George Sharman of the \textsc{noaa} National Geophysical Data Center.
\label{fig:depth-r}
\vspace{-3ex}
\end{figure}
\begin{figure}[b!]
\centering
\vspace{-2ex}
\makebox[121 mm] [c]{\includegraphics{bathysketch}}
\footnotesize
Figure 3.6 Schematic section through the ocean showing\rule{0pt}{4ex} principal
features of the sea floor. Note that the slope of the sea floor is greatly
exaggerated in the figure.
\label{fig:bathysketch}
%\vspace{-4ex}
\end{figure}
The volume of the water in the ocean exceeds the volume of the ocean basins, and
some water spills over on to the low lying areas of the continents. These shallow
seas are the continental shelves. Some, such as the South China Sea,
are more than 1100 km wide. Most are relatively shallow, with typical depths of
50--100 m. A few of the more important shelves are: the East China Sea, the Bering
Sea, the North Sea, the Grand Banks, the Patagonian Shelf, the Arafura Sea and
Gulf of Carpentaria, and the Siberian Shelf. The shallow seas help dissipate
tides, they are often areas of high biological productivity, and they are usually included in
the exclusive economic zone of adjacent countries.
The crust is broken into large plates that move relative to each other. New
crust is created at the mid-ocean ridges, and old crust is lost at trenches. The
relative motion of crust, due to plate tectonics, produces the distinctive
features of the sea floor sketched in figure 3.6, including mid-ocean ridges,
trenches, island arcs, and basins.
\index{ocean!features of|(}The names of the sub-sea features have been defined by the
International Hydrographic Organization\index{International Hydrographic Bureau} (1953), and
the following definitions are taken from Sverdrup, Johnson, and Fleming (1942), Shepard
(1963), and Dietrich et al. (1980).
\textit{Basins} \index{basins|textbf}are deep depressions of the sea floor of more or
less circular or oval form.
\textit{Canyons} \index{canyon|textbf}are relatively narrow, deep furrows with
steep slopes, cutting across the continental shelf and slope, with bottoms sloping
continuously downward.
\textit{Continental shelves} \index{continental shelves|textbf}are zones adjacent
to a continent (or around an island) and extending from the low-water line to the depth,
usually about 120 m, where there is a marked or rather steep descent toward great
depths. (figure 3.7)
\begin{figure}[b!]
\vspace{-2ex}
\makebox[121 mm] [c]{\includegraphics{canyon}}
\footnotesize
Figure 3.7 An example of a continental shelf,\rule{0pt}{3ex} the
shelf offshore of Monterey California showing the Monterey and other
canyons. Canyons are common on shelves, often extending across the shelf and
down the continental slope to deep water. Figure copyright Monterey
Bay Aquarium Research Institute (\textsc{mbari}).
\label{fig:canyon}
%\vspace{-3ex}
\end{figure}
\textit{Continental slopes} \index{continental slopes|textbf}are the declivities
seaward from the shelf edge into greater depth.
\textit{Plains} \index{plains|textbf}are very flat surfaces found in many deep ocean
basins.
\textit{Ridges} \index{ridges|textbf}are long, narrow elevations of the sea floor with steep
sides and rough topography.
\textit{Seamounts} \index{seamounts|textbf}are isolated or comparatively isolated
elevations rising 1000 m or more from the sea floor and with small summit area (figure 3.8).
\textit{Sills} \index{sills|textbf}are the low parts of the ridges separating ocean basins from one another or from
the adjacent sea floor.
\textit{Trenches} \index{trenches|textbf}are long, narrow, and deep depressions of the sea floor, with relatively
steep sides (figure 3.9).\index{ocean!features of|)}
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{wildeguyot}}
\footnotesize
Figure 3.8 An example of a seamount, the Wilde Guyot.\rule{0pt}{4ex} A guyot is
a seamount with a flat top created by wave action when the seamount
extended above sea level. As the seamount is carried by plate motion, it
gradually sinks deeper below sea level. The depth was contoured from echo
sounder data collected along the ship track (thin straight lines) supplemented
with side-scan sonar data. Depths are in units of 100 m. From William Sager, Texas A\&M
University.
\label{fig:wildeguyot}
\vspace{-3ex}
\end{figure}
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{aleutiantrench}}
\footnotesize
Figure 3.9 An example of a trench, \rule{0pt}{3ex}the Aleutian Trench; an island
arc, the Alaskan Peninsula; and a continental shelf, the Bering Sea. The island arc
is composed of volcanos produced when oceanic crust carried deep into a trench
melts and rises to the surface. \textbf{Top:} Map of the Aleutian region of the
North Pacific.
\textbf{Bottom:} Cross-section through the region.
\label{fig:aleutiantrench}
\vspace{-4ex}
\end{figure}
Sub-sea features strongly influences the ocean circulation.
Ridges separate deep waters of the ocean into distinct basins. Water deeper than the
sill\index{sills} between two basins cannot move from one to the other. Tens of thousands of
seamounts are scattered throughout the ocean basins. They interrupt ocean currents, and produce
turbulence\index{turbulence!in deep ocean} leading to vertical mixing\index{mixing!vertical} in
the ocean.
\section{Measuring the Depth of the Ocean}
The depth of the ocean is usually measured two ways: 1) using acoustic
echo-sounders on ships, or 2) using data from satellite altimeters.
\paragraph{Echo Sounders} \index{echo sounders|(}Most maps of the ocean are based
on measurements made by echo sounders. The instrument transmits a burst of 10--30 kHz
sound\index{sound!used to measure depth} and listens for the echo from the sea floor. The time
interval between transmission of the pulse and reception of the echo, when multiplied by the
velocity of sound, gives twice the depth of the ocean (figure 3.10).
The first transatlantic echo soundings were made by the U.S. Navy
Destroyer \textit{Stewart} in 1922. This was quickly followed by
the first systematic survey of an ocean basin, made by the German
research and survey ship \textit{Meteor} during its expedition to
the south Atlantic from 1925 to 1927. Since then, oceanographic
and naval ships have operated echo sounders almost continuously
while at sea. Millions of miles of ship-track data recorded on
paper have been digitized to produce data bases used to make maps.
The tracks are not well distributed. Tracks tend to be far apart
in the southern hemisphere, even near Australia (figure 3.11) and
closer together in well mapped areas such as the North
Atlantic.\index{echo sounders|)}
\begin{figure}[t!]
\makebox[121 mm] [c] {\includegraphics{Sonar}}
%\centering
\footnotesize
Figure 3.10 \textbf{Left:} Echo \rule{0ex}{5ex}sounders measure depth of the ocean
by transmitting pulses of sound\index{sound!used to measure depth} and observing the time
required to receive the echo from the bottom. \textbf{Right:} The time is recorded by a spark
burning a mark on a slowly moving roll of paper. After Dietrich et al. (1980: 124).
\label{fig:Sonar}
\vspace{-3ex}
\end{figure}
\begin{figure}[t!]
%\vspace{-3ex}
\makebox[121 mm] [c]{\includegraphics{shiptracks10}}
\footnotesize
%\centering
Figure 3.11 Locations of \rule{0pt}{5ex}echo-sounder data
used for mapping the ocean floor near Australia. Note the large areas
where depths have not been measured from ships. From David Sandwell, Scripps Institution of
Oceanography.
\label{fig:shiptracks10}
\vspace{-3ex}
\end{figure}
Echo sounders \index{echo sounders!errors in measurement}make the most accurate measurements
of ocean depth. Their accuracy\index{accuracy!echo sounders} is $\pm$1\%.
\paragraph{Satellite Altimetry} \index{satellite altimetry!use in measuring
depth}Gaps in our knowledge of ocean depths between ship tracks have now been filled
by satellite-altimeter data. Altimeters profile the shape of the sea surface, and
its shape is very similar to the shape of the sea floor (Tapley and Kim, 2001;
Cazenave and Royer, 2001; Sandwell and Smith, 2001). To see this,
we must first consider how gravity influences sea level.
\textit{The Relationship Between Sea Level and the Ocean's Depth}
Excess mass at the sea floor, for example the mass of a seamount, increases local
gravity because the mass of the seamount is larger than the mass of water it
displaces. Rocks are more than three times denser than water. The excess mass
increases local gravity, which attracts water toward the seamount. This changes
the shape of the sea surface (figure 3.12).
\begin{figure} [t!]
\fbox{\parbox{12cm}{
\centering
\vspace{-0.5 em}
\section*{The Geoid}
\begin{minipage}{11.5cm}
\vspace{0.5 em} \hspace*{1 em}The \index{geoid|textbf}level surface\index{level surface} that corresponds to the surface of an ocean at rest is a special surface, the \textit{geoid}. To a first
approximation, the geoid\index{geoid} is an ellipsoid that corresponds to the surface of a
rotating, homogeneous fluid in solid-body rotation, which means that the fluid has
no internal flow. To a second approximation, the geoid differs from the ellipsoid
because of local variations in gravity. The deviations are called
\textit{geoid undulations}. \index{geoid!undulations|textbf}The maximum amplitude of the
undulations is roughly $\pm 60$ m. To a third approximation, the geoid deviates from the sea
surface because the ocean is not at rest. The deviation of sea level from the geoid\index{geoid} is defined
to be the \textit{topography}. \index{topography|textbf}The definition is identical to the
definition for land topography, for example the heights given on a topographic map.
\hspace*{1 em}The ocean's topography is caused by tides, heat content of the water, and ocean surface
currents. I will return to their influence in chapters 10 and 17. The maximum amplitude of the topography is roughly $\pm 1$ m, so it is small compared to the geoid\index{geoid} undulations.
\hspace*{1 em}Geoid undulations are caused by local variations in gravity due to the uneven distribution of mass at the sea floor.
Seamounts have an excess of mass because they are more dense than water. They produce an
upward bulge in the geoid (see below). Trenches have a deficiency of mass. They produce a downward deflection of the geoid\index{geoid}. Thus the geoid\index{geoid} is closely
related to sea-floor topography. Maps of the oceanic geoid\index{geoid} have a remarkable
resemblance to the sea-floor topography.
\vspace{5ex}
\makebox[118mm][c]{\includegraphics{geoidsketch}}
\footnotesize
Figure 3.12 Seamounts \rule{0mm}{6ex}are more dense than sea water. They increase local gravity, causing a plumb line at the sea surface (arrows)
to be deflected toward the seamount. Because the surface of an ocean at rest must
be perpendicular to gravity, the sea surface and the local geoid\index{geoid} must have a
slight bulge as shown. Such bulges are easily measured by satellite altimeters.
As a result, satellite altimeter data can be used to map the sea floor. Note, the
bulge at the sea surface is greatly exaggerated, a two-kilometer high seamount
would produce a bulge of approximately 10 m.
\label{fig:geoidsketch}
\vspace{0.7ex}
\end{minipage}
}}
\vspace{-4ex}
\end{figure}
Let's make the concept more exact. To a very good approximation, the sea surface is
a particular \textit{level surface} \index{level surface|textbf}called the
\textit{geoid} (see box). By definition a level surface is a surface of constant
gravitational potential, and it is everywhere perpendicular to gravity. In particular, it must be perpendicular to the local vertical determined by a plumb line, which is ``a line or cord having at one end a metal weight for determining vertical direction'' (Oxford English Dictionary).
The excess mass of the seamount attracts the plumb line's weight, causing the
plumb line to point a little toward the seamount instead of toward earth's center
of mass. Because the sea surface must be perpendicular to gravity, it must
have a slight bulge above a seamount as shown in figure 3.12. If there were no
bulge, the sea surface would not be perpendicular to gravity. Typical seamounts
produce a bulge that is 1--20 m high over distances of 100--200 kilometers.
This bulge is far too small to be seen from a ship, but it is easily
measured by satellite altimeters. Oceanic trenches have a deficit of mass, and they
produce a depression of the sea surface.
The correspondence between the shape of the sea surface and the depth of the water
is not exact. It depends on the strength of the sea floor, the age of the
sea-floor feature, and the thickness of sediments. If a seamount floats on the sea floor like ice on water, the gravitational signal is much weaker than it would be if the seamount rested on
the sea floor like ice resting on a table top. As a result, the relationship
between gravity and sea-floor topography varies from region to region.
Depths measured by acoustic echo sounders are used to determine the regional
relationships. Hence, altimetry is used to interpolate between acoustic echo
sounder measurements (Smith and Sandwell, 1994).
\textit{Satellite-altimeter systems} \index{satellite altimetry!systems|textbf}Now
let's see how altimeters measure the shape of the sea surface. Satellite
altimeter systems include a radar to measure the height of the satellite above the
sea surface and a tracking system to determine the height of the satellite in
geocentric coordinates. The system measures the height of the sea surface relative
to the center of mass of earth (figure 3.13). This gives the shape of the sea
surface.
\begin{figure}[t!]
\makebox[121 mm][c]{\includegraphics{altimetersketch}}
\footnotesize
Figure 3.13 A satellite altimeter \rule{0mm}{3ex}measures the height
of the satellite above the sea surface. When this is subtracted from the height
$r$ of the satellite's orbit, the difference is sea level relative to the center
of earth. The shape of the surface is due to variations in gravity, which
produce the geoid undulations\index{geoid!undulations}, and to ocean currents which produce the oceanic topography, the departure of the sea surface from the geoid\index{geoid}. The reference
ellipsoid is the best smooth approximation to the geoid. The variations in the geoid, geoid
undulations, and topography are greatly exaggerated in the figure. From Stewart (1985).
\label{fig:altimetersketch}
\vspace{-4ex}
\end{figure}
Many altimetric satellites have flown in space. All observed the marine geoid\index{geoid} and the influence of sea-floor features on the geoid\index{geoid}. The altimeters that produced the most useful data include Seasat (1978)\index{Seasat},
\textsc{geosat} (1985--1988), \textsc{ers}--1\index{ERS satellites} (1991--1996),
\textsc{ers}--2 (1995-- ), Topex/Poseidon\index{Topex/Poseidon} (1992--2006), Jason\index{Jason} (2002--), and Envisat (2002)\index{Envisat}.
Topex/Poseidon and Jason were specially designed to make extremely accurate measurements of sea-surface height. They measure sea-surface height with an accuracy of $\pm 0.05$ m\index{Jason!accuracy of}\index{Topex/Poseidon!accuracy of}.
\textit{Satellite Altimeter Maps of the Sea-floor Topography} Seasat\index{Seasat}, \textsc{geosat}\index{Geosat}, \textsc{ers}--1,
and \textsc{ers}--2\index{ERS satellites} were \index{satellite altimetry!maps of the sea-floor
topography}operated in orbits with ground tracks spaced 3--10 km apart, which was sufficient to map the
geoid\index{geoid}. By combining data from echo-sounders with data from \textsc{geosat} and \textsc{ers}--1 altimeter systems, Smith and Sandwell (1997) produced maps of the sea floor with horizontal resolution of 5--10 km and a global average depth accuracy of $\pm 100$ m.
\section{Sea Floor Charts and Data Sets}
\index{ocean!maps of}Almost all echo-sounder data have been digitized and combined to make
sea-floor charts. Data have been further processed and edited to produce digital data
sets which are widely distributed in \textsc{cd-rom} format. These data have
been supplemented with data from altimetric satellites to produce
maps of the sea floor with horizontal resolution around 3 km.
The British Oceanographic Data Centre publishes the General Bathymetric Chart of
the ocean (\textsc{gebco})\index{bathymetric charts!GEBCO} Digital Atlas on behalf of the Intergovernmental Oceanographic Commission of \textsc{unesco} and the International Hydrographic
Organization\index{International Hydrographic Organization}. The atlas consists primarily of the location of depth
contours, coastlines, and tracklines from the \textsc{gebco} 5th Edition published at a scale of 1:10 million. The original contours were drawn by hand based on digitized echo-sounder data plotted on base maps.
The U.S. National Geophysical Data Center\index{bathymetric charts!ETOPO-2} publishes the \textsc{etopo-2 cd-rom}
containing digital values of oceanic depths from echo sounders and altimetry and land
heights from surveys. Data are interpolated to a 2-minute (2 nautical mile)
grid. Ocean data between 64\degrees N and 72\degrees S are from the work of Smith and Sandwell (1997), who combined echo-sounder data with altimeter data from
\textsc{geosat} and \textsc{ers--1}. Seafloor data northward of 64\degrees N are from the International Bathymetric Chart of the Arctic Ocean. Seafloor data southward of 72\degrees S are from are from the US Naval Oceanographic Office's Digital Bathymetric Data Base Variable Resolution. Land data are from the \textsc{globe} Project, that produced a digital elevation model with 0.5-minute (0.5 nautical mile) grid spacing using data from many nations.
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{worldbathym}}
\centering
\footnotesize
Figure 3.14 The sea-floor \rule{0pt}{3ex}topography of the ocean with 3 km
resolution produced from satellite altimeter observations of the shape of the sea
surface. From Smith and Sandwell.
\label{fig:worlsbathym}
\vspace{-4ex}
\end{figure}
National governments publish coastal and harbor maps. In the USA, the
\textsc{noaa} National Ocean Service publishes nautical charts useful for
navigation of ships in harbors and offshore waters.
\section{Sound in the Ocean}
Sound\index{sound!in ocean} provides the only convenient means for transmitting
information over great distances in the ocean.
Sound\index{sound!use of} is used to measure the properties of the sea floor,
the depth of the ocean, temperature, and currents. Whales and other ocean animals use sound to
navigate, communicate over great distances, and find food.
\paragraph{Sound Speed}
The sound speed \index{sound!speed}in the ocean varies with
temperature, salinity, and pressure (MacKenzie, 1981; Munk
et al. 1995: 33):
\begin{align}
C & = 1448.96 + 4.591\,t - 0.05304\,t^2 + 0.0002374\,t^3+ 0.0160\,Z \\
&+ (1.340 - 0.01025\,t) (S - 35) + 1.675 \times 10^{-7}\,Z^2 - 7.139 \times
10^{-13}\,t\,Z^3 \notag
\end{align}
where $C$ is speed in m/s, $t$ is temperature in Celsius, $S$ is salinity (see Chapter 6 for a definition of salinity), and $Z$ is
depth in meters. The equation has an accuracy\index{accuracy!equation!sound speed} of
about 0.1 m/s (Dushaw et al. 1993). Other sound-speed equations have been widely used,
especially an equation proposed by Wilson (1960) which has been widely used by
the U.S. Navy.
For typical oceanic conditions, $C$ is usually between 1450 m/s and 1550 m/s (figure
3.15). Using (3.1), we can calculate the sensitivity of $C$ to changes of
temperature, depth, and salinity typical of the ocean. The approximate values are:
40 m/s per 10\degrees C rise of temperature, 16 m/s per 1000 m increase in depth,
and 1.5 m/s per 1 increase in salinity. Thus the primary causes of variability
of sound speed\index{sound!speed!variation of} is temperature and depth (pressure).
Variations of salinity are too small to have much influence.
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{sound_profile}}
\footnotesize
Figure 3.15 Processes \rule{0pt}{4ex}producing the sound channel\index{sound!channel}
in the ocean.
\textbf{Left:} Temperature $T$ and salinity $S$ measured as a function of depth during the R.V.
\textit{Hakuho Maru} cruise KH-87-1, station JT, on 28 January 1987 at Latitude 33\degrees
52.90$'$ N, Long 141\degrees 55.80$'$ E in the North Pacific.
\textbf{Center:} Variations in sound speed\index{sound!speed!variation of} due to
variations in temperature, salinity, and depth. \textbf{Right:} Sound
speed\index{sound!speed!as function of depth} as a function of depth showing the velocity
minimum near 1 km depth which defines the sound channel\index{sound!channel} in
the ocean. (Data from \textsc{jpots} Editorial Panel, 1991).
\label{fig:soundprofile}
\vspace{-3ex}
\end{figure}
If we plot sound speed\index{sound!speed!as function of depth} as a function of depth, we find
that the speed usually has a minimum at a depth around 1000 m (figure 3.16). The depth of
minimum speed is called the \textit{sound channel}\index{sound!channel|textbf}. It occurs in
all ocean, and it usually reaches the surface at very high latitudes.
The sound channel\index{sound!channel} is important because sound in the
channel can travel very far, sometimes half way around the earth. Here is how the channel
works: Sound\index{sound!rays} rays that begin to travel out of the channel
are refracted back toward the center of the channel. Rays propagating upward at small angles
to the horizontal are bent downward, and rays propagating downward at small angles to the
horizontal are bent upward (figure 3.16). Typical depths of the chan\-nel vary from 10 m to
1200 m depending on geographical area.
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{raypaths}}
\footnotesize
\centering
Figure 3.16 Ray paths of sound\index{sound!rays} in the ocean for a
\rule{0pt}{3ex}source near\\the axis of the sound channel. After Munk et al. (1995).
\label{fig:raypaths}
\vspace{-3ex}
\end{figure}
\paragraph{Absorption of Sound}
Absorption of sound \index{sound!absorption of}per unit distance
depends on the intensity $I$ of the sound:
\begin{equation}
dI = -k I_0 \, dx
\end{equation}
where $I_0$ is the intensity before absorption and $k$ is an absorption
coefficient which depends on frequency of the sound. The equation has the
solution:
\begin{equation}
I = I_0 \exp(-kx)
\end{equation}
Typical values of $k$ (in decibels dB per kilometer) are: 0.08 dB/km at 1000 Hz,
and 50 dB/km at 100,000 Hz. Decibels are calculated from: $dB = 10 \log(I/I_0)$,
where $I_0$ is the original acoustic power, $I$ is the acoustic power after
absorption.
For example, at a range of 1 km a 1000 Hz signal is attenuated by only 1.8\%: $I =
0.982 I_0$. At a range of 1 km a 100,000 Hz signal is reduced to $I = 10^{-5}
I_0$. The 30,000 Hz signal used by typical echo sounders to
map the ocean's depths are little attenuated going from the surface to the bottom
and back.
Very low frequency sounds in the sound channel\index{sound!channel}, those with
frequencies below 500 Hz have been detected at distances of megameters. In 1960 15-Hz sounds
from explosions set off in the sound channel\index{sound!channel} off Perth
Australia were heard in the sound channel near Bermuda, nearly halfway around the world. Later
experiment showed that 57-Hz signals transmitted in the sound channel near Heard
Island (75\degrees E, 53\degrees S) could be heard at Bermuda in the Atlantic and
at Monterey, California in the Pacific (Munk et al. 1994).
\paragraph{Use of Sound}
Because low frequency sound \index{sound!use of}can be heard at
great distances, the US Navy, in the 1950s, placed arrays of
microphones on the sea floor in deep and shallow water and
connected them to shore stations. The Sound Surveillance System
\textsc{sosus}, although designed to track submarines, has found
many other uses. It has been used to listen to and track whales up
to 1,700 km away, and to find the location of sub-sea volcanic
eruptions.
\section{Important Concepts}
\begin{enumerate}
\item If the ocean were scaled down to a width of 8 inches it would have
depths about the same as the thickness of a piece of paper. As a result, the
velocity field in the ocean is nearly 2-dimensional. Vertical velocities are much
smaller than horizontal velocities.
\vitem There are only three official ocean.
\vitem The volume of ocean water exceeds the capacity of the ocean basins, and
the ocean overflows on to the continents creating continental shelves.
\vitem The depths of the ocean are mapped by echo sounders which measure the time
required for a sound\index{sound!used to measure depth} pulse to travel from the surface to
the bottom and back. Depths measured by ship-based echo sounders have been used to produce
maps of the sea floor. The maps have poor horizontal resolution in some regions
because the regions were seldom visited by ships and ship tracks are far
apart.
\vitem The depths of the ocean are also measured by satellite altimeter systems
which profile the shape of the sea surface. The local shape of the surface is
influenced by changes in gravity due to sub-sea features. Recent maps based
on satellite altimeter measurements of the shape of the sea surface combined with
ship data have depth accuracy of $ \pm $100 m and horizontal
resolutions of $ \pm $3 km.
\vitem Typical sound\index{sound!speed!typical} speed in the ocean is 1480 m/s. Speed
depends primarily on temperature, less on pressure, and very little on salinity. The
variability of sound speed as a function of pressure and temperature produces a horizontal
sound channel in the ocean. Sound in the channel can travel great distances.
Low-frequency sounds below 500 Hz can travel halfway around the world provided the
path is not interrupted by land.
\end{enumerate}
\chapter{Atmospheric Influences}
The sun \index{sun}and the atmosphere drive directly or indirectly almost all dynamical
processes in the ocean. The dominant external sources and sinks of energy are
sunlight, evaporation, infrared emissions from the sea surface, and sensible
heating of the sea by warm or cold winds. Winds drive the ocean's surface
circulation down to depths of around a kilometer. Wind and tidal mixing\index{mixing!tidal}
drive the deeper currents in the ocean.
The ocean, in turn, is the dominant source of heat that drives the atmospheric circulation.\index{atmospheric circulation!driven by ocean} The uneven distribution of heat loss and gain by the ocean leads to winds in the atmosphere.
Sunlight warms the tropical ocean, which evaporate, transferring heat in the
form of water vapor to the atmosphere. The heat is released when the vapor condenses as rain. Winds and ocean currents carry heat poleward,
where it is lost to space.
Because the atmosphere drives the ocean, and the ocean drives the
atmosphere, we must consider the ocean and the atmosphere as a coupled dynamic
system. In this chapter we will look mainly at the exchange of momentum between the
atmosphere and the ocean. In the next chapter, we will look at heat exchanges. In chapter 14 we
will look at how the ocean and the atmosphere interact in the Pacific to produce El
Ni\~{n}o.
\section{The Earth in Space}
Earth's \index{earth!in space}orbit about the sun\index{sun} is nearly circular at a mean
distance of \(1.5 \times 10^8\) km. The eccentricity of the orbit is small, 0.0168.
Thus earth is 3.4\% further from the Sun\index{sun} at aphelion than at perihelion, the time
of closest approach to the sun. Perihelion occurs every year in January, and
the exact time changes by about 20 minutes per year. In 1995, it occurred on 3
January. Earth's axis of rotation is inclined 23.45\degrees\ to the plane of
earth's orbit around the sun\index{sun} (figure 4.1). The orientation is such that the
sun\index{sun!equinox} is directly overhead at the Equator on the vernal and autumnal
equinoxes, which occur on or about 21 March and 21 September each year.
The latitudes of 23.45\degrees\ North and South are the Tropics of Cancer
and Capricorn respectively. The tropics lie equatorward of these latitudes. As a
result of the eccentricity of earth's orbit, maximum solar insolation\index{insolation!maximum}
averaged over the surface of the earth occurs in early January each year. As a result of the
inclination of earth's axis of rotation, the maximum insolation at any location
outside the tropics occurs around 21 June in the northern hemisphere, and around 21
December in the southern hemisphere.
\begin{figure}[t!]
\makebox[121 mm] [c] {\includegraphics{earthinspace}}
\footnotesize
Figure 4.1 The earth in space. The \rule{0mm}{3ex}ellipticity of earth's orbit
around the sun\index{sun} and the tilt of earth's axis of rotation relative to the plane of
earth orbit leads to an unequal distribution of heating and to the seasons. Earth is closest
to the sun at perihelion.
\label{fig:earthinspace}
\vspace{-4ex}
\end{figure}
\begin{figure}[b!]
\vspace{-2ex}
\makebox[121 mm] [c] {\includegraphics{surfacewind}}
\footnotesize
Figure 4.2 Map of mean annual \index{wind!global map}wind velocity \rule{0 pt}{3 ex}
calculated from Trenberth et al. (1990) and sea-level pressure for 1989 from the \textsc{nasa}
Goddard Space Flight Center's Data Assimilation Office (Schubert et al. 1993). The winds near
140\degrees W in the equatorial Pacific are about 8 m/s.
\label{fig:surfacewinds} %\vspace{-3ex}
%\vspace{-3ex}
\end{figure}
If solar heat was rapidly redistributed over earth, maximum
temperature would occur in January. Conversely, if heat were poorly redistributed,
maximum temperature in the northern hemisphere would occur in summer. So it is
clear that heat is not rapidly redistributed by winds and currents.
\section{Atmospheric Wind Systems}
Figure 4.2 shows the distribution of sea-level winds and pressure averaged
over the year 1989. The map shows strong winds from the west between 40\degrees\
to 60\degrees\ latitude, the roaring forties, weak winds in the subtropics near
30\degrees\ latitude, trade winds from the east in the tropics, and weaker winds
from the east along the Equator. The strength and direction of winds in the
atmosphere is the result of uneven distribution of solar heating and continental
land masses and the circulation of winds in a vertical plane in the atmosphere.
\begin{figure}[b!]
\vspace{-2ex}
\makebox [121mm][c]{\includegraphics{atmosphereA}}
\makebox [121mm][c]{\includegraphics{atmosphereB}}
\footnotesize
Figure 4.3 Sketch of earth's atmospheric \rule{0mm}{3ex}circulation
driven by solar heating in the tropics and cooling at high latitudes.
\textbf{Upper:} The meridional cells in the atmosphere and the influence of
earth's rotation on the winds. \textbf{Bottom:} Cross-section through the
atmosphere showing the two major cells of meridional circulation. After The Open
University (1989a: 14).
\label{fig:atmosphere}
%\vspace{-5ex}
\end{figure}
A cartoon of the distribution of winds in the atmosphere (figure 4.3) shows
that the surface winds are influenced by equatorial convection and other processes
higher in the atmosphere. The mean value \index{wind!global mean}of winds over the
ocean is (Wentz et al. 1984):
\begin{equation}
U_{10} = 7.4 \text{ m/s}
\end{equation}
\begin{figure}[b!]
\vspace{-1ex}
%\centering
\makebox[121 mm] [c] {\includegraphics{seasonalwinds}}
\footnotesize
Figure 4.4 Mean, sea-surface \rule{0pt}{3ex}winds for July and January calculated from
the Trenberth et al. (1990) data set, which is based on the \textsc{ecmwf} reanalyses of
weather data from 1980 to 1989. The winds near 140\degrees W in the equatorial Pacific are
about 8 m/s.
\label{fig:seasonalwinds}
%\vspace{-3ex}
\end{figure}
Maps of surface winds change somewhat with the seasons. The largest
changes are in the Indian Ocean and the western Pacific Ocean (figure 4.4). Both
regions are strongly influenced by the Asian monsoon. In winter, the cold air
mass over Siberia creates a region of high pressure at the surface, and cold air
blows southeastward across Japan and on across the hot Kuroshio\index{Kuroshio},
extracting heat from the ocean. In summer, the thermal low over Tibet draws warm,
moist air from the Indian Ocean leading to the rainy season over India.
\section{The Planetary Boundary Layer}
The atmosphere within 100 m of the sea surface is influenced by the turbulent drag
of the wind on the sea and the fluxes of heat through the surface. This is the
\textit{atmospheric boundary layer}. \index{atmospheric boundary layer|textbf}It's
thickness $Z_i$ varies from a few tens of meters for weak winds blowing over water
colder than the air to around a kilometer for stronger winds blowing over water
warmer than the air.
The lowest part of the atmospheric boundary layer is the surface layer. Within
this layer, which has thickness of $\approx 0.1 Z_i$, vertical fluxes of
heat and momentum are nearly constant.
Wind speed varies as the logarithm of height within the surface layer for
neutral stability. See ``The Turbulent Boundary Layer Over a Flat Plate'' in
Chapter 8. Hence, the height of a wind measurement is important. Usually, winds
are reported as the value of wind at a height 10 m above the sea $U_{10}$.
\section{Measurement of Wind}
\index{wind!measurement of|(}Wind at sea has been measured for centuries. Maury (1855)
was the first to systematically collect and map wind reports. Recently, the US
National Atmospheric and Oceanic Administration
\textsc{noaa} has collected, edited, and digitized millions of observations going back
over a century. The resulting
\textit{International Comprehensive Ocean, Atmosphere Data Set} \textsc{icoads} \index{ICOADS
(international comprehensive ocean-atmosphere data set)}discussed in \S 5.5 is widely used for
studying atmospheric forcing of the ocean.
Our knowledge of winds at the sea surface come from many sources. Here are the more
important, listed in a crude order of relative importance:
\paragraph{Beaufort Scale}
\index{wind!Beaufort scale}By far the most common source of wind data up to 1991 have been
reports of speed based on the Beaufort scale. The scale is based on features,
such as foam coverage and wave shape, seen by an observer on a ship (table 4.1).
The scale was originally proposed by Admiral Sir F. Beaufort in 1806 to give the
force of the wind on a ship's sails. It was adopted by the British Admiralty in
1838 and it soon came into general use.
\begin{table}[t!] {\textbf{\footnotesize{Table 4.1 Beaufort Wind Scale and State of the Sea}}}
\index{wind!Beaufort scale}
\\[1ex]
\begin{footnotesize}
\begin{tabular}{@{}clrp{70mm}@{}} \hline
Beaufort & Descriptive & m/s & Appearance \rule{0ex}{2.5ex}of the Sea \\
Number & term \ & & \\[0.5ex]
\hline %\\
0 & Calm \rule{0ex}{2.5ex} & 0 & Sea like a mirror. \\
1 & Light Air& 1.2 & Ripples with appearance of scales; no foam crests.\\
2 & Light Breeze & 2.8 & Small wavelets; crests of glassy appearance, \\
\ &\ &\ & \hspace{1em}not breaking. \\
3 & Gentle breeze& 4.9 & Large wavelets; crests begin to break; scattered\\
\ &\ &\ & \hspace{1em}whitecaps.\\
4 & Moderate breeze & 7.7 & Small waves, becoming longer; numerous whitecaps. \\
5 & Fresh breeze & 10.5 & Moderate waves, taking longer to form; many\\
\ &\ &\ & \hspace{1em}whitecaps; some spray. \\
6 & Strong breeze & 13.1 & Large waves forming; whitecaps everywhere; \\
\ &\ &\ &\hspace{1em}more spray. \\
7 & Near gale & 15.8 & Sea heaps up; white foam from breaking waves begins\\
\ &\ &\ & \hspace{1em}to be blown into streaks.\\
8 & Gale & 18.8 & Moderately high waves of greater length; edges of \\
\ &\ &\ & \hspace{1em}crests begin to break into spindrift; foam is blown \\
\ &\ &\ & \hspace{1em}in well-marked streaks.\\
9 & Strong gale & 22.1 & High waves; sea begins to roll; dense streaks of foam;
\\
\ &\ &\ & \hspace{1em}spray may reduce visibility. \\
10 & Storm & 25.9 & Very high waves with overhanging crests; sea takes \\
\ &\ &\ & \hspace{1em}white appearance as foam is blown in very dense \\
\ &\ &\ & \hspace{1em}streaks; rolling is heavy and visibility reduced. \\
11 & Violent storm & 30.2 & Exceptionally high waves; sea covered with white \\
\ &\ &\ & \hspace{1em}foam patches; visibility still more reduced. \\
12 & Hurricane & 35.2 & Air is filled with foam; sea completely white\\
\ &\ &\ & \hspace{1em}with driving \rule[-2.5ex]{0ex}{0.5ex}spray;
visibility greatly reduced.\\
\hline
\end{tabular} \\[0.5ex]
From Kent and Taylor (1997)
\end{footnotesize}
\vspace{-4ex}
\end{table}
The International Meteorological Committee adopted the force scale for
international use in 1874. In 1926 they adopted a revised scale giving
the wind speed at a height of 6 meters corresponding to the Beaufort Number. The
scale was revised again in 1946 to extend the scale to higher wind speeds and to
give the equivalent wind speed at a height of 10 meters. The 1946 scale was based
on the equation $U_{10} = 0.836 B^{3/2}$, where $B = $ Beaufort Number
and $U_{10}$ is the wind speed in meters per second at a height of 10 meters
(List, 1966). More recently, various groups have revised the Beaufort
scale by comparing Beaufort force with ship measurements of winds. Kent and
Taylor (1997) compared the various revisions of the scale with winds
measured by ships having anemometers at known heights. Their recommended values
are given in table 4.1.
Observers on ships everywhere in the world usually report weather observations,
including Beaufort force, at the same four times every day. The times are at 0000Z,
0600Z, 1200Z and 1800Z, where Z indicates Greenwich Mean Time. The reports are coded
and reported by radio to national meteorological agencies. The biggest error in
the reports is the sampling error\index{sampling error}. Ships are unevenly distributed over
the ocean. They tend to avoid high latitudes in winter and hurricanes in summer, and few
ships cross the southern hemisphere (figure 4.5). Overall, the
accuracy\index{accuracy!winds!Beaufort} is around 10\%.
\begin{figure}[t!]
\centering
\includegraphics{shiplocations}
\footnotesize
Figure 4.5 Location of surface observations \rule{0mm}{3ex}made from volunteer
observing ships and\\ reported to national meteorological agencies. From
\textsc{noaa}, National Ocean Service.
\label{fig:shiplocations}
\vspace{-4ex}
\end{figure}
\paragraph{Scatterometers}
\index{wind!from scatterometers} \index{scatterometers}Observations of winds at sea now come
mostly from scatterometers on satellites (Liu, 2002). The scatterometer is a instrument very
much like a radar that measures the scatter of centimeter-wavelength radio waves from small,
centimeter-wavelength waves on the sea surface. The area of the sea covered by small waves, their amplitude, and their orientation, depend on wind speed and direction. The scatterometer measures scatter from 2--4 directions, from which wind speed and direction are calculated.
The scatterometers on \textsc{ers-1} and 2 have made global measurements of
winds from space since 1991. The \textsc{nasa} scatterometer\index{scatterometers} on
\textsc{adeos} measured winds for a six-month period beginning November 1996 and ending with
the premature failure of the satellite. It was replaced by another scatterometer on QuikScat, launched on 19 June 1999. Quikscat\index{scatterometer!Quikscat} views
93\% of the ocean every 24 hr with a resolution of 25 km.
Freilich and Dunbar (1999) report that, overall, the \textsc{nasa}
scatterometer\index{scatterometers!accuracy of} on \textsc{adeos} measured wind speed with an
accuracy\index{accuracy!winds!scatterometer} of
$\pm 1.3$ m/s. The error in wind direction was $\pm17$\degrees. Spatial resolution was 25 km.
Data from QuikScat\index{QuikScat} has an accuracy of $\pm 1$ m/s.
Because scatterometers\index{scatterometers} view a specific oceanic area only once a day, the
data must be used with numerical weather models to obtain 6-hourly wind maps required for some
studies.
\paragraph{Windsat}
Windsat\index{Windsat} is an experimental, polarimetric, microwave radiometer developed by the US Navy that measures the amount and polarization of microwave radiation emitted from the sea at angles between 50\degrees\ to 55\degrees\ relative to the vertical and at five radio frequencies. It was launched on 6 January 2003 on the Coriolis satellite. The received radio signal is a function of wind speed, sea-surface temperature, water vapor in the atmosphere, rain rate, and the amount of water in cloud drops. By observing several frequencies simultaneously, data from the instrument are used for calculating the
surface wind speed and direction, sea-surface temperature, total precipitable water, integrated cloud liquid water, and rain rate over the ocean regardless of time of day or cloudiness.
Winds are calculated over most of the ocean on a 25-km grid once a day. Winds measured by Windsat have an accuracy of $\pm 2$ m/s in speed and $\pm 20$\degrees\ in direction over the range of 5--25 m/s.
\paragraph{Special Sensor Microwave SSM/I}
Another satellite instrument that is used to measure wind speed is the
Special-Sensor Microwave/Imager (\textsc{ssm/i}) carried since 1987 on the
satellites of the U.S. Defense Meteorological Satellite Program in orbits similar
to the \textsc{noaa} polar-orbiting meteorological satellites. The instrument
measures the microwave radiation emitted from the sea at an angle near 60\degrees\
from the vertical. The radio signal is a function of wind speed, water vapor in the
atmosphere, and the amount of water in cloud drops. By observing several
frequencies simultaneously, data from the instrument are used for calculating the
surface wind speed, water vapor, cloud water, and rain rate.
Winds measured by \textsc{ssm/i} have an accuracy\index{accuracy!winds!SSM/I} of $\pm$ 2 m/s
in speed. When combined with \textsc{ecmwf} 1000 mb wind analyses, wind direction
can be calculated with an accuracy of $\pm 22$\degrees\ (Atlas, Hoffman, and
Bloom, 1993). Global, gridded data are available since July 1987 on a 0.25\degrees\ grid every 6 hours. But remember, the instrument views a specific oceanic area only once a day, and
the gridded, 6-hourly maps have big gaps.
\paragraph{Anemometers on Ships}
Satellite observations are supplemented by winds reported to meteorological agencies by
observers reading ane\-mom\-eters on ships. The anemometer is read four times a day at the standard
Greenwich times and reported via radio to meteorological agencies.
Again, the biggest error is the sampling error\index{sampling error}. Very few ships carry
calibrated anemometers. Those that do tend to be commercial ships participating in the
Volunteer Observing Ship program (figure 4.5). These ships are met in port by scientists who
check the instruments and replace them if necessary, and who collect the data measured at sea.
The accuracy\index{accuracy!winds!ship} of wind measurements from these ships is about $\pm 2$
m/s.
\paragraph{Calibrated Anemometers on Weather Buoys}
The most accurate measurements of winds at sea are made by calibrated anemometers
on moored weather buoys. Unfortunately there are few such buoys, perhaps only a
hundred scattered around the world. Some, such as Tropical Atmosphere Ocean
\textsc{tao} array in the tropical Pacific (figure 14.14) provide data from remote
areas rarely visited by ships, but most tend to be located just offshore of coastal
areas.
\textsc{noaa} operates buoys offshore of the United States and the \textsc{tao}
array in the Pacific. Data from the coastal buoys are averaged for eight minutes
before the hour, and the observations are transmitted to shore via satellite
links.
The best accuracy of anemometers on buoys operated by the \textsc{us} National Data
Buoy Center is the greater of \(\pm\)1 m/s or 10\% for wind speed and $\pm
10$\degrees\ for wind direction (Beardsley et al. 1997).\index{wind!measurement of|)}
\section{Calculations of Wind}
\index{numerical models!numerical weather models}Satellites, ships, and buoys measure winds
at various locations and times of the day. If you wish to use the observations to
calculate monthly averaged winds over the sea, then the observations can be
averaged and gridded. If you wish to use wind data in numerical models of the
ocean's currents, then the data will be less useful. You are faced with a very
common problem: How to take all observations made in a six-hour period and determine
the winds over the ocean on a fixed grid?
One source of gridded winds over the ocean is the \textit{surface analysis}\index{surface analysis|textbf}
calculated by numerical weather models\index{wind!from numerical weather models}. The strategy
used to produce the six-hourly gridded winds is called \textit{sequential estimation techniques}
\index{sequential estimation techniques|textbf}or
\textit{data assimilation}\index{data assimilation|textbf}. ``Measurements are used
to prepare initial conditions for the model, which is then integrated forward in
time until further measurements are available. The model is thereupon
re-initialized'' (Bennett, 1992: 67). The initial condition is called the \textit{analysis}.
Usually, all available measurements are used in the analysis, including observations from
weather stations on land, pressure and temperature reported by ships and buoys, winds
from scatterometers\index{scatterometers}\index{wind!from scatterometers} in space, and data
from meteorological satellites. The model interpolates the measurements to produce an analysis
consistent with previous and present observations. Daley (1991) describes the techniques in
considerable detail.
\paragraph{Surface Analysis from Numerical Weather Models}
Perhaps the most widely used weather model is that run by the European Centre for Medium-range
Weather Forecasts \textsc{ecmwf}. It calculates a surface analysis\index{surface analysis},
including surface winds and heat fluxes\index{heat flux} (see Chapter 5) every six hours on a
1\degrees\ $
\times $ 1\degrees\ grid from an explicit boundary-layer model. Calculated
values are archived on a 2.5\degrees grid. Thus the wind maps from the numerical weather
models lack the detail seen in maps from scatterometer data, which have a
1/4\degrees\ grid.
\textsc{ecmwf} calculations of winds have relatively good
accuracy\index{accuracy!winds!calculated}. Freilich and Dunbar (1999) estimated that
the accuracy for wind speed at 10 meters is
$\pm 1.5$ m/s, and $\pm 18$\degrees\ for direction.
Accuracy in the southern hemisphere is probably as good as in the northern
hemisphere because continents do not disrupt the flow as much as in the northern
hemisphere, and because scatterometers\index{scatterometers} give accurate positions of storms
and fronts over the ocean.
The \textsc{noaa} National Centers for Environmental Prediction and the US Navy also produces global analyses and forecasts every six hours.
\paragraph{Reanalyzed Data from Numerical Weather Models}
\index{numerical models!numerical weather models!reanalysis from}\index{wind!from
numerical weather models}Surface analyses of weather over some regions have been produced for more than a hundred years, and over the whole earth since about 1950. Surface analyses calculated by numerical models of the atmospheric circulation have been available for decades. Throughout this period, the methods for calculating surface analyses have constantly changed as meteorologists worked to make ever more accurate forecasts. Fluxes calculated from the analyses are therefore not consistent in time. The changes can be larger than the interannual variability of the fluxes (White, 1996). To minimize this problem, meteorological agencies have taken all archived weather data and reanalyzed them using the best numerical models to produce a uniform, internally-consistent, surface analysis\index{surface
analysis}.
The reanalyzed data are used to study oceanic and atmospheric processes in the past. Surface
analyses\index{surface analysis} issued every six hours from weather agencies are used only for problems that require up-to-date information. For example, if you are designing an offshore structure, you will probably use decades of reanalyzed data. If you are operating an offshore structure, you will watch the surface analysis and forecasts put out every six hours by meteorological agencies.
\paragraph{Sources of Reanalyzed Data}
\index{numerical models!numerical weather models!sources of reanalyzed data}Reanalyzed surface
flux data are available from national meteorological centers operating numerical weather
prediction models.
\begin{enumerate}
\vitem
The U.S. National Centers for Environmental Predictions, working with the
National Center for Atmospheric Research have produced the \textsc{ncep/ ncar} reanalysis
based on 51 years of weather data from 1948 to 2005 using the 25 January 1995
version of their forecast model. The reanalysis period is being extended forward to include all
date up to the present with about a three-day delay in producing data sets. The
reanalysis uses surface and ship observations plus sounder data from satellites.
Reanalysis products are available every six hours on a T62 grid having
$192 \times 94$ grid points with a spatial resolution of 209 km and with 28
vertical levels. Important subsets of the reanalysis, including surface
fluxes, are available on
\textsc{cd--rom} (Kalnay et al. 1996; Kistler et al. 2000).
\vitem
The European Centre for Medium-range Weather Forecasts \textsc{ecmwf} has
reanalyzed 45 years of weather data from September 1957 to August 2002 (\textsc{era}-40) using their forecast model of 2001 (Uppala et al. 2005). The reanalysis uses mostly the same surface and ship data used by the \textsc{ncep/ncar}
reanalysis plus data from the \textsc{ers}-1 and \textsc{ers}-2\index{ERS satellites} satellites and
\textsc{ssm/i}. The \textsc{era}-40 full-resolution products are available every six hours on a N80 grid having
$160 \times 320$ grid points with a spatial resolution of 1.125\degrees\ and with 60
vertical levels. The \textsc{era}-40 basic-resolution products are available every six hours with a spatial resolution of 2.5\degrees\ and with 23 vertical levels. The reanalysis includes an ocean-wave model that calculates ocean wave heights and wave spectra every six hours on a 1.5\degrees\ grid.
\end{enumerate}
\section{Wind Stress}
\index{wind stress|textbf}The wind, by itself, is usually not very interesting. Often we are
much more interested in the force of the wind, or the work done by the wind. The horizontal
force of the wind on the sea surface is called the \textit{wind stress}. Put another way, it
is the vertical transfer of horizontal momentum. Thus momentum is transferred from the
atmosphere to the ocean by wind stress.
Wind stress $T$ is calculated from:
\begin{equation}
T = \rho_a \,C_D U_{10}^2
\end{equation}
where $\rho_a = 1.3$ kg/m$^3$ is the density of air, $U_{10}$ is wind speed at 10
meters, and $C_D$ is the \textit{drag coefficient}\index{drag!coefficient|textbf}.
$C_D$ is measured using the techniques described in \S5.6. Fast response
instruments measure wind fluctuations within 10--20 m of the sea surface, from
which $T$ is directly calculated. The correlation of $T$ with $U_{10}^2$ gives
$C_D$ (figure 4.6).
Various measurements of $C_D$ have been published based on careful measurements
of turbulence\index{turbulence!measurement of} in the marine boundary layer. Trenberth et al.
(1989) and Harrison (1989) discuss the accuracy\index{accuracy!drag coefficient} of an
effective drag coefficient\index{drag!coefficient} relating wind
stress\index{wind stress!and drag coefficient} to wind velocity on a global scale. Perhaps the
best of the recently published values are those of Yelland and Taylor (1996) and Yelland et al.
(1998) who give:
\begin{figure}[t!]
%\vspace{-1ex}
\makebox[121mm] [c] {\includegraphics{dragcoefficient}}
\footnotesize
Figure 4.6 The drag\rule{0mm}{3ex} coefficient as a function of wind speed $U_{10}$ ten meters
above the sea. Circles: Measured values from Smith (1980). Triangles: Measured values from
Powell, Vickery, and Reinhold (2003). The solid line is from eq (4.3) proposed by Yelland
and Taylor (1996). The dashed line is from Jarosz (2007).
\label{fig:dragcoefficient}
\vspace{-3ex}
\end{figure}
\begin{subequations}
\begin {align}
1000 \, C_D = & \,0.29 + \frac{3.1}{U_{10}} + \frac{7.7}{U_{10}^2} & \left( 3 \le
U_{10}
\le 6 \text{ m/s}\right) \\
1000 \, C_D = & \,0.60 + 0.071 \, U_{10} & \left( 6 \le U_{10} \le 26 \text{ m/s}
\right)
\end{align}
\end{subequations}
for neutrally stable boundary layer. Other values are listed in their table 1 and
in figure 4.6.
\section{Important Concepts}
\begin{enumerate}
\item
Sunlight is the primary energy source driving the atmosphere and ocean.
\vitem
There is a boundary layer at the bottom of the atmosphere where wind speed
decreases with as the boundary is approached, and in which fluxes of heat and momentum are
constant in the lower 10--20 meters.
\vitem
Wind is measured many different ways. The most common until 1995 was from observations
made at sea of the Beaufort force\index{wind!Beaufort scale} of the wind.
\vitem
Since 1995, the most important source of wind measurements is from
scatterometers\index{scatterometers}\index{wind!from scatterometers} on satellites. They
produce daily global maps with 25 km resolution.
\vitem
The surface analysis from numerical models of the atmosphere\index{wind!from numerical weather models} is
the most useful source of global, gridded maps of wind velocity for dates before 1995. It
also is a useful source for 6-hourly maps. Resolution is 100-250 km.
\vitem
The flux of momentum from the atmosphere to the ocean, the wind stress\index{wind stress}, is
calculated from wind speed using a drag coefficient\index{drag!coefficient}.
\end{enumerate}
\chapter{The Oceanic Heat Budget}
About half the solar energy reaching earth is absorbed by the ocean and land, where it is temporarily stored near the surface. Only about a fifth of the available solar energy is directly absorbed by the atmosphere. Of the energy absorbed by the ocean, most is released locally to the atmosphere, mostly by evaporation and infrared radiation. The remainder is transported \index{transport!heat}by currents to other areas especially mid latitudes.
Heat lost by the tropical ocean is the major source of heat needed to drive the atmospheric circulation.
And, solar energy stored in the ocean from summer to winter helps ameliorate earth's climate. The thermal energy transported by ocean currents is not steady, and significant changes in the transport, particularly in the Atlantic, may have been important for the development of the ice ages. For these reasons, oceanic heat budgets and transports are important for understanding earth's climate and its short and long term variability.
\section{The Oceanic Heat Budget}
\index{heat budget|textbf}Changes in energy stored in the upper ocean result from an imbalance between input and output of heat through the sea surface. This transfer of heat across or through a surface is called a \textit{heat flux}\index{heat flux|textbf}. The flux of heat and water also changes the density of surface waters, and hence their buoyancy. As a result, the sum of the heat and water fluxes is often called the \textit{buoyancy flux}\index{buoyancy flux|textbf}\index{flux!buoyancy}.
The flux of energy to deeper layers is usually much smaller than the flux through the surface. And,
the total flux of energy into and out of the ocean must be zero, otherwise the ocean as a whole would heat up or cool down. The sum of the heat fluxes into or out of a volume of water is the \textit{heat
budget}.\index{heat budget!terms of}
The major terms in the budget at the sea surface are:
\begin{enumerate}
\vitem \textit{Insolation} $Q_{SW}$, \index{insolation|textbf}the flux of solar energy into the sea;
\vitem \textit{Net Infrared Radiation} $Q_{LW}$, \index{net infrared radiation|textbf}net flux of infrared radiation from the sea;
\vitem \textit{Sensible Heat Flux} $Q_S$, \index{sensible heat flux|textbf}the flux
of heat out of the sea due to conduction;
\vitem \textit{Latent Heat Flux}
$Q_L$, \index{latent heat flux|textbf}the flux of energy carried by evaporated water;
and
\vitem \textit{Advection} $Q_V$, \index{advection|textbf}heat carried away by
currents.
\end{enumerate}
Conservation of heat requires:
\begin{equation}Q = Q_{SW} + Q_{LW} + Q_S + Q_L + Q_V \end{equation}
where $Q$ is the resultant heat gain or loss. Units for heat fluxes\index{heat flux!units of}
are watts/m$^2$. The product of flux times surface area times time is energy in
joules. The change in temperature $\Delta t$ of the water is related to change in
energy $\Delta E$ through:
\begin{equation}
\Delta E = C_{p} \, m \, \Delta t
\end{equation}
where $m$ is the mass of water being warmed or cooled, and $C_p$ is the specific
heat of sea water at constant pressure.
\begin{equation}
C_{p} \approx 4.0\times 10^{3} \mbox{ J}\cdot \mbox{kg}^{-1} \cdot \, ^\circ
\mbox{C}^{-1}
\end{equation}
Thus, 4,000 joules of energy are required to heat 1.0 kilogram of sea water by
1.0$^{\circ}$C (figure 5.1).
\begin{figure}[t!]
\includegraphics{Cp}
\footnotesize Figure 5.1 Specific heat of \rule{0pt}{3ex}sea water at
atmospheric pressure
$C_{p}$ in joules per gram per degree Celsius as a function of temperature in
Celsius and salinity, calculated from the empirical formula given by Millero
et al. (1973) using algorithms in Fofonoff and Millard (1983). The
lower line is the freezing point of salt water.
\label{fig:Cp}
\vspace{-3ex}
\end{figure}
\paragraph{Importance of the Ocean in Earth's Heat Budget}
\index{heat budget!importance of}To understand the importance of
the ocean in earth's heat budget, let's make a comparison
of the heat stored in the ocean with heat stored on land during an
annual cycle. During the cycle, heat is stored in summer and
released in the winter. The point is to show that the ocean store
and release much more heat than the land.
To begin, use (5.3) and the heat capacity of soil and rocks
\begin{equation}
C_{p(rock)} = 800 \mbox{ J}\cdot \mbox{kg}^{-1} \cdot \, ^\circ \mbox{C}^{-1}
\end{equation}
to obtain $C_{p(rock)} \approx 0.2 \, C_{p(water)}$.
The volume of water which exchanges heat with the atmosphere on a seasonal cycle
is 100 m$^3$ per square meter of surface, i.e. that mass from the surface to a
depth of 100 meters. The density of water is 1000 kg/m$^3$, and the mass in
contact with the atmosphere is density $\times$ volume = $m_{water} = 100,000$
kg. The volume of land which exchanges heat with the atmosphere on a seasonal
cycle is 1 m$^3$. Because the density of rock is 3,000 kg/m$^3$, the mass of the
soil and rock in contact with the atmosphere is 3,000 kg.
The seasonal heat storage\index{heat storage!seasonal} values for the ocean and land are
therefore:
\begin{align}
\Delta E_{ocean} & = C_{p(water)} \, m_{water}\, \Delta t && \Delta t =
10^{\circ} \text{C} \notag \\
& = (4000) (10^5) (10^{\circ}) \text{
Joules} \notag \\
& = 4.0 \times 10^9 \, \text{Joules} \notag \\
\Delta E_{land} & = C_{p(rock)} \, m_{rock}\, \Delta t & & \Delta t = 20^{\circ
} \text{C} \notag \\
& = (800) (3000) (20^{\circ}) \text{
Joules} \notag \\
& = 4.8 \times 10^7 \, \text{Joules} \notag \\
\frac{\Delta E_{ocean}}{\Delta E_{land}} & = 100 \notag
\end{align}
where $\Delta t$ is the typical change in temperature from summer to winter.
The large storage of heat in the ocean compared with the land has important
consequences. The seasonal range of air temperatures on land increases with
distance from the ocean, and it can exceed 40\degrees C in the center of
continents, reaching 60\degrees C in Siberia. Typical range of temperature over
the ocean and along coasts is less than 10\degrees C. The variability of water
temperatures is still smaller (see figure 6.3, bottom).
\section{Heat-Budget Terms}
\index{heat budget!terms of}Let's look at the factors influencing
each term in the heat budget.
\paragraph{Factors Influencing Insolation}
\index{insolation!factors influencing}Incoming solar radiation is primarily determined
by latitude, season, time of day, and cloudiness. The polar
regions are heated less than the tropics, areas in winter are
heated less than the same area in summer, areas in early morning
are heated less than the same area at noon, and cloudy days have
less sun than sunny days.
The following factors are important:
\begin{enumerate}
\vitem
The height of the sun\index{sun!height above horizon} above the horizon, which depends on
latitude, season, and time of day. Don't forget, there is no insolation\index{insolation} at
night!
\vitem
The length of day, which depends on latitude and season.
\vitem
The cross-sectional area of the surface absorbing sunlight, which depends on
height of the sun above the horizon.
\vitem
Attenuation, which depends on:
i) Clouds, which absorb and scatter radiation. ii) Path length through the atmosphere, which varies as
$\csc \varphi$, where $\varphi$ is angle of the sun above the horizon. iii) Gas molecules which absorb
radiation in some bands (figure 5.2). H$_2$O, O$_3$, and CO$_2$ are all important. iv) Aerosols which
scatter and absorb radiation. Both volcanic and marine aerosols are important. And v) dust, which scatters
radiation, especially Saharan dust over the Atlantic.
\vitem
Reflectivity of the surface, which depends on solar elevation angle and roughness
of sea surface.
\end{enumerate}
Solar inclination and cloudiness dominate. Absorption by ozone, water
vapor, aerosols, and dust are much weaker.
\begin{figure}[t!]
\makebox [121 mm] [c] {\includegraphics{insolation}}
\footnotesize
Figure 5.2 Insolation\index{insolation!at top of atmosphere} \rule{0pt}{3ex}(spectral
irradiance) of sunlight at top of the atmosphere and at the sea surface on a clear day. The
dashed line is the best-fitting curve of blackbody radiation the size and distance of the sun.
The number of standard atmospheric masses is designated by $m$. Thus $m = 2$ is
applicable for sunlight when the sun\index{sun!height above horizon} is 30\degrees above the
horizon. After Stewart (1985: 43).
\label{fig:insolation}
\vspace{-3ex}
\end{figure}
\begin{figure}[t!]
\includegraphics{QswDown}
\footnotesize
Figure 5.3 Monthly average of \rule{0pt}{3ex}downward flux of sunlight through a
cloud-free sky and into the ocean in W/m$^2$ during 1989 calculated
by the Satellite Data Analysis Center at the \textsc{nasa} Langley Research Center
(Darnell et al. 1992) using data from the International Satellite Cloud Climatology
Project.
\label{fig:QswDown}
\vspace{-3ex}
\end{figure}
The average annual value for insolation\index{insolation!annual average} (figure 5.3) is in the
range:
\begin{equation}
30 \text{\ W/m$^2$} < Q_{SW} < 260 \text{\ W/m$^2$}
\end{equation}
\paragraph{Factors Influencing Infrared Flux}
\index{infrared flux!factors influencing}The sea surface radiates as a blackbody having the
same temperature as the water, which is roughly 290 K. The distribution of radiation as
a function of wavelength is given by Planck's equation. Sea water at 290 K radiates
most strongly at wavelengths near 10 $\mu$m. These wavelengths are strongly
absorbed by clouds, and somewhat by water vapor. A plot of atmospheric
transmittance as a function of wavelength for a clear atmosphere but with varying
amounts of water vapor (figure 5.4) shows the atmosphere is nearly transparent
in some wavelength bands called windows.
\begin{figure}[t!]
\makebox [121 mm] [c] {\includegraphics{transmittance}}
\footnotesize
Figure 5.4 Atmospheric \rule{0pt}{3ex}transmittance\index{atmospheric transmittance} for a vertical path to space
from sea level for six model atmospheres with very clear, 23 \textit{km},
visibility, including the influence of molecular and aerosol scattering. Notice
how water vapor modulates the transparency of the 10-14 $\mu$m atmospheric
window, hence it modulates $Q_{LW}$, which is a maximum at these wavelengths.
After Selby and McClatchey (1975).
\label{fig:transmittance}
\vspace{-3ex}
\end{figure}
The transmittance on a cloud-free day through the window from 8 $\mu$m to 13
$\mu$m is determined mostly by water vapor. Absorption in other bands, such as
those at 3.5
$\mu$m to 4.0 $\mu$m depends on CO$_2$ concentration in the atmosphere. As the
concentration of CO$_2$ increases, these windows close and more radiation is
trapped by the atmosphere.
Because the atmosphere is mostly transparent to incoming sunlight, and somewhat opaque to outgoing infrared radiation, the atmosphere traps radiation. The trapped radiation, coupled with convection in the atmosphere, keeps earth's surface 33\degrees\ warmer than it would be in the absence of a convecting, wet atmosphere but in thermal equilibrium with space. The atmosphere acts like the panes of glass on a greenhouse, and the effect is known as the \textit{greenhouse effect}\index{greenhouse effect|textbf}. See Hartmann (1994: 24--26) for a simple discussion of the radiative balance of a planet. CO$_2$, water vapor, methane, and ozone are all important greenhouse gasses.
The net infrared flux\index{infrared flux!net} depends on:
\begin{enumerate}
\vitem Clouds thickness. The thicker the cloud deck, the less heat escapes to
space.
\vitem Cloud height, which determines the temperature at which the
cloud radiates heat back to the ocean. The rate is proportional to $t^4$, where
$t$ is the temperature of the radiating body in Kelvins. High clouds are colder
than low clouds.
\vitem Atmospheric water-vapor content. The more humid the atmosphere the less
heat escapes to space.
\vitem Water Temperature. The hotter the water the more heat is radiated.
Again, radiation depends of $t^4$.
\vitem Ice and snow cover. Ice emits as a black body, but it cools much faster
than open water. Ice-covered seas are insulated from the atmosphere.
\end{enumerate}
Water vapor and clouds influence the net loss of infrared radiation more than
surface temperature. Hot tropical regions lose less heat than cold polar regions. The
temperature range from poles to equator is
$0^{\circ}\rm{C} < t < 25^{\circ}\rm{C}$ or
$273\rm{K} < t < 298\rm{K}$, and the ratio of maximum to minimum temperature in
Kelvins is 298/273 = 1.092. Raised to the fourth power this is 1.42. Thus
there is a 42\% increase in emitted radiation from pole to equator. Over the same
distance water vapor can change the net emitted radiance by 200\%.
The average annual value for net infrared flux\index{infrared flux!annual average} is in the
narrow range:
\begin{equation}
-60 \text{\ W/m$^2$} < Q_{LW} < -30 \text{\ W/m$^2$}
\end{equation}
\paragraph{Factors Influencing Latent-Heat Flux}
\index{latent heat flux}Latent heat flux is influenced primarily by wind speed and
relative humidity. High winds and dry air evaporate much more water than weak winds
with relative humidity near 100\%. In polar regions, evaporation from ice covered
ocean is much less than from open water. In the arctic, most of the heat lost from the
sea is through leads (ice-free areas). Hence the percent open water is
very important for the arctic heat budget.
The average annual value for latent-heat flux is in the range:
\begin{equation}
-130 \text{\ W/m$^2$} < Q_{L} < -10 \text{\ W/m$^2$}
\end{equation}
\paragraph{Factors Influencing Sensible-Heat Flux}
\index{sensible heat flux}Sensible heat flux is influenced by wind speed and air-sea
temperature difference. High winds and large temperature differences cause high
fluxes. Think of this as a wind-chill factor for the ocean.
The average annual value for sensible-heat flux is in the range:
\begin{equation}
-42 \text{\ W/m$^2$} < Q_{S} < -2 \text{\ W/m$^2$}
\end{equation}
\section[Direct Calculation of Fluxes]{Direct Calculation of Fluxes}
Before we can describe the geographical distribution of fluxes
into and out of the ocean, we need to know how they are measured
or calculated.
\paragraph{Gust-Probe Measurements of Turbulent Fluxes}
\index{flux!direct calculation of!gust probe measurement}There is
only one accurate method for calculating fluxes of sensible and
latent heat and momentum at the sea surface: from direct
measurement of turbulent quantities in the atmospheric boundary
layer made by gust probes on low-flying aircraft or offshore
platforms. Very few such measurements have been made. They are
expensive, and they cannot be used to calculate heat fluxes\index{heat flux!measurements of}
averaged over many days or large areas. The gust-probe
measurements are used only to calibrate other methods of
calculating fluxes.
\begin{enumerate}
\vitem Measurements must be made in the surface layer of the atmospheric boundary
layer (See \S 4.3), usually within 30 m of the sea surface, because fluxes are
independent of height in this layer.
\vitem Measurements must be made by fast-response instruments (gust probes) able
to make several observations per second on a tower, or every meter from a plane.
\vitem Measurements include the horizontal and vertical components of the wind,
the humidity, and the air temperature.
\end{enumerate}
Fluxes are calculated from the correlation of vertical wind and horizontal
wind, humidity, or temperature: Each type of flux is calculated from
different measured variables, $u'$, $w'$, $t'$, and $q'$:
\begin{subequations}
\begin{align}
T &= \langle \rho_a \,{u'w'}\rangle = \rho_a \, \langle {u'w'}\rangle \equiv \rho_a \,u_*^2\\
Q_S &= C_p\,\langle\rho_a\,{w't'}\rangle = \rho_a \, {C_p} \, \langle{w't'}\rangle \\
Q_L &= L_E \, \langle{w'q'}\rangle
\end{align}
\end{subequations}
where the brackets denotes time or space averages, and the notation is given in table 5.1. Note that \textit{specific
humidity}\index{specific humidity|textbf} mentioned in the table is the mass of water vapor per unit mass of air.
\begin{table}[t!]\small
\begin{tabular*}{121mm}{@{}llc@{}}
\multicolumn{3}{@{}l@{}}{\bfseries Table 5.1 Notation \rule[-1ex]{0mm}{1ex}Describing Fluxes} \\
\hline
Symbol & Variable \rule{0mm}{2.5ex} & Value and Units \\
\hline
$C_p$ & Specific heat capacity of air\rule{0mm}{2.5ex} & 1030 J$\cdot$kg$^{-1}\cdot$K$^{-1}$ \\
$C_D$ & Drag coefficient (see 4.3) & $(0.50 + 0.071 \, U_{10})\times
10^{-3}$ \\
$C_L$ & Latent heat transfer coefficient & $1.2 \times 10^{-3}$ \\
$C_S$ & Sensible heat transfer coefficient & $1.0 \times 10^{-3}$ \\
$L_E$ & Latent heat of evaporation & $2.5 \times 10^6$ J/kg \\
$q$ & Specific humidity of air & kg (water vapor)/kg (air) \\
$q_a$ & Specific humidity of air 10 m above the sea & kg (water vapor)/kg (air) \\
$q_s$ & Specific humidity of air at the sea surface & kg (water vapor)/kg (air) \\
$Q_S$ & Sensible heat flux & W/m$^{2}$ \\
$Q_L$ & Latent heat flux & W/m$^{2}$ \\
$T$ & Wind stress & Pascals \\
$t_a$ & Temperature of the air 10 m above the sea & K or
$^{\circ}$C \\
$t_s$ & Sea-surface temperature & K or
$^{\circ}$C \\
$t'$ & Temperature fluctuation & $^{\circ}C$ \\
$u'$ & Horizontal component of fluctuation of wind & m/s \\
$u_*$ & Friction velocity & m/s \\
$U_{10}$ & Wind speed at 10 m above the sea & m/s \\
$w'$ & Vertical component of wind fluctuation & m/s \\
$\rho_a$ & Density of air & 1.3 kg/m$^{3}$ \\
$T$ & Vector wind stress &
Pa \\ [0.5ex]
\hline
\end{tabular*} \\ [0.5ex]
$C_S$ and $C_L$ from Smith (1988).
\vspace {-3ex}
\end{table}
\paragraph{Radiometer Measurements of Radiative Fluxes}
\index{flux!direct calculation of!radiometer measurements}Radiometers on ships,
offshore platforms, and even small islands are used to make direct measurements of
radiative fluxes. Wideband radiometers sensitive to radiation from 0.3
$\mu$m to 50 $\mu$m can measure incoming solar and infrared radiation with an
accuracy\index{accuracy!fluxes!radiative} of around 3\% provided they are well calibrated
and maintained. Other, specialized radiometers can measure the incoming solar radiation, the
downward infrared radiation, and the upward infrared radiation.
\section{Indirect Calculation of Fluxes: Bulk Formulas}
\index{flux!indirect calculation of!bulk formulas}The use of gust-probes is very
expensive, and radiometers must be carefully maintained. Neither can be used to
obtain long-term, global values of fluxes. To calculate these fluxes from practical
measurements, we use observed correlations between fluxes and variables that can
be measured globally.
For fluxes of sensible and latent heat and momentum, the correlations are called \textit{bulk formulas}. \index{bulk
formulas|textbf}They are:
\begin{subequations}
\begin{align}
T & = \rho_a \,C_D \, U^{2}_{10} \\
Q_S & = \rho_a \, C_p \,C_S \, U_{10} \, (t_s - t_a) \\
Q_L & = \rho_a \, L_E \, C_L \, U_{10} \,(q_s - q_a)
\end{align}
\end{subequations}
Air temperature $t_a$ is measured using thermometers on ships. It cannot
be measured from space using satellite instruments. $t_s$ is measured using
thermometers on ships or from space using infrared radiometers such as the
\textsc{avhrr}\index{Advanced Very High Resolution Radiometer (AVHRR)}.
The specific humidity of air at 10 m above the sea surface
$q_a$ is calculated from measurements of relative humidity made from ships. Gill
(1982: pp: 39--41, 43--44, \& 605--607) describes equations relating water vapor
pressure, vapor density, and specific heat capacity of wet air. The specific
humidity at the sea surface $q_s$ is calculated from $t_s$ assuming the air at
the surface is saturated with water vapor. $U_{10}$ is measured or calculated
using the instruments or techniques described in Chapter 4. Note that wind
stress\index{wind stress!is a vector} is a vector with magnitude and direction. It is parallel
to the surface in the direction of the wind.
The problem now becomes: How to calculate the fluxes across the sea surface
required for studies of ocean dynamics? The fluxes include: 1) stress; 2)
solar heating; 3) evaporation; 4) net infrared radiation; 5) rain; 5) sensible
heat; and 6) others such as CO$_2$ and particles (which produce marine
aerosols). Furthermore, the fluxes must be accurate. We need an accuracy\index{accuracy!heat
fluxes} of approximately $\pm$15 W/m$^2$. This is equivalent to the flux of heat which would
warm or cool a column of water 100 m deep by roughly 1\degrees{C} in one year. Table 5.2 lists
typical accuracies of fluxes measured globally from space. Now, let's look at each variable.
\begin{table}[t!]\small \centering
\begin{tabular*}{120mm}{@{}lll@{}}
\multicolumn{3}{@{}l@{}}{\bfseries Table 5.2 Accuracy of \rule[-1ex]{0mm}{1ex}Wind
and Fluxes Observed Globally From Space} \\
\hline
Variable & Accuracy & Comments \rule{0mm}{2.5ex} \\
\hline
Wind Speed & $\pm$1.5 m/s & Instrument Error\rule{0ex}{2.5ex} \\
& $\pm$1.5 m/s & Sampling Error (Monthly Average) \\
Wind Stress & $\pm$10 \% & Drag Coefficient Error\rule{0ex}{2.5ex} \\
& $\pm$14 Pa & Assuming 10 m/s Wind Speed \\
Insolation & $\pm$5 \% & Monthly Average\rule{0ex}{2.5ex} \\
& $\pm$15 W/m$^2$ & Monthly Average \\
& $\pm$10 \% & Daily Average \\
Rain Rate & $\pm$50 \% & \rule{0ex}{2.5ex} \\
Rainfall & $\pm$10 \% & $5^{\circ} \times 5^{\circ}$ area for \textsc{trmm}\rule{0ex}{2.5ex} \\
Net Long Wave Radiation & $\pm$4--8 \% & Daily Average\rule{0ex}{2.5ex} \\
& $\pm$15--27 W/m$^2$ & \\
Latent Heat Flux & $\pm$35 W/m$^2$ & Daily Average\rule{0ex}{2.5ex} \\
& $\pm$15 W/m$^2$ & Monthly Average \\ [0.5ex]
\hline
\end{tabular*} \\[0.5ex]
\vspace{-3ex}
\end{table}
\paragraph{Wind Speed and Stress}
\index{wind stress!calculation of}\index{wind!speed}Stress is calculated from wind
observations made from ships at sea and from
scatterometers\index{scatterometers}\index{wind!from scatterometers} in space as described in
the last chapter.
\paragraph{Insolation}
\index{insolation!calculation of} is calculated from cloud observations made from
ships and from visible-light radiometers on meteorological satellites. Satellite
measurements are far more accurate than the ship data because it's very hard to
measure cloudiness from below the clouds. Satellite measurements processed
by the International Satellite Cloud Climatology Project \textsc{isccp} are the
basis for maps of insolation\index{insolation!maps of} and its variability from month to month
(Darnell et al. 1988; Rossow and Schiffer 1991).
The basic idea behind the calculation of insolation\index{insolation!calculation of} is this. Sunlight at the top of the atmosphere is accurately known from the solar
constant\index{solar constant!and insolation}, latitude, longitude, and time. Sunlight is
either reflected back to space by clouds, or it eventually reaches the sea surface. Only a
small and nearly constant fraction is absorbed in the atmosphere. But, recent work by Cess et
al. (1995) and Ramanathan et al. (1995) suggest that this basic idea may be incomplete, and
that atmospheric absorption may be a function of cloudiness. Assuming atmospheric absorption is
constant, insolation\index{insolation!calculation of} is calculated from:
\[
\text{Insolation} = S (1 - A) - C
\]
where $S = 1365$ W/m$^2$ is the solar constant\index{solar constant!value}, $A$ is albedo,
the ratio of incident to reflected sunlight, and $C$ is a constant which includes absorption
by ozone and other atmospheric gases and by cloud droplets. Insolation is
calculated from cloud data (which also includes reflection from aerosols)
collected from instruments such as the \textsc{avhrr} \index{Advanced Very High Resolution
Radiometer (AVHRR)}on meteorological satellites.
Ozone and gas absorption are calculated from known distributions of the gases
in the atmosphere. Q$_{SW}$ is calculated from satellite data with an
accuracy\index{accuracy!short-wave radiation} of 5--7\%.
\paragraph{Water Flux In (Rainfall)}
\index{flux!water flux!calculation of}\index{water
flux!calculation of}\index{rainfall!calculation of}Rain rate is another variable
that is very difficult to measure from ships. Rain collected from gauges at
different locations on ships and from gauges on nearby docks all differ by more
than a factor of two. Rain at sea falls mostly horizontally because of wind, and
the ship's superstructure distorts the paths of raindrops. Rain in many areas falls
mostly as drizzle, and it is difficult to detect and measure.
The most accurate measurements of rain rate in the tropics ($\pm 35$\degrees) are
calculated from microwave radiometers and radar observations of rain at several
frequencies using instruments on the Tropical Rain Measuring Mission
\textsc{trmm} launched in 1997. Rain for other times and latitudes can be
calculated accurately by combining microwave data with infrared observations of
the height of cloud tops and with rain gauge data (figure 5.5). Rain is also
calculated from the reanalyses weather data by numerical models of the atmospheric circulation (Schubert,
Rood, and Pfaendtner, 1993), and by combining ship and satellite observations with analyses from
numerical weather-prediction models (Xie and Arkin, 1997).
The largest source of error is due to conversion of rain rate to cumulative
rainfall\index{rainfall!cumulative}, a sampling error\index{sampling error}. Rain is very rare, it is log-normally
distributed, and most rain comes from a few storms. Satellites tend to miss storms, and data
must be averaged over areas up to 5\degrees\ on a side to obtain useful
values of rainfall.
\begin{figure}[t!]
\includegraphics{precip}
\footnotesize Figure 5.5 Rainfall\index{rainfall!map}
\rule{0pt}{3ex}in m/year calculated \index{global
precipitation!map of}from data compiled by the Global
Precipitation Climatology Project at \textsc{nasa}'s Goddard Space
Flight Center using data from rain gauges, infrared radiometers on
geosynchronous meteorological satellites, and the \textsc{ssm/i}.
Contour interval is 0.5 m/yr, light shaded areas exceed 2 m/yr,
heavy shaded areas exceed 3 m/yr. \label{fig:precip}
\vspace{-4ex}
\end{figure}
\paragraph{Net Long-Wave Radiation}
\index{flux!net long-wave radiation}\index{net long-wave radiation}Net Long-wave
radiation is not easily calculated because it depends on the height and thickness
of clouds, and the vertical distribution of water vapor in the atmosphere. It is
calculated by numerical weather-prediction models or from observations of the
vertical structure of the atmosphere from atmospheric sounders.
\paragraph{Water Flux Out (Latent Heat Flux)}
\index{flux!latent heat flux!calculation of}\index{latent heat flux!calculation of}
Latent heat flux is calculated from ship observations of relative humidity, water
temperature, and wind speed using bulk formulas (5.10c) and ship data accumulated
in the \textsc{icoads} \index{ICOADS (international comprehensive ocean-atmosphere data set)}described below. The fluxes are not calculated from satellite data because satellite instruments are not very sensitive to water vapor close to the sea. Perhaps the best fluxes are those calculated from
numerical weather models.
\paragraph{Sensible Heat Flux}
\index{flux!sensible heat flux!calculation of}\index{sensible heat
flux!calculation of}Sensible heat flux is calculated from observations of air-sea
temperature difference and wind speed made from ships, or by numerical weather models. Sensible fluxes are
small almost everywhere except offshore of the east coasts of continents in winter when cold, Arctic air
masses extract heat from warm, western, boundary currents. In these areas, numerical
models give perhaps the best values of the fluxes. Historical ship report give the
long-term mean values of the fluxes.
\section{Global Data Sets for Fluxes}
\index{flux!global data sets for}Ship and satellite data have been processed to
produce global maps of fluxes. Ship measurements made over the
past 150 years yield maps of the long-term mean values of the fluxes, especially in
the northern hemisphere. Ship data, however, are sparse in time and space, and
they are being replaced more and more by fluxes calculated by numerical weather models and by
satellite data.
The most useful maps are those made by combining level 3 and 4
satellite data sets with observations from ships, using numerical weather
models. Let's look first at the sources of data, then at a few of the more widely
used data sets.
\paragraph{International Comprehensive Ocean-Atmosphere Data Set}
\index{ICOADS (international comprehensive ocean-atmosphere data set)|textbf}Data collected by
observers on ships are the richest source of marine information. Slutz et al.
(1985) describing their efforts to collect, edit, and publish all marine
observations write:
\begin{quotation} \small
Since 1854, ships of many countries have been taking regular observations
of local weather, sea surface temperature, and many other characteristics
near the boundary between the ocean and the atmosphere. The observations
by one such ship-of-opportunity at one time and place, usually incidental
to its voyage, make up a marine report. In later years fixed research
vessels, buoys, and other devices have contributed data. Marine reports
have been collected, often in machine-readable form, by various agencies and
countries. That vast collection of data, spanning the ocean from
the mid-nineteenth century to date, is the historical ocean-atmosphere record.
\end{quotation}
These marine reports have been edited and published as the
\textit{International Comprehensive Ocean-Atmos\-phere Data Set} \textsc{icoads} \index{ICOADS
(international comprehensive ocean-atmosphere data set)}(Woodruff et al.
1987) available through the National Oceanic and Atmospheric Administration.
The \textsc{icoads} release 2.3 includes 213 million reports of marine surface conditions collected from 1784--2005 by buoys, other platform types, and by observers on merchant ships. The data set include fully quality-controlled (trimmed) reports and summaries. Each unique report contains 22 observed and derived variables, as well as flags indicating which observations were statistically trimmed or subjected to adaptive quality control. Here, statistically trimmed means outliers were removed from the data set. The summaries included in the data set give 14 statistics, such as the median and mean, for each of eight observed variables: air and sea surface temperatures, wind velocity, sea-level pressure, humidity, and cloudiness, plus 11 derived variables.
The data set consists of an easily-used data base at three principal resolutions: 1) individual reports, 2) year-month summaries of the individual reports in 2\degrees \ latitude by 2\degrees \ longitude boxes from 1800 to 2005 and 1\degrees \ latitude by 1\degrees \ longitude boxes from 1960 to 2005, and 3) decade-month summaries. Note that data from 1784 through the early 1800s are extremely sparse--based on scattered ship voyages.
Duplicate reports judged inferior by a first quality control process designed by
the National Climatic Data Center \textsc{ncdc} were eliminated or flagged, and
``untrimmed'' monthly and decadal summaries were computed for acceptable data
within each 2\degrees\ latitude by 2\degrees\ longitude grid. Tighter,
median-smoothed limits were used as criteria for statistical rejection of
apparent outliers from the data used for separate sets of \textit{trimmed}
monthly and decadal summaries. Individual observations were retained in report
form but flagged during this second quality control process if they fell outside
2.8 or 3.5 estimated standard- deviations about the smoothed median applicable to
their 2\degrees\ latitude by 2\degrees\ longitude box, month, and 56--, 40--, or
30--year period (\textit{i.e.}, 1854--1990, 1910--1949, or 1950--1979).
The data are most useful in the northern hemisphere, especially the North
Atlantic. Data are sparse in the southern hemisphere and they are not reliable
south of 30\degrees\ S. Gleckler and Weare (1997) analyzed the
accuracy\index{accuracy!fluxes!ICOADS} of the \textsc{icoads} data for calculating global maps
and zonal averages of the fluxes from 55\degrees N to 40\degrees S. They found that systematic
errors dominated the zonal means. Zonal averages of insolation\index{insolation!zonal average}
were uncertain by about 10\%, ranging from $\pm 10$ W/m$^2$ in high latitudes to $\pm 25$
W/m$^2$ in the tropics. Long wave fluxes were uncertain by about $\pm 7$ W/m$^2$. Latent heat
flux uncertainties ranged from
$\pm 10$ W/m$^2$ in some areas of the northern ocean to $\pm 30$ W/m$^2$ in the western
tropical ocean to $\pm 50$ W/m$^2$ in western boundary currents. Sensible heat
flux\index{sensible heat flux!uncertainty} uncertainties tend to be around $\pm 5- 10$ W/m$^2$.
Josey et al (1999) compared averaged fluxes calculated from \textsc{icoads}
with fluxes calculated from observations made by carefully calibrated instruments
on some ships and buoys. They found that mean flux into the ocean, when
averaged over all the seas surface had errors of $\pm 30$ W/m$^2$. Errors vary
seasonally and by region, and global maps of fluxes require corrections such as
those proposed by DaSilva, Young, and Levitus (1995) shown in figure 5.7.
\paragraph{Satellite Data}
Raw data are available from satellite projects, but we need processed data. Various
levels of processed data from satellite projects are produced (table 5.3):
\begin{table}[h!]\small \centering \vspace{-1ex}
\begin{tabular*}{120mm}{@{}ll@{}}
\multicolumn{2}{@{}l@{}}{\bfseries Table 5.3 Levels of
\rule[-1ex]{0mm}{1ex}Processed Satellite Data}
\\
\hline
Level & Level of Processing\rule{0mm}{2.5ex} \\
\hline
Level 1 & Data from the satellite in engineering units (volts)\rule{0ex}{2.5ex} \\
Level 2 & Data processed into geophysical units (wind speed) at the time and place \\
& the satellite instrument made the observation \\
Level 3 & Level 2 data interpolated to fixed coordinates in time and space \\
Level 4 & Level 3 data averaged in time and space or further processed \\[0.5ex]
\hline
\end{tabular*} \\[0.5ex]
\vspace{-2ex}
\end{table}
The operational meteorological satellites that observe the ocean include:
\begin{enumerate}
\vitem \textsc{noaa} series of polar-orbiting, meteorological satellites;
\vitem U.S. Defense Meteorological Satellite Program \textsc{dmsp} polar-orbiting
satellites, which carry the Special Sensor Microwave/ Imager \textsc{(ssm/i)};
\vitem Geostationary meteorological satellites operated by \textsc{noaa}
(\textsc{goes}), Japan (\textsc{gms}) and the European Space Agency
(\textsc{meteosats}).
\end{enumerate}
Data are also available from instruments on experimental satellites such
as:
\begin{enumerate}
\vitem Nimbus-7, Earth Radiation Budget Instruments;
\vitem Earth Radiation Budget Satellite, Earth Radiation Budget Experiment;
\vitem The European Space Agency's \textsc{ers}--1 \& 2\index{ERS satellites};
\vitem The Japanese ADvanced Earth Observing System (\textsc{adeos}) and Midori;
\vitem QuikScat\index{QuikScat};
\vitem The Earth-Observing System satellites Terra, Aqua, and Envisat;
\vitem The Tropical Rainfall Measuring Mission (\textsc{trmm}); and,
\vitem Topex/Poseidon\index{Topex/Poseidon} and its replacement Jason-1\index{Jason}.
\end{enumerate}
Satellite data are collected, processed, and archived by government
organizations. Archived data are further processed to produce useful flux data
sets.
\paragraph{International Satellite Cloud Climatology Project}
\index{International Satellite Cloud Climatology
Project}The International Sat\-ellite Cloud Climatology Project is an ambitious
project to collect observations of clouds made by dozens of
meteorological satellites from 1983 to 2000, to calibrate the the
satellite data, to calculate cloud cover using carefully verified
techniques, and to calculate surface insolation\index{insolation!calculation of} and net
surface infrared fluxes (Rossow and Schiffer, 1991). The clouds were observed with
visible-light instruments on polar-orbiting and geostationary satellites.
\paragraph{Global Precipitation Climatology Project} This project uses three sources
of
\index{Global Precipitation Climatology Project}data to calculate rain rate (Huffman, et al. 1995, 1997):
\begin{enumerate}
\vitem
Infrared observations of the height of cumulus clouds from \textsc{goes}
satellites. The basic idea is that the more rain produced by cumulus clouds,
the higher the cloud top, and the colder the top appears in the infrared. Thus
rain rate at the base of the clouds is related to infrared temperature.
\vitem
Measurements by rain gauges on islands and land.
\vitem
Radio emissions from water drops in the atmosphere observed by \textsc{ssm--i}.
\end{enumerate}
Accuracy\index{accuracy!rainfall} is about 1 mm/day. Data from the project are available on a
2.5\degrees\ latitude by 2.5\degrees\ longitude grid from July 1987 to December 1995 from the
Global Land Ocean Precipitation Analysis at the \textsc{nasa} Goddard Space
Flight Center.
Xie and Arkin (1997) produced a 17-year data set based on seven types of
satellite and rain-gauge data combined with the rain calculated from the
\textsc{ncep/ncar} reanalyzed data from numerical weather models. The data set
has the same spatial and temporal resolution as the Huffman data set.
\paragraph{Reanalyzed Output From Numerical Weather Models}
\index{numerical models!numerical weather models!reanalyzed data from}Surface heat
flux\index{heat flux!from numerical models} has been calculated from weather data using
numerical weather models by various reanalysis projects described in \S 4.5. The fluxes are
consistent with atmospheric dynamics, they are global, they are calculated every six hours,
and they are available for many years on a uniform grid. For example, the \textsc{ncar/ncep}
reanalysis, available on a \textsc{cd-rom}, include daily averages of wind
stress\index{wind stress!daily averages of}, sensible and latent heat fluxes, net long and
short wave fluxes, near-surface temperature, and precipitation.
\paragraph{Accuracy of Calculated Fluxes}
Recent studies of the accuracy\index{accuracy!fluxes!from models} of fluxes computed by
numerical weather models and reanalysis projects suggest:
\begin{enumerate}
\vitem
Heat fluxes from the \textsc{ncep} and \textsc{ecmwf} reanalyses have similar global average values, but the fluxes have important regional differences. Fluxes from the Goddard Earth Observing System reanalysis are much less accurate (Taylor, 2000: 258). Chou et al (2004) finds large differences in fluxes calculated by different groups.
\vitem
The fluxes are biased because they were calculated using numerical models
optimized to produce accurate weather forecasts. The time-mean values of the
fluxes may not be as accurate as the time-mean values calculated directly from
ship observations.
\vitem
The simulation of boundary-layer clouds is a significant source of error in calculated
fluxes. The poor vertical resolution of the numerical models does not adequately resolve the
low-level cloud structure (Taylor, 2001).
\vitem
The fluxes have zonal means that differ significantly from the same
zonal means calculated from \textsc{icoads} \index{ICOADS
(international comprehensive ocean-atmosphere data set)}data. The differences can exceed 40
W/m$^2$.
\vitem
The atmospheric models do not require that the net heat flux\index{heat flux} averaged over
time and earth's surface be zero. The \textsc{ecmwf} data set averaged over fifteen
years gives a net flux of 3.7 W/m$^2$ into the ocean. The \textsc{ncep} reanalysis gives a net
flux of 5.8 W/m$^2$ out of the ocean (Taylor, 2000: 206). \textsc{Icoads} data give a net flux
of 16 W/m$^2$ into the ocean (figure 5.7).
\end{enumerate}
Thus reanalyzed fluxes are most useful for forcing climate models needing actual heat fluxes\index{heat flux!from numerical models}
and wind stress\index{wind stress!from numerical models}.
\textsc{icoads} \index{ICOADS
(international comprehensive ocean-atmosphere data set)}data are most useful for calculating time-mean fluxes except
perhaps in the southern hemisphere. Overall, Taylor (2000) notes that there are no ideal data sets, all
have significant and unknown errors.
\paragraph{Output From Numerical Weather Models}
Some projects require fluxes a few hours after after observations are
collected. The surface analysis\index{surface analysis} from numerical weather models is a
good source for this type of flux.
\section[Geographic Distribution of Terms]{Geographic Distribution of Terms in
the Heat Budget} \index{heat budget!geographical distribution of
terms}Various groups have used ship and satellite data in numerical weather models
to calculate globally averaged values of the terms for earth's heat budget. The
values give an overall view of the importance of the various terms (figure 5.6).
Notice that insolation\index{insolation!at top of atmosphere} balances infrared radiation at
the top of the atmosphere. At the surface, latent heat flux and net infrared radiation tend to
balance insolation\index{insolation!at surface}, and sensible heat flux\index{sensible heat
flux!global average} is small.
\begin{figure}[t!]
%\vspace{-2ex}
\makebox [121 mm] [c] {\includegraphics{heatbudget}}
\centering
\footnotesize
Figure 5.6 The mean \rule{0mm}{4ex}annual radiation and heat
balance of the earth. \\After Houghton et al. (1996: 58), which used data from
Kiehl and Trenberth (1996).
\label{fig:heatbudget}
\vspace{-2ex}
\end{figure}
Note that only 20\% of insolation\index{insolation!absorption of} reaching earth is absorbed
directly by the atmosphere while 49\% is absorbed by the ocean and land. What then warms
the atmosphere and drives the atmospheric circulation? The
answer is rain and infrared radiation from the ocean absorbed by the moist tropical
atmosphere. Here's what happens. Sunlight warms the tropical ocean which evaporates water to keep from warming up. The ocean also radiates heat to the atmosphere, but the net radiation term is smaller than the evaporative term. Trade winds carry the heat in the form of water vapor to the tropical convergence zone. There the vapor condenses as rain, releasing its latent heat, and heating the atmosphere by as much as 125 W/m$^2$ averaged over a year (See figure 14.1).
\begin{figure}[b!]
\vspace{-2ex}
\makebox [121 mm] [c] {\includegraphics{zonalaveheat}}
\footnotesize
Figure 5.7 \textbf{Upper:} Zonal averages \rule{0mm}{4ex}of heat
transfer to the ocean by insolation\index{insolation!zonal average} $Q_{SW}$, and loss by
infrared radiation
$Q_{LW}$, sensible heat flux\index{sensible heat
flux!zonal average} $Q_S$, and latent heat flux $Q_L$, calculated by
DaSilva, Young, and Levitus (1995) using the \textsc{icoads} data set.
\textbf{Lower:} Net heat flux through the sea surface calculated from the data
above (solid line) and net heat flux\index{heat flux!zonal average} constrained to give heat
and other transports that match independent calculations of these transports. The area under the lower curves ought to be zero, but it is 16 W/m$^2$ for the unconstrained case and -3 W/m$^2$ for the constrained case.
\label{fig:zonalaveheat}
%\vspace{-3ex}
\end{figure}
At first it may seem strange that rain heats the air. After
all, we are familiar with summertime thunderstorms cooling the air at ground
level. The cool air from thunderstorms is due to downdrafts. Higher in the
cumulus cloud, heat released by rain warms the mid-levels of the atmosphere
causing air to rise rapidly in the storm. Thunderstorms are large heat engines
converting the energy of latent heat into kinetic energy of winds.
The zonal average of the oceanic heat-budget terms (figure 5.7) shows that
insolation\index{insolation!zonal average} is greatest in the tropics, that evaporation
balances insolation\index{insolation!balanced by evaporation}, and that sensible heat flux\index{sensible heat
flux!zonal average} is
small.
\textit{Zonal average}\index{heat budget!zonal average|textbf} is an average along lines of
constant latitude. Note that the terms in figure 5.7 don't sum to zero. The areal-weighted
integral of the curve for total heat flux\index{heat flux!global average} is not zero. Because
the net heat flux\index{heat flux!global average} into the ocean averaged over several years
must be less than a few watts per square meter, the non-zero value must be due to errors in the
various terms in the heat budget.
\begin{figure}[t!]
\includegraphics{globalsw}
\includegraphics{globalLW}
\footnotesize
Figure 5.8 \rule{0mm}{3ex}Annual-mean
insolation\index{insolation!annual average} $Q_{SW}$ (\textbf{top}) and infrared radiation
$Q_{LW}$ (\textbf{bottom}) through the sea surface during 1989 calculated by the
Satellite Data Analysis Center at the \textsc{nasa} Langley Research Center
(Darnell et al., 1992) using data from the International Satellite Cloud
Climatology Project. Units are W/m$^2$, contour interval is 10 W/m$^2$.
\label{fig:globalSW}
\vspace{-4ex}
\end{figure}
Errors in the heat budget terms can be reduced by using additional information. For example, we know roughly how much heat and other quantities are transported\index{transport!heat} by the ocean and atmosphere, and the known values for these transports can be used to constrain the calculations of net heat fluxes\index{heat flux!net} (figure 5.7). The constrained fluxes show that the heat gained by
the ocean in the tropics is balanced by heat lost by the ocean at high latitudes.
\begin{figure}[t!]
\includegraphics{globallatent}
\footnotesize
Figure 5.9 \rule{0mm}{3ex}Annual-mean latent heat
flux\index{heat flux!mean annual} from the sea surface $Q_{L}$ in W/m$^2$ during 1989
calculated from data compiled by the Data Assimilation\index{data assimilation} Office of
\textsc{nasa}'s Goddard Space Flight Center using reanalyzed data from the \textsc{ecmwf}
numerical weather prediction model. Contour interval is 10 W/m$^2$.
\label{fig:globallatent}
\vspace{-3ex}
\end{figure}
\begin{figure}[t!]
\includegraphics{globalsensible}
\includegraphics{globalflux}
\footnotesize
Figure 5.10 \rule{0mm}{3ex}Annual-mean upward sensible heat flux\index{sensible heat
flux!annual average}
$Q_{S}$ (\textbf{top}) and constrained, net, downward heat flux (\textbf{bottom})
through the sea surface in W/m$^2$ calculated by DaSilva, Young, and Levitus
(1995) using the \textsc{icoads} data set from 1945 to 1989. Contour interval is 2
W/m$^2$ (top) and 20 W/m$^2$ (bottom).
\label{fig:globalsensible}
\vspace{-5ex}
\end{figure}
Maps of the regional distribution of fluxes give clues to the processes
producing the fluxes. Clouds regulate the amount of sunlight reaching the sea
surface (figure 5.8 top), and solar heating is everywhere positive.
The net infrared heat flux\index{infrared flux} (figure 5.8 bottom) is largest in regions with
the least clouds, such as the center of the ocean and the eastern central Pacific. The
net infrared flux is everywhere negative. Latent heat fluxes (figure 5.9) are
dominated by evaporation in the trade wind regions and the offshore flow of cold
air masses behind cold fronts in winter offshore of Japan and North America.
Sensible heat fluxes\index{sensible heat
flux!maps of} (figure 5.10 top) are dominated by cold air blowing off
continents. The net heating gain (figure 5.10 bottom) is largest in equatorial
regions and net heat loss is largest downwind on Asia and North America.
Heat fluxes change substantially from year to year, especially in the topics, especially due to El Ni\~{n}o. See Chapter 14 for more on tropical variability.
\section{Meridional Heat Transport}
\index{meridional transport|textbf}\index{heat
transport!meridional|textbf}\index{transport!meridional} Overall, earth gains heat at the top
of the tropical atmosphere, and it loses heat at the top of the polar atmosphere. The
atmospheric and oceanic circulation together must transport heat from low to high latitudes to
balance the gains and losses. This north-south transport is called the \textit{meridional heat
transport}\index{heat transport!meridional}.
The sum of the meridional heat transport in the ocean and atmosphere is
calculated from the zonal average of the net heat flux\index{heat flux!net through the top of
atmosphere} through the top of the atmosphere measured by satellites. In making the
calculation, we assume that transports averaged over a few years are steady. Thus any
long-term, net heat gain or loss through the top of the atmosphere must be balanced by a
meridional transport and not by heat storage\index{heat storage} in the ocean or atmosphere.
\paragraph{Net Heat Flux at the Top of the Atmosphere}
\index{heat budget!through the top of the atmosphere}Heat flux through
the top of the atmosphere is measured very accurately by radiometers on satellites.
\begin{enumerate}
\vitem
Insolation is calculated from the solar constant\index{solar constant!and insolation} and
observations of reflected sunlight made by meteorological satellites and by
special satellites of the Earth Radiation Budget Experiment.
\vitem
Infrared radiation is measured by infrared radiometers on the satellites.
\vitem
The difference between insolation\index{insolation!at top of atmosphere} and net infrared
radiation is the net heat flux\index{heat flux!net through the top of atmosphere} across the
top of the atmosphere.
\end{enumerate}
\paragraph{Net Meridional Heat Transport}
To calculate the meridional heat transport\index{transport!meridional} in the atmosphere and
the ocean, we first average the net heat flux\index{heat flux!net through the top of
atmosphere} through the top of the atmosphere in a zonal band. Because the meridional
derivative of the transport is the zonal-mean flux, we calculate the transport from the
meridional integral of the zonal-mean flux. The integral must be balanced by the heat
transported by the atmosphere and the ocean across each latitude band.
Calculations by Trenberth and Caron (2001) show that the total, annual-mean, meridional heat
transport\index{transport!meridional} by the ocean and atmosphere peaks at 6 PW toward each
pole at 35\degrees\ latitude.
\paragraph{Oceanic Heat Transport}
\index{heat transport!calculation of}The meridional heat transport\index{transport!meridional}
in the ocean can be calculated three ways:
\begin{enumerate}
\vitem \textit{Surface Flux Method} \index{heat transport!calculation
of!surface flux method|textbf}calculates the heat flux\index{heat flux!net} through the sea
surface from measurements of wind, insolation\index{insolation}, air, and sea
temperature, and cloudiness. The fluxes are integrated to obtain the zonal average of the heat
flux\index{heat flux!zonal average} (figure 5.7). Finally, we calculate the transport from the
meridional integral of the zonal-mean flux just as we did at the top of the atmosphere.
\vitem
\textit{Direct Method} \index{heat transport!calculation of!direct method|textbf}
calculates the heat transport from values of current velocity and temperature
measured from top to bottom along a zonal section spanning an ocean basin. The flux
is the product of northward velocity and heat content derived from the temperature
measurement.
\vitem \textit{Residual Method}
\index{heat transport!calculation of!residual method|textbf} \index{transport!heat}first
calculates the atmospheric heat transport from atmospheric measurements or the output of
numerical weather models. This is the direct method applied to the atmosphere. The
atmospheric transport is subtracted from the total meridional transport calculated
from the top-of-the-atmosphere heat flux\index{heat flux!net through the top of atmosphere} to
obtain the oceanic contribution as a residual (figure 5.11).
\end{enumerate}
\begin{figure}[t!]
\makebox [120mm][c]{\includegraphics{heattransport}}
\footnotesize
Figure 5.11 Northward \rule{0mm}{3ex}heat transport\index{transport!heat} for 1988 in
each ocean and the total transport summed over all ocean calculated by
the residual method using atmospheric heat transport from \textsc{ecmwf} and
top of the atmosphere heat fluxes\index{heat flux!net through the top of atmosphere} from the
Earth Radiation Budget Experiment satellite. After Houghton et al. (1996: 212), which used data
from Trenberth and Solomon (1994). 1 PW $=$ 1 petawatt $= 10^{15}$ W.
\label{fig:heattransport}
\vspace{-4ex}
\end{figure}
\begin{figure}[b!]
\vspace{-5ex}
\includegraphics{solarinfluence}
\footnotesize
Figure 5.12 Changes in \rule{0mm}{3ex}solar constant\index{solar constant!variability of}
(total solar irradiance) and global mean temperature of earth's surface over the past 400
years. Except for a period of enhanced volcanic activity in the early 19th
century, surface temperature is well correlated with solar variability. After
Lean, personal communication.
\label{fig:solarinfluence}
%\vspace{-3ex}
\end{figure}
Various calculations of oceanic heat transports, \index{transport!heat}such as those shown in
figure 5.11, tend to be in agreement, and the error bars shown in the figure are realistic.
The total meridional transport of heat by the ocean is small compared with the total
meridional heat transport by the atmosphere except in the tropics. At 35\degrees, where the
total meridional heat transport is greatest, the ocean carries only 22\% of the heat in the
northern hemisphere, and 8\% in the southern (Trenberth and Caron, 2001).
\section{Variations in Solar Constant}
\index{solar constant!variability of}We have assumed so far that the solar constant, the output
of light and heat from the sun, is steady. Recent evidence based on variability of
sunspots and faculae (bright spots) shows that the output varied by $\pm 0.2$\% over
centuries (Lean, Beer, and Bradley, 1995), and that this variability is correlated
with changes in global mean temperature of earth's surface of $\pm 0.4$\degrees{C}.
(figure 5.12). In addition, White and Cayan (1998) found a small 12 yr, 22 yr, and
longer-period variations of sea-surface temperature measured by
bathythermographs\index{bathythermograph (BT)} and ship-board thermometers over the past
century. The observed response of earth to solar variability is about that calculated from
numerical models of the coupled ocean-atmosphere climate system. Many other changes in climate
and weather have been attributed to solar variability. The correlations are somewhat
controversial, and much more information can be found in Hoyt and Schatten's (1997) book on the
subject.
\section{Important Concepts}
\begin{enumerate}
\item Sunlight is absorbed primarily in the tropical ocean. The amount of
sunlight changes with season, latitude, time of day, and cloud cover.
\vitem Most of the heat absorbed by the ocean in the tropics is released as water
vapor which heats the atmosphere when water is condenses as rain. Most of the
rain falls in the tropical convergence zones, lesser amounts fall in
mid-latitudes near the polar front.
\vitem Heat released by rain and absorbed infrared radiation from the ocean are
the primary drivers for the atmospheric circulation.
\vitem The net heat flux\index{heat flux!net} from the ocean is largest in mid-latitudes and
offshore of Japan and New England.
\vitem Heat fluxes can be measured directly using fast response instruments on
low-flying aircraft, but this is not useful for measuring heat fluxes\index{heat
flux!measurement of} over large oceanic regions.
\vitem Heat fluxes through large regions of the sea surface can be calculated
from bulk formula. Seasonal, regional, and global maps of fluxes are
available based on ship and satellite observations.
\vitem The most widely used data sets for studying heat fluxes\index{heat flux!from
\textsc{icoads}} are the International Comprehensive Ocean-Atmosphere Data Set and the reanalysis of
meteorological data by numerical weather prediction models.
\vitem The atmosphere transports most of the heat\index{transport!heat} needed to warm
latitudes higher than 35\degrees. The oceanic meridional transport is comparable to the
atmospheric meridional transport only in the tropics.
\vitem Solar output is not constant, and the observed small variations in
output of heat and light from the sun seem to produce the changes in
global temperature observed over the past 400 years.
\end{enumerate}
\chapter{Temperature, Salinity, and Density}
Heat fluxes, evaporation, rain, river inflow, and freezing and melting of sea
ice all influence the distribution of temperature and salinity at the ocean's
surface. Changes in temperature and salinity can increase or decrease the density
of water at the surface, which can lead to convection. If water from the
surface sinks into the deeper ocean, it retains a distinctive relationship between
temperature and salinity which helps oceanographers track the movement of deep
water. In addition, temperature, salinity, and pressure are used to calculate
density. The distribution of density inside the ocean is directly related to the
distribution of horizontal pressure gradients and ocean currents. For all these
reasons, we need to know the distribution of temperature, salinity, and density in
the ocean.
Before discussing the distribution of temperature and salinity, let's first define
what we mean by the terms, especially salinity.
\section{Definition of Salinity}
At the simplest level, salinity\index{salinity|textbf} is the total amount of dissolved material in grams in one kilogram of sea water. Thus salinity is a dimensionless quantity. It has no units. The variability of dissolved salt is very small, and we must be very careful to define salinity in ways that are accurate and practical. To better understand the need for accuracy\index{accuracy!salinity}, look at figure
6.1. Notice that the range of salinity for most of the ocean's water is from 34.60 to 34.80 parts per thousand, which is 200 parts per million. The variability in the deep North Pacific is even smaller, about 20 parts per million. If we want to classify water with different salinity, we need definitions and instruments accurate to about one part per million. Notice that the range of temperature is much larger, about 1\degrees{C}, and temperature is easier to measure.
Writing a practical definition of salinity that has useful accuracy is difficult (see Lewis, 1980, for the details), and various definitions have been used.\\
\begin{figure}[t!]
\makebox [120mm][c]{\includegraphics{salhistogram}}
\footnotesize
Figure 6.1 Histogram of temperature \rule{0mm}{3ex}and salinity of ocean water colder than 4\degrees C. Height is proportional to
volume. Height of highest peak corresponds to a volume of 26 million cubic kilometers per bivariate class of 0.1\degrees{C} and 0.01. After Worthington (1981: 47).
\label{fig:salhistogram}
\vspace{-3ex}
\end{figure}
\paragraph{A Simple Definition} Originally salinity was defined to be the ``Total\index{salinity|textbf} amount of dissolved material in grams in one kilogram of sea water.'' This is not useful because the dissolved material is almost impossible to measure in practice. For example, how do we measure volatile material like gasses? Nor can we evaporate sea-water to dryness because chlorides are lost in the last stages of drying (Sverdrup, Johnson, and Fleming, 1942: 50).
\paragraph{A More Complete Definition} To avoid these difficulties, the \index{salinity!simple vs. complete}International Council for the Exploration of the Sea set up a commission in 1889 which recommended that salinity be defined as the ``Total amount of solid materials in grams dissolved in one kilogram of sea water when all the carbonate has been converted to oxide, the bromine and iodine replaced by chlorine and all organic matter completely oxidized.'' The definition was published in 1902. \textit{This is useful but difficult to use routinely}.
\paragraph{Salinity Based on Chlorinity} Because the above definition was difficult \index{salinity!based on chlorinity}to implement in practice, because salinity is directly proportional to the amount of chlorine in sea water, and because chlorine can be measured
accurately by a simple chemical analysis, salinity $S$ was redefined using chlorinity:
\begin{equation}
S = 0.03 + 1.805\, Cl
\end{equation}
where \textit{chlorinity}\index{chlorinity|textbf} $Cl$ is defined as ``the mass of silver required to precipitate completely the halogens in 0.328 523 4 kg of the sea-water sample.''
As more and more accurate measurements were made, (6.1) turned out to be too inaccurate. In 1964 \textsc{unesco} and other international organizations appointed a Joint Panel on Oceanographic Tables and Standards to produce a more accurate definition. The Joint Panel recommended in 1966 (Wooster, Lee, and Dietrich, 1969) that salinity and chlorinity be related using:
\begin{equation}
S = 1.806\,55\,Cl
\end{equation}
This is the same as (6.1) for $S=35$.
\paragraph{Salinity Based on Conductivity} At the same time (6.2) was \index{salinity!based on conductivity}adopted, ocean\-ographers had began using conductivity meters to measure salinity. The meters were very precise and relatively easy to use compared with the chemical techniques used to measure chlorinity. As a result, the Joint Panel also recommended that salinity be related to conductivity\index{conductivity} of sea water using:
\begin{subequations}
\begin{align}
S = &-0.089\,96 + 28.297\,29\,R_{15} + 12.808\,32\,R^2_{15} \notag \\
&-10.678\,69\,R^3_{15} + 5.986\,24\,R^4_{15} - 1.323\,11\,R^5_{15} \\
R_{15} = \: &C(S,15,0)/C(35,15,0)
\end{align}
\end{subequations}
where $C(S, 15, 0)$ is the conductivity of the sea-water sample at 15\degrees C and atmospheric pressure, having a salinity $S$ derived from (6.4), and $C(35, 15, 0)$ is the conductivity of standard ``Copenhagen'' sea water\index{Copenhagen sea water}. Millero (1996) points out that (6.3) is not a new definition of salinity, it merely gives chlorinity as a function of conductivity of seawater relative to standard seawater.
\paragraph{Practical Salinity Scale of 1978} By the early 1970s, accurate conductivity meters could be deployed from ships to measure conductivity at depth. The need to re-evaluate the salinity scale led the Joint Panel to recommend in 1981 (\textsc{jpots}, 1981; Lewis, 1980) that salinity be defined using only conductivity, breaking the link with chlorinity. All water samples with the same conductivity ratio have the same salinity even though the their chlorinity may differ.
The \textit{Practical Salinity Scale of 1978}\index{salinity!Practical Salinity Scale|textbf} is now the official definition:
\begin{subequations}
\begin{align}
S = &\:0.0080 -0.1692\,K^{1/2}_{15} + 25.3851\,K_{15} + 14.0941\,K^{3/2}_{15} \notag \\
&- 7.0261\,K^{2}_{15} + 2.7081\,K^{5/2}_{15} \\
K_{15} = &\:C(S,15,0)/C(KCl,15,0) \\
2 \leq S &\:\leq 42 \notag
\end{align}
\end{subequations}
where $C(S, 15, 0)$ is the conductivity of the sea-water sample at a temperature of 14.996\degrees C on the International Temperature Scale of 1990 (\textsc{its}-90, see \S 6.2) and standard atmospheric pressure of 101 325 Pa\index{pressure!standard atmospheric}. $C(KCl, 15, 0)$ is the conductivity of the standard potassium chloride (KCl) solution at a temperature of 15\degrees C and standard atmospheric pressure. The standard KCl solution contains a mass of $32.435\,6$ grams of KCl in a mass of $1.000\,000$ kg of solution. Millero (1996: 72) and Lewis (1980) gives equations for calculating salinity at other pressures and temperatures.
\paragraph{Comments} The various definitions of salinity work well because the ratios of the various ions in sea water are nearly independent of salinity and location in the ocean (table 6.1). Only very fresh waters, such as are found in estuaries, have significantly different ratios. The result is based on Dittmar's (1884) chemical analysis of 77 samples of sea water collected by the \textit{Challenger} Expedition and further studies by Carritt and Carpenter (1959).
\begin{quote} \small
The importance of this result cannot be over emphasized, as upon it depends the validity of the chlorinity: salinity: density relationships and, hence, the accuracy\index{accuracy!density} of all conclusions based on the distribution of density where the latter is determined by chemical or indirect physical methods such as electrical conductivity$\ldots$---Sverdrup, Johnson, Fleming (1942).
\end{quote}
The relationship between conductivity and salinity has an accuracy\index{accuracy!salinity} of around $\pm 0.003$ in salinity. The very small error is caused by variations in constituents such as SiO$_2$ which cause small changes in density but no change in conductivity.
\begin{table}[h!]\centering \small
\begin{tabular*}{72mm}{@{}clcl@{}}
\multicolumn{4}{@{}l@{}}{\bfseries Table 6.1 Major \rule[-1ex]{0mm}{1ex}Constituents of Sea Water} \\
\hline
& Ion & & \rule{0ex}{2.5ex}Atoms \\
\hline
55.3\% & \rule{0ex}{2.5ex}Chlorine & 55.3\% & Chlorine \\
30.8\% & Sodium & 30.8\% & Sodium \\
7.7\% & Sulfate & 3.7\% & Magnesium \\
3.7\% & Magnesium & 2.6\% & Sulfur \\
1.2\% & Calcium & 1.2\% & Calcium \\
1.1\% & Potassium & 1.1\% & Potassium \\[0.5ex]
\hline
\end{tabular*} \\[0.5ex]
\vspace{-3ex}
\end{table}
\paragraph{Reference Seawater and Salinity}
The Practical Salinity Scale of 1978 introduced several small problems. It led to confusion about units and to the use of ``practical salinity units" that are not part of the definition of Practical Salinity. In addition, absolute salinity differs from salinity by about 0.5\%. And, the composition of seawater differs slightly from place to place in the ocean, leading to small errors in measuring salinity.
To avoid these and other problems, Millero et al (2008) defined a new measure of salinity, the Reference Salinity, that accurately represents the Absolute Salinity of an artificial seawater solution. It is based on a Reference Composition of seawater that is much more accurate than the values in Table 6.1 above. The \textit{Reference Composition}\index{salinity!Reference|textbf} of the artificial seawater is defined by a list of solutes and their mole fractions given in Table 4 of their paper. From this, they defined artificial \textit{Reference Seawater}\index{Reference Seawater|textbf} to be seawater having a Reference Composition solute dissolved in pure water as the solvent, and adjusted to its thermodynamic equilibrium state. Finally, the \textit{Reference Salinity} of Reference Seawater was defined to be exactly 35.16504 g kg$^{-1}$.
With these definitions, plus many details described in their paper, Millero et al (2008) show Reference Salinity is related to Practical Salinity\index{salinity!practical} by:
\begin{equation}
S_R \approx (35.16504/35) \text{g kg}^{-1} \times S
\end{equation}
The equation is exact at $S = $ 35. Reference Salinity is approximately 0.47\% larger than Practical Salinity. Reference Salinity $S_R$ is intended to be used as an SI-based extension of Practical Salinity.
\section{Definition of Temperature}
\index{temperature|textbf}Many physical processes depend on temperature.\index{temperature} A few can be used to define absolute
temperature $T$. The unit of $T$ is the kelvin, which has the symbol K. The fundamental processes used for defining an absolute temperature scale over the range of temperatures found in the ocean include (Soulen and Fogle, 1997): 1) the gas laws relating pressure to temperature of an ideal gas with corrections for the density of the gas; and 2) the voltage noise of a resistance $R$.
The measurement of temperature using an absolute scale\index{temperature!absolute} is difficult and the measurement is usually made by national standards laboratories. The absolute measurements are used to define a practical temperature scale\index{temperature!practical scale} based on the temperature of a few fixed points and interpolating devices which are calibrated at the fixed points.
For temperatures commonly found in the ocean, the interpolating device is a platinum-resistance thermometer. It consists of a loosely wound, strain-free, pure platinum wire whose resistance is a function of temperature. It is calibrated at fixed points between the triple point of equilibrium hydrogen at 13.8033 K and the freezing point of silver at 961.78 K, including the triple point of water at 0.060\degrees C, the melting point of Gallium at 29.7646\degrees C, and the freezing point of Indium at 156.5985\degrees C (Preston-Thomas, 1990). The triple point of water is the temperature at which ice, water, and water vapor are in equilibrium. The temperature scale in kelvin $T$ is related to the temperature scale in degrees Celsius $t$[\degrees C] by:
\begin{equation}
t \text{ [\degrees C]} = T \text{ [K]} -273.15
\end{equation}
\begin{figure}[t!]
%\vspace{-4ex}
\includegraphics{sst_climatology}
\footnotesize
Figure 6.2 Mean sea-surface \rule{0mm}{3ex}temperature calculated from the optimal interpolation technique (Reynolds and Smith, 1995) using ship reports and \textsc{avhrr} measurements of temperature. Contour interval is 1\degrees C with heavy contours every 5\degrees C. Shaded areas exceed 29\degrees C.
\label{fig:sst_climatology}
\vspace{-4ex}
\end{figure}
The practical temperature scale was revised in 1887, 1927, 1948, 1968, and 1990
as more accurate determinations of absolute temperature became accepted. The most
recent scale is the International Temperature Scale of 1990 (\textsc{its}-90)\index{temperature!International Temperature Scale}. It
differs slightly from the International Practical Temperature Scale of
1968 \textsc{ipts}-68. At 0\degrees C they are the same, and above 0\degrees C
\textsc{its}-90 is slightly cooler. $t_{90}-t_{68} = -0.002$ at 10\degrees
C, $-0.005$ at 20\degrees C, $-0.007$ at 30\degrees C and $-0.010$ at 40\degrees
C.
\begin{figure}[b!]
\vspace{-3ex}
\makebox[121mm][c]{\includegraphics{SSTvariability}}
\footnotesize
Figure 6.3 \textbf{Top:} Sea-surface \rule{0mm}{3ex}temperature anomaly for January 1996 relative to mean temperature shown in figure 6.2 using data published by Reynolds and Smith (1995) in the \textit{Climate Diagnostics Bulletin} for February 1995. Contour interval is 1\degrees C. Shaded areas are
positive. \textbf{Bottom:}Annual range of sea-surface temperature in \degrees{C} calculated from the Reynolds and Smith (1995) mean sea-surface temperature data set. Contour interval is 1\degrees C with heavy contours at 4\degrees C and 8\degrees C. Shaded areas exceed 8\degrees C.
\label{fig:SSTvariability}
%\vspace{-3ex}
\end{figure}
Notice that while oceanographers use thermometers calibrated with an
accuracy\index{accuracy!temperature} of a millidegree, which is 0.001\degrees C, the
temperature scale itself has uncertainties of a few millidegrees.
\section[Geographical Distribution]{Geographical Distribution of Surface
Temperature and Salinity} \index{salinity!geographical distribution
of}\index{temperature!geographical distribution of}The distribution of temperature
at the sea surface tends to be \textit{zonal}, \index{zonal|textbf}that is, it is
independent of longitude (figure 6.2). Warmest water is near the equator, coldest
water is near the poles. The deviations from zonal are small. Equatorward of
40\degrees, cooler waters tend to be on the eastern side of the basin. North of this
latitude, cooler waters tend to be on the western side.
The \textit{anomalies} \index{anomalies!sea-surface temperature|textbf}of sea-surface
temperature, the deviation from a long term average, are small, less than 1.5\degrees C
(Harrison and Larkin, 1998) except in the equatorial Pacific where the deviations can be
3\degrees{C} (figure 6.3: top).
\begin{figure}[t!]
\makebox[121mm][c]{\includegraphics{salinity}}
\footnotesize
Figure 6.4 \textbf{Top}: Mean sea-surface \rule{0pt}{3ex} salinity. \rule{0mm}{2ex}Contour interval is 0.25. Shaded areas exceed a salinity of 36. From Levitus (1982).
\textbf{Bottom}: Precipitation minus evaporation in meters per year calculated from global rainfall\index{rainfall!global} by the Global Precipitation Climatology Project and latent heat flux calculated by the Data Assimilation Office, both at \textsc{nasa}'s Goddard Space Flight Center. Precipitation
exceeds evaporation in the shaded regions, contour interval is 0.5 m.
\label{fig:salinity}
\vspace{-3ex}
\end{figure}
The annual range of surface temperature is highest at mid-latitudes,
especially on the western side of the ocean (figure 6.3: bottom). In the west, cold
air blows off the continents in winter and cools the ocean. The cooling dominates
the heat budget. In the tropics the temperature range is mostly less than
2\degrees{C}.
The distribution of sea-surface salinity also tends to be zonal. The saltiest waters are at mid-latitudes where evaporation is high. Less salty waters are near the equator where rain freshens the surface, and at high latitudes where melted sea ice freshens the surface (figure 6.4). The zonal (east-west) average of salinity shows a close correlation between salinity and evaporation minus precipitation plus river input (figure 6.5).
Because many large rivers drain into the Atlantic and the Arctic Sea, why is the
Atlantic saltier than the Pacific? Broecker (1997) showed that 0.32 Sv of the
water evaporated from the Atlantic does not fall as rain on land. Instead, it is
carried by winds into the Pacific (figure 6.6). Broecker points out that the
quantity is small, equivalent to a little more than the flow in the Amazon
River, but ``were this flux not compensated by an exchange of more salty
Atlantic waters for less salty Pacific waters, the salinity of the entire
Atlantic would rise about 1 gram per liter per millennium.''
\begin{figure}[t!]
%\vspace{-2ex}
\makebox [121mm][c]{\includegraphics{zonalsalinity}}
\footnotesize
Figure 6.5 Zonal average of sea-surface \rule{0mm}{3ex}salinity calculated for all the ocean from Levitus (1982) and the difference between evaporation and precipitation ($E - P$) calculated from data shown in figure 6.4 (bottom).
\label{fig:zonalsalinity}
\vspace{-5ex}
\end{figure}
\begin{figure}[b!]
\vspace{-3ex}
\makebox[121mm][c]{\includegraphics{BroeckerPlot}}
\footnotesize
Figure 6.6 Water \rule{0pt}{3ex} transported \index{transport!atmospheric}by the atmosphere
into and out of the Atlantic. Basins draining into the Atlantic are black, deserts
are white, and other drainage basins are shaded. Arrows give direction of water
transport by the atmosphere, and values are in Sverdrups. Bold numbers give the
net transport for the Atlantic at each latitude band. Overall, the Atlantic loses 0.32 Sv, an
amount approximately equal to the flow in the Amazon River. After Broecker (1997).
\label{fig:BroeckerPlot}
%\vspace{-4ex}
\end{figure}
\begin{figure}[t!]
%\vspace{-2ex}
\makebox[121 mm][c]{\includegraphics{seasonalthermo}}
\footnotesize
Figure 6.7 Growth and decay \rule{0mm}{4ex} of the mixed layer\index{mixed layer!seasonal
growth and decay} and seasonal thermocline\index{thermocline!seasonal} from November 1989 to
September 1990 at the Bermuda Atlantic Time-series Station (\textsc{bats}) at 31.8\degrees N
64.1\degrees W. Data were collected by the Bermuda Biological Station for Research, Inc. Note
that pressure in decibars is nearly the same as depth in meters (see \S 6.8 for a definition of
decibars).
\label{fig:seasonalthermo}
\vspace{-4ex}
\end{figure}
\textit{Mean Temperature and Salinity of the Ocean} \index{ocean!mean
temperature|textbf}\index{ocean!mean salinity|textbf}The mean temperature of the ocean's
waters is: t = 3.5\degrees{C}. The mean salinity is S = 34.7. The distribution about the mean
is small: 50\% of the water is in the range:
\begin{align}
1.3^{\circ}\text{C} < &\:t < 3.8^{\circ}\text{C} \notag \\
34.6 < &\:S < 34.8 \notag
\end{align}
\section{The Oceanic Mixed Layer and Thermocline}
Wind blowing on the ocean stirs the upper layers leading to a thin \textit{mixed
layer} \index{mixed layer|textbf}at the sea surface having constant temperature and salinity
from the surface down to a depth where the values differ from those at the surface. The
magnitude of the difference is arbitrary, but typically the temperature at the bottom of the
layer must be no more than 0.02--0.1\degrees\ colder than at the surface. Note that both
temperature and salinity must be constant in the mixed layer. We will see later that mean
velocity is not constant. The mixed layer is roughly 10--200 m thick over most of the tropical
and mid-latitude belts.
The depth and temperature of the mixed layer\index{mixed layer!external forcing of} varies from
day to day and from season to season in response to two processes:
\begin{enumerate}
\vitem Heat fluxes through the surface heat and cool the surface waters. Changes
in temperature change the density contrast between the mixed layer and deeper
waters. The greater the contrast, the more work is needed to mix the layer
downward and visa versa.
\vitem Turbulence in the mixed layer mixes heat downward. The turbulence\index{turbulence!in
mixed layer} depends on the wind speed and on the intensity of breaking waves. Turbulence mixes
water in the layer, and it mixes the water in the layer with water in the
thermocline\index{thermocline}.
\end{enumerate}
The mid-latitude mixed layer\index{mixed layer!mid-latitude} is thinnest in late summer when
winds are weak, and sunlight warms the surface layer (figure 6.7). At times, the heating is so
strong, and the winds so weak, that the layer is only a few meters thick. In fall,
the first storms of the season mix the heat down into the ocean thickening
the mixed layer, but little heat is lost. In winter, heat is lost, and the mixed
layer continues to thicken, becoming thickest in late winter. In spring, winds
weaken, sunlight increases, and a new mixed layer forms.
\begin{figure}[t!]
%\vspace{-2ex}
\makebox[121 mm] [c] {\includegraphics{TandSProfile}}
%\centering
\footnotesize
Figure 6.8 Typical \rule{0mm}{4ex}temperature and salinity profiles in the
open ocean. AAC: At 62.0\degrees\ S, 170.0\degrees\ E in the Antarctic Circumpolar
Current\index{Antarctic Circumpolar Current} on 16 January 1969 as measured by the \textit{R/V
Hakuho Maru}. Warm Pool: At 9.5\degrees\ N 176.3\degrees\ E in the tropical west Pacific warm
pool on 12 March 1989 as measured by Bryden and Hall on the \textit{R/V Moana Wave}.
\textsc{bats}: At 31.8\degrees\ N 64.1\degrees\ W near Bermuda on 17 April and 10
September 1990 as measured by the Bermuda Biological Station for Research, Inc.
Data are included with Java OceanAtlas.
\label{fig:TandSProfile}
\vspace{-5ex}
\end{figure}
Below the mixed layer\index{mixed layer!mid-latitude}, water temperature decreases rapidly with
depth except at high latitudes. The range of depths where the rate of change, the gradient of
temperature, is large is called the
\textit{thermocline}\index{thermocline|textbf}. Because density is closely related
to temperature, the thermocline also tends to be the layer where density gradient is greatest,
the \textit{pycnocline}\index{pycnocline|textbf}.
The shape of the thermocline varies slightly with the seasons (figure 6.7). This is the \textit{seasonal
thermocline}.
\index{thermocline!seasonal|textbf}
The \textit{permanent thermocline}
\index{thermocline!permanent|textbf}extends from below the seasonal thermocline to
depths of 1500--2000 meters (figure 6.8). At high latitudes, such as at the
\textsc{aac} station in the figure, there may be a cooler, fresher layer above the
permanent thermocline.
The mixed layer tends to be saltier than the thermocline\index{thermocline} between
10\degrees\ and 40\degrees\ latitude, where evaporation exceeds precipitation. At high
latitudes the mixed layer\index{mixed layer!high latitude} is fresher because rain and melting
ice reduce salinity. In some tropical regions, such as the warm pool in the western tropical
Pacific, rain also produces a thin fresher mixed layer\index{mixed layer!tropical Pacific}.
\section[Density]{Density, Potential Temperature, and Neutral Density}
During winter, cold water formed at the surface sinks to a depth determined by its
density relative to the density of the deeper water. Currents then carry the
water to other parts of the ocean. At all times, the water parcel moves to stay
below less dense water and above more dense water. The distribution of currents
within the ocean depends on the distribution of pressure, which depends
on the variations of density inside the ocean as outlined in \S10.4. So, if we
want to follow water movement within the ocean, we need to know the distribution
of density within the ocean.
\paragraph{Density and sigma-t}
\index{density}The calculation of water movement requires
measurements of density with an accuracy\index{accuracy!density} of a few parts per
million. This is not easy.
\textit{Absolute Density} \index{density!absolute|textbf}of water can only be
measured in special laboratories, and only with difficulty. The best accuracy is
$1\,:\, 2.5 \times 10^5 = 4$ parts per million.
To avoid the difficulty of working with absolute density, oceanographers use
density relative to density of pure water. Density $\rho (S, t, p)$ is now
defined using Standard Mean Ocean Water of known isotopic composition, assuming
saturation of dissolved atmospheric gasses. Here $S, t, p$ refers to salinity,
temperature, and pressure.
In practice, density is not measured, it is calculated from \textit{in situ} \index{in situ}
measurements of pressure, temperature, and conductivity using the equation of
state for sea water. This can be done with an accuracy\index{accuracy!density} of two parts per
million.
Density of water at the sea surface is typically 1027 kg/m$^3$. For
simplification, physical oceanographers often quote only the last 2 digits of the
density, a quantity they call \textit{density anomaly} or \textit{Sigma
(S,t,p)}\index{density!anomaly or sigma|textbf}:
\begin{equation}
\sigma(S,t,p) = \rho (S, t, p) - 1000 \text{\ kg/m$^3$}
\end{equation}
The Working Group on Symbols, Units and Nomenclature in Physical Oceanography (\textsc{sun}, 1985) recommends that $\sigma$ be replaced by $\gamma$ because $\sigma$ was originally defined relative to pure water and it was dimensionless. Here, however, I will follow common practice and use $\sigma$.
If we are studying surface layers of the ocean, we can ignore compressibility, and
we use a new quantity sigma-t (written $\sigma _t$):
\begin{equation}
\sigma _t = \sigma(S,t,0)
\end{equation}
This is the density anomaly of a water sample when the total pressure on it
has been reduced to atmospheric pressure (\textit{i.e.} zero water pressure), but
the temperature and salinity are \textit{in situ} values.\index{in situ}
\begin{figure}[t!]
\makebox[121 mm] [c] {\includegraphics{thetaprofile}}
\footnotesize
Figure 6.9 Profiles \rule{0mm}{4ex}of \textbf{Left} \textit{in situ}\index{in situ} $t$ and potential $\theta$
temperature and \textbf{Right} sigma-t and sigma-theta in the Kermadec Trench in the
Pacific measured by the R/V Eltanin during the Scorpio Expedition on 13 July 1967
at 175.825\degrees\ E and 28.258\degrees\ S. Data from Warren (1973).
\label{fig:thetaprofile}
\vspace{-4ex}
\end{figure}
\paragraph{Potential Temperature}
\index{temperature!potential}\index{potential!temperature}As a water parcel moves within the ocean below the mixed layer\index{mixed layer}, its salt and heat content can change only by mixing with other water. Thus we can use measurements of temperature and salinity to trace the path of the water. This is best done if we remove the effect of compressibility.
As water sinks, pressure increases, the water is compressed, and the compression does work on the water. This increases the internal energy of the water. To understand how compression increases energy, consider a cube containing a fixed mass of water. As the cube sinks, its sides move inward as the cube is compressed. Recalling that work is force times distance, the work is the distance the side moves times the force exerted on the side by pressure. The change in internal energy may or may not result in a change in temperature (McDougall and Feistel, 2003). The internal energy of a fluid is the sum of molecular kinetic energy (temperature) and molecular potential energy. In sea water, the later term dominates, and the change of internal energy produces the temperature change shown in figure 6.9. At a depth of 8 km, the increase in temperature is almost 0.9\degrees\ C.
To remove the influence of compressibility from measurements of temperature, oceanographers (and meteorologists who have the same problem in the atmosphere) use the concept of potential temperature. \textit{Potential temperature}\index{temperature!potential|textbf}\index{potential!temperature|textbf} $\Theta$ is defined as the temperature of a parcel of water at the sea surface after it has been raised adiabatically from some depth in the ocean. Raising the parcel \textit{adiabatically} \index{adiabatically|textbf}means that it is raised in an insulated container so it does not exchange heat with its surroundings. Of course, the parcel is not actually brought to the surface. Potential temperature is calculated from the temperature in the water at depth, the \textit{in situ}\index{in situ} temperature.
\paragraph{Potential Density}
\index{density!potential|textbf}\index{potential!density|textbf}If we are studying intermediate layers of the ocean, say at depths near a kilometer, we cannot ignore compressibility. Because changes in pressure primarily influence the temperature of the water, the influence of pressure can be removed, to a first approximation, by using the \textit{potential density}.
\textit{Potential density} $\rho _{\Theta}$ is the density a parcel of water would have if it were raised adiabatically to the surface without change in salinity. Written as sigma,
\begin{equation}
\sigma _{\Theta} = \sigma(S, \Theta, 0)
\end{equation}
$\sigma _{\Theta}$ is especially useful because it is a conserved thermodynamic property.
Potential density is not useful for comparing density of water at great depths. If we bring water parcels to the surface and compare their densities, the calculation of potential density ignores the effect of pressure on the coefficients for thermal and salt expansion. As a result, two water samples having the same density but different temperature and salinity at a depth of four kilometers can have noticeably different potential density. In some regions the use of $\rho(\Theta)$ can lead to an apparent decrease of density with depth (figure 6.10) although we know that this is not possible because such a column of water would be unstable.
\begin{figure}[b!]
\vspace{-2ex}
\includegraphics{atlsection}
\footnotesize
Figure 6.10 Vertical \rule{0mm}{4ex}sections of density in the western Atlantic.
Note that the depth scale changes at 1000 m depth. \textbf{Upper}: $\sigma
_{\Theta}$, showing an apparent density inversion below 3,000 m. \textbf{Lower}:
$\sigma _4$ showing continuous increase in density with depth. After Lynn and Reid
(1968).
\label{fig:atlsection4}
%\vspace{-4ex}
\end{figure}
To compare samples from great depths, it is better to bring both samples to a nearby depth instead of to the surface $p = 0$. For example, we can bring both parcels to a pressure of 4,000 decibars, which is near a depth of 4 km:
\begin{equation}
\sigma_4 = \sigma(S, \Theta, 4000)
\end{equation}
where $\sigma_4$ is the density of a parcel of water brought adiabatically to a pressure of 4,000 decibars. More generally, oceanographers sometimes use $\rho_r$
\begin{equation}
\sigma _r = \sigma(S, \Theta, p, p_r)
\end{equation}
where $p$ is pressure, and $p_r$ is pressure at some reference level. In (6.8) the level is $p_r = 0$ decibars, and in (6.9) $p_r = 4000$ decibars.
The use of $\sigma_r$ leads to problems. If we wish to follow parcels of water
deep in the ocean, we might use $\sigma_3$ in some areas, and $\sigma_4$ in others.
But what happens when a parcel moves from a depth of 3 km in one area to a depth of
4 km in another? There is a small discontinuity between the density of the
parcel expressed as $\sigma_3$ compared with density expressed as $\sigma_4$. To
avoid this difficulty, Jackett and McDougall (1997) proposed a new variable they
called neutral density.
\paragraph{Neutral Surfaces and Density}
\index{neutral surfaces}\index{density!neutral surfaces}A parcel of water moves
locally along a path of constant density so that it is always below less dense water and above
more dense water. More precisely, it moves along a path of constant potential
density $\sigma_r$ referenced to the local depth $r$. Such a path is called a
\textit{neutral path}\index{neutral path|textbf} (Eden and Willebrand, 1999). A
\textit{neutral surface element} \index{neutral surface element|textbf}is the
surface tangent to the neutral paths through a point in the water. No work is
required to move a parcel on this surface because there is no buoyancy\index{buoyancy} force
acting on the parcel as it moves (if we ignore friction).
Now let's follow the parcel as it moves away from a local region. At first we might think that because we know the tangents to the surface everywhere, we can define a surface that is the envelope
of the tangents. But an exact surface is not mathematically possible in the real
ocean, although we can come very close.
Jackett and McDougall (1997) developed a practical neutral density variable
$\gamma^n$ and surface that stays within a few tens meters of an ideal surface
anywhere in the world. They constructed their variables using data in the Levitus
(1982) atlas. The neutral density values were then used to label the data in the
Levitus atlas. This prelabeled data set is used to calculate $\gamma^n$ at new
locations where $t, S$ are measured as a function of depth by interpolation to
the four closest points in the Levitus atlas. Through this practice, neutral
density $\gamma^n$ is a function of salinity $S$, \textit{in situ}\index{in situ} temperature $t$,
pressure $p$, longitude, and latitude.
The neutral surface defined above differs only slightly from an ideal neutral
surface. If a parcel moves around a gyre on the neutral surface and returns to its
starting location, its depth at the end will differ by around 10 meters from the
depth at the start. If potential density surfaces are used, the difference can be
hundreds of meters, a far larger error.
\paragraph{Equation of state of sea water}
\index{density!equation!of state|textbf}\index{equation of state|textbf}Density
of sea water is rarely measured.
\textit{Density is calculated from measurements of temperature, conductivity,
or salinity, and pressure using the equation of state of sea water}. The
\textit{equation of state} is an equation relating density to temperature, salinity,
and pressure.
The equation is derived by fitting curves through laboratory measurements of
density as a function of temperature, pressure, and salinity, chlorinity, or
conductivity. The International Equation of State (1980) published by the Joint
Panel on Oceanographic Tables and Standards (1981) is now used. See also
Millero and Poisson (1981) and Millero et al (1980). The equation has an
accuracy\index{accuracy!equation!of state} of 10 parts per million, which is 0.01 units of
$\sigma(\Theta)$.
I have not actually written out the equation of state because it consists of
three polynomials with 41 constants (\textsc{jpots}, 1991).
\paragraph{Accuracy of Temperature, Salinity, and Density}
\index{accuracy}\index{temperature!accuracy of}
\index{salinity!accuracy of} \index{density!accuracy of}If we want
to distinguish between different water masses in the ocean, and if
the total range of temperature and salinity is as small as the
range in figure 6.1, then we must measure temperature, salinity,
and pressure very carefully. We will need an accuracy of a few
parts per million.
Such accuracy can be achieved only if all quantities are carefully
defined, if all measurements are made with great care, if all
instruments are carefully calibrated, and if all work is done
according to internationally accepted standards. The standards are
laid out in \textit{Processing of Oceanographic Station
Data}\index{JPOTS (Processing of Oceanographic Station Data)}
(\textsc{jpots}, 1991) published by \textsc{unesco}. The book
contains internationally accepted definitions of primary variables
such as temperature and salinity and methods for the measuring the
primary variables. It also describes accepted methods for
calculating quantities derived from primary variables, such as
potential temperature, density, and stability.
\section{Measurement of Temperature}
\index{temperature!measurement at surface}Temperature in the ocean
is measured many ways. Thermistors and mercury thermometers
are commonly used on ships and buoys. These are calibrated in the
laboratory before being used, and after use if possible, using
mercury or platinum thermometers with accuracy\index{accuracy!thermometers!platinum} traceable
to national standards laboratories. Infrared radiometers on
satellites measure the ocean's surface temperature.
\paragraph{Mercury Thermometer} This is the most widely used,
\index{temperature!measurement at surface!by mercury
thermometers}\index{thermometer!mercury}non-electronic thermometer. It was widely
used in buckets dropped over the side of a ship to measure the
temperature of surface waters, on Nansen bottles to measure sub-sea
temperatures, and in the laboratory to calibrate other
thermometers. Accuracy\index{accuracy!thermometers!mercury} of the best thermometers is about
$\pm$0.001\degrees{C} with very careful calibration.
\begin{figure}[t!]
\makebox [120mm][c]{\includegraphics{thermometer}}
\footnotesize
Figure 6.11 \textbf{Left}: Protected \rule{0mm}{4ex} and
unprotected reversing thermometers\index{thermometer!reversing} is set position, before
reversal.
\textbf{Right}: The constricted part of the capillary in set and reversed
positions. After von Arx (1962: 259).
\label{fig:thermometer}
\vspace{-3ex}
\end{figure}
One very important mercury thermometer is the reversing
thermometer\index{thermometer!reversing} (figure 6.11) carried on Nansen bottles, which are
described in the next section. It is a thermometer that has a constriction in the mercury
capillary that causes the thread of mercury to break at a precisely determined point when the
thermometer is turned upside down. The thermometer is lowered deep into the ocean in the
normal position, and it is allowed to come to equilibrium with the water. Mercury
expands into the capillary, and the amount of mercury in the capillary is
proportional to temperature. The thermometer is then flipped upside down, the
thread of mercury breaks trapping the mercury in the capillary, and the
thermometer is brought back. The mercury in the capillary of the reversed
thermometer is read on deck along with the temperature of a normal thermometer,
which gives the temperature at which the reversed thermometer is read. The two
readings give the temperature of the water at the depth where the thermometer
was reversed.
The reversing thermometer\index{thermometer!reversing} is carried inside a glass tube which
protects the thermometer from the ocean's pressure because high pressure can squeeze
additional mercury into the capillary. If the thermometer is unprotected, the
apparent temperature read on deck is proportional to temperature and pressure at
the depth where the thermometer was flipped. A pair of protected and unprotected
thermometers gives temperature and pressure of the water at the depth the
thermometer was reversed.
Pairs of reversing thermometers\index{thermometer!reversing} carried on Nansen bottles were the
primary source of sub-sea measurements of temperature as a function of pressure from around 1900
to 1970.
\paragraph{Platinum Resistance Thermometer} This is the standard for temperature.
\index{temperature!measurement at surface!by platinum resistance
thermometers}It is used by national standards laboratories to
interpolate between defined points on the practical temperature
scale. It is used primarily to calibrate other temperature sensors.
\paragraph{Thermistor} A thermistor is a semiconductor having
\index{temperature!measurement at surface!by thermistors}\index{thermistor}resistance
that varies rapidly and predictably with temperature. It has been widely
used on moored instruments and on instruments deployed from ships since about 1970.
It has high resolution and an accuracy\index{accuracy!temperature!thermistor} of about
$\pm$0.001\degrees{C} when carefully calibrated.
\paragraph{Bucket temperatures} The temperature of surface waters has been
\index{temperature!measurement at surface!by bucket thermometers}routinely measured at sea by
putting a mercury thermometer into a bucket which is lowered into the water, letting
it sit at a depth of about a meter for a few minutes until the
thermometer comes to equilibrium, then bringing it aboard and
reading the temperature before water in the bucket has time to
change temperature. The accuracy\index{accuracy!temperature!bucket} is around 0.1\degrees{C}.
This is a very common source of direct surface temperature measurements.
\paragraph{Ship Injection Temperature} The temperature of the water drawn into
\index{temperature!measurement at surface!from ship injection
temperatures}the ship to cool the engines has been recorded
routinely for decades. These recorded values of temperature are called
injection temperatures. Errors are due to ship's structure warming
water before it is recorded. This happens when the temperature
recorder is not placed close to the point on the hull where water
is brought in. Accuracy\index{accuracy!temperature!ship-injection} is 0.5\degrees --1\degrees
C.
\paragraph{Advanced Very High Resolution Radiometer} The most commonly
\index{temperature!measurement at surface!by Advanced Very High
Resolution Radiometer (AVHRR)}\index{Advanced Very High Resolution Radiometer
(AVHRR)|textbf}used instrument to measure sea-surface temperature from space is the Advanced
Very High Resolution Radiometer \textsc{avhrr}. The instrument has been carried on all
polar-orbiting meteorological satellites operated by \textsc{noaa}
since Tiros-N was launched in 1978.
The instrument was originally designed to measure cloud temperatures and
hence cloud height. The instrument had, however, sufficient accuracy\index{accuracy!AVHRR
temperature} and precision that it was soon used to measure regional and global temperature
patterns at the sea surface.
The instrument is a radiometer that converts infrared radiation into an electrical voltage. It includes a mirror that scans from side to side across the
sub-satellite track and reflects radiance from the ground into a telescope, a
telescope that focuses the radiance on detectors, detectors sensitive to
different wavelengths that convert the radiance at those wavelengths into
electrical signals, and electronic circuitry to digitize and store the radiance
values. The instruments observes a 2700-km wide swath centered on the
sub-satellite track. Each observation along the scan is from a pixel that is
roughly one kilometer in diameter near the center of the scan and that increases
in size with distance from the sub-satellite track.
The radiometers measures infrared radiation emitted from the surface in five
wavelength bands: three infrared bands: 3.55--3.99 $\mu$m, 10.3--11.3 $\mu$m, and
11.5--12.5
$\mu$m; a near-infrared band at 0.725--1.10 $\mu$m; and a visible-light band at
0.55--0.90 $\mu$m. All infrared bands include radiation emitted from the sea
and from water vapor in the air along the path from the satellite to the ground.
The 3.7 $\mu$m band is least sensitive to water vapor and other errors, but it
works only at night because sunlight has radiance in this band. The two longest
wavelength bands at 10.8 $\mu$m and 12.0 $\mu$m are used to observe sea-surface
temperature and water vapor along the path in daylight.
Data with 1-km resolution are transmitted directly to ground stations that view
the satellite as it passes the station. This is the Local
Area Coverage mode. Data are also averaged to produce observations from 4
$\times$ 4 km pixels. These data are stored by the satellite and later
transmitted to \textsc{noaa} receiving stations. This is the Global Area
Coverage mode.
The swath width is sufficiently wide that the satellite views the entire earth
twice per day, at approximately 09:00 AM and 9:00 PM local time. Areas at high
latitudes may be observed as often as eight or more times per day.
\begin{figure}[b!]
\vspace{-2ex}
\makebox [121mm][c]{\includegraphics{cloudalgo}}
\footnotesize
Figure 6.12 The influence \rule{0pt}{4ex}of clouds on infrared observations.
\textbf{Left:} The standard deviation of the radiance from small, partly cloudy
areas each containing 64 pixels. The feet of the arch-like distribution of
points are the sea-surface and cloud-top temperatures. After Coakley and
Bretherton (1982). \textbf{Right:} The maximum difference between local values of
$T_{11} - T_{3.7}$ and the local mean values of the same quantity. Values inside
the dashed box indicate cloud-free pixels. $T_{11}$ and $T_{3.7}$ are the
apparent temperatures at 11.0 and 3.7 $\mu$m (data from K. Kelly). After Stewart
(1985: 137).
\label{fig:cloudalgo}
%\vspace{-3ex}
\end{figure}
The most important errors are due to:\index{temperature!measurement at
surface!errors in|(}
\begin{enumerate}
\vitem Unresolved or undetected clouds: Large, thick clouds are
obvious in the images of water temperature Thin clouds such as low
stratus and high cirrus produce much small errors that are
difficult or almost impossible to detect. Clouds smaller in
diameter than 1 km, such as trade-wind cumuli, are also difficult
to detect. Special techniques have been developed for detecting
small clouds (figure 6.12).
\vitem Water vapor, which absorbs part
of the energy radiated from the sea surface: Water vapor reduces
the apparent temperature of the sea surface. The influence is
different in the 10.8 $\mu$m and 12.0 $\mu$m channels, allowing
the difference in the two signals to be used to reduce the error.
\vitem Aerosols, which absorb infrared radiation. They radiate at
temperatures found high in the atmosphere. Stratospheric aerosols
generated by volcanic eruptions can lower the observed
temperatures by up to a few degrees Celsius. Dust particles
carried over the Atlantic from Saharan dust storms can also cause
errors.
\vitem Skin temperature errors. The infrared radiation seen by the instrument
comes from a layer at the sea surface that is only a few
micrometers thick. The temperature in this layer is not quite the same
as temperature a meter below the sea surface. They can differ by
several degrees when winds are light (Emery and Schussel,
1989).\index{temperature!measurement at surface!errors in|)} This error is greatly
reduced when \textsc{avhrr} \index{Advanced Very High Resolution Radiometer (AVHRR)}data are
used to interpolate between ship measurements of surface temperature.
\end{enumerate}
Maps of temperature processed from Local Area Coverage of cloud-free regions
show variations of temperature with a precision of 0.1\degrees{C}. These maps
are useful for observing local phenomena including patterns produced by local
currents. Figure 10.16 shows such patterns off the California coast.
Global maps are made by the U.S. Naval Oceanographic Office, which receives the global \textsc{avhrr} \index{Advanced Very High Resolution Radiometer (AVHRR)}data directly from \textsc{noaa}'s National Environmental Satellite, Data and Information Service in near-real time each day. The data are carefully processed to remove the influence of clouds, water vapor, aerosols, and other sources of error. Data are then used to produce global maps between $\pm 70$\degrees\ with an accuracy\index{accuracy!AVHRR temperature!maps} of $\pm 0.6$\degrees{C} (May et al 1998). The maps of sea-surface temperature are sent to the U.S. Navy and to \textsc{noaa}'s National Centers for Environmental Prediction. In addition, the office produces daily 100-km global and 14-km regional maps of temperature.
\paragraph{Global Maps of Sea-Surface Temperature} Global, monthly maps of \index{temperature!global maps of}surface temperature are produced by the National Centers for Environmental Prediction using Reynolds et al (2002) optimal-interpolation method. The technique blends ship and buoy measurements of sea-surface temperature with \textsc{avhrr} \index{Advanced Very High Resolution Radiometer (AVHRR)}data processed by the Naval Oceanographic Office in 1\degrees\ areas for a month. Essentially, \textsc{avhrr} data are interpolated between buoy and ship reports using previous information about the temperature field. Overall accuracy\index{accuracy!temperature!sea-surface maps} ranges from approximately $\pm 0.3$\degrees\ C in the tropics to $\pm 0.5$\degrees\ C near western boundary currents in the northern hemisphere where temperature gradients are large.
Maps are available from November 1981. Figures 6.2--6.4 were made by \textsc{noaa} using Reynolds' technique. Other data sets have been produced by the \textsc{noaa/nasa} Pathfinder program (Kilpatrick, Podesta, and Evans, 2001).
Maps of mean temperature have also been made from \textsc{icoads} \index{ICOADS (international comprehensive ocean-atmosphere data set)}data (Smith and Reynolds, 2004). Because the data are poorly distributed in time and space, errors also vary in time and space. Smith and Reynolds (2004) estimated the error in the global mean temperature and found the 95\% confidence uncertainty for the near-global average is 0.48\degrees\ C or more in the nineteenth century, near 0.28\degrees\ C for the first half of the twentieth century, and 0.18\degrees\ C or less after 1950. Anomalies\index{anomalies!sea-surface temperature} of sea-surface temperature were calculated using mean sea-surface temperature from the period 1854--1997 using \textsc{icoads}\index{ICOADS (international comprehensive ocean-atmosphere data set)} supplemented with satellite data since 1981.
\section{Measurement of Conductivity or Salinity}
\index{conductivity!measurement of}Conductivity is measured by placing platinum electrodes in
seawater and measuring the current that flows when there is a known voltage between
the electrodes. The current depends on conductivity, voltage,
and volume of sea water in the path between electrodes. If the electrodes are in a tube of
non-conducting glass, the volume of water is accurately known, and the current is independent
of other objects near the conductivity cell (figure 6.13). The best measurements of salinity
from conductivity give salinity with an accuracy\index{accuracy!salinity} of
$\pm$0.005.
Before conductivity measurements were widely used, salinity
\index{salinity!measurement of}was measured using chemical titration of the
water sample with silver salts. The best measurements of salinity from titration
give salinity with an accuracy\index{accuracy!salinity!from titration} of $\pm$0.02.
\begin{figure}[t!]
%\vspace{-2ex}
\makebox[120mm] [c] {\includegraphics{conductivity}}
\footnotesize
Figure 6.13 A conductivity \rule{0pt}{4ex}cell. Current flows through the seawater between
platinum electrodes in a cylinder of borosilicate glass 191 mm long with an inside diameter
between the electrodes of 4 mm. The electric field lines (solid lines) are confined to the
interior of the cell in this design making the measured conductivity (and instrument
calibration) independent of objects near the cell. This is the cell used to measure
conductivity and salinity shown in figure 6.15. From Sea-Bird Electronics.
\label{fig:conductivity}
\vspace{-3ex}
\end{figure}
Individual salinity measurements \index{salinity!measurement of}are calibrated using standard
seawater. Long-term studies of accuracy\index{accuracy!salinity} use data from measurements of
deep water masses of known, stable, salinity. For example, Saunders (1986) noted that
temperature is very accurately related to salinity for a large volume of water contained in
the deep basin of the northwest Atlantic under the Mediterranean outflow. He used the
consistency of measurements of temperature and salinity made at many hydrographic
stations\index{hydrographic stations!used for salinity} in the area to estimate the
accuracy\index{accuracy!salinity} of temperature, salinity and oxygen measurements. He
concluded that the most careful measurements made since 1970 have an accuracy of 0.005 for
salinity and 0.005\degrees{C} for temperature. The largest source of salinity error was the
error in determination of the standard water used for calibrating the salinity measurements.
\begin{figure}[t!]
%\vspace{-3ex}
\includegraphics{salinityaccuracy}
\footnotesize
Figure 6.14. Standard deviation \rule{0mm}{3ex}of salinity measurements below 1500 m in the south Atlantic. Each point is the average for the decade centered on the point. The value for 1995 is an estimate of the accuracy\index{accuracy!salinity} of recent measurements. From Gouretski and Jancke (1995).
\label{fig:salinityaccuracy}
\vspace{-4ex}
\end{figure}
\begin{figure}[b!]
\vspace{-4ex}
\makebox [120mm][c]{\includegraphics{911data}}
\footnotesize
Figure 6.15. Results from \rule{0pt}{3ex}a test of the Sea-Bird Electronics 911
Plus CTD\index{CTD} in the North Atlantic Deep Water\index{North Atlantic Deep Water} in 1992. Data were collected at
43.17\degrees\ N and 14.08\degrees\ W from the R/V Poseidon. From Sea-Bird
Electronics (1992).
\label{fig:911data}
%\vspace{-3ex}
\end{figure}
Gouretski and Jancke (1995) estimated accuracy\index{accuracy!salinity}\index{salinity!accuracy of}of salinity measurements as a function of time. Using high quality measurements from 16,000 hydrographic stations\index{hydrographic stations!used for salinity} in the south Atlantic from 1912 to 1991, they estimated accuracy by plotting salinity as a function of temperature using all data collected below 1500 m in twelve regions for each decade from 1920 to 1990. A plot of accuracy as a function of time since 1920 shows consistent improvement in accuracy since 1950 (figure 6.14). Recent measurements of salinity are the most accurate. The standard deviation of salinity data collected from all areas in the south Atlantic from 1970 to 1993 adjusted as described by Gouretski and Jancke (1995) was 0.0033. Recent instruments such as the Sea-Bird Electronics Model 911 Plus have an accuracy\index{accuracy!salinity} of better than 0.005 without adjustments. A comparison of salinity measured at 43\degrees\ 10\'{}N, 14\degrees\ 4.5\'{}W
by the 911 Plus with historic data collected by Saunders (1986) gives an accuracy of 0.002 (figure 6.15).
\section{Measurement of Pressure}
\index{pressure!measurement of}Pressure is routinely measured by
many different types of instruments. The SI unit of pressure is
the pascal (Pa), but oceanographers normally report pressure in
decibars (dbar), where:
\begin{equation}
1 \text{ dbar} = 10^4 \text{ Pa}
\end{equation}
because the pressure in decibars is almost exactly equal to the depth in meters.
Thus 1000 dbar is the pressure at a depth of about 1000 m.
\paragraph{Strain Gage}\index{pressure!measurement of!strain gage}This is the simplest and
cheapest instrument,
\index{strain gage}and it is widely used. Accuracy\index{accuracy!pressure} is about
$\pm$1\%.
\paragraph{Vibratron} Much more accurate measurements of pressure can be
\index{pressure!measurement of!vibratron}\index{vibratron}made by
measuring the natural frequency of a vibrating tungsten wire
stretched in a magnetic field between diaphragms closing the ends
of a cylinder. Pressure distorts the diaphragm, which changes the
tension on the wire and its frequency. The frequency can be
measured from the changing voltage induced as the wire vibrates in
the magnetic field. Accuracy\index{accuracy!pressure} is about $\pm$0.1\%, or better when
temperature controlled. Precision is 100--1000 times better than
accuracy. The instrument is used to detect small changes in
pressure at great depths. Snodgrass (1964) obtained a precision
equivalent to a change in depth of $\pm$0.8 mm at a depth of 3 km.
\paragraph{Quartz crystal} Very accurate measurements of pressure can also
\index{pressure!measurement of!quartz crystal}be made by measuring
the natural frequency of a quartz crystal cut for minimum
temperature dependence. The best accuracy\index{accuracy!pressure} is obtained when the
temperature of the crystal is held constant. The accuracy is
$\pm$0.015\%, and precision is $\pm$0.001\% of full-scale values.
\paragraph{Quartz Bourdon Gage} has accuracy\index{accuracy!pressure} and stability comparable
to
\index{pressure!measurement of!quartz bourdon gage}quartz
crystals. It too is used for long-term measurements of pressure in
the deep sea.
\section[Temperature and Salinity With Depth]{Measurement of Temperature and
Salinity with Depth} \index{temperature!measurement with
depth}\index{salinity!measurement with depth} Temperature,
salinity, and pressure are measured as a function of depth using
various instruments or techniques, and density is calculated from
the measurements.
\paragraph{Bathythermograph (BT)} was a mechanical device that measured
\index{temperature!measurement with depth!by bathythermograph
(BT)} \index{bathythermograph (BT)}temperature vs depth on a smoked
glass slide. The device was widely used to map the thermal
structure of the upper ocean, including the depth of the mixed
layer\index{mixed layer!measured by bathythermograph} before being replaced by the expendable
bathythermograph in the 1970s.
\paragraph{Expendable Bathythermograph (XBT)} is an electronic device that
\index{temperature!measurement with depth!by expendable
bathythermograph (XBT)}\index{bathythermograph (BT)!expendable
(XBT)}measures temperature vs depth using a thermistor on a
free-falling streamlined weight. The thermistor is connected to an
ohm-meter on the ship by a thin copper wire that is spooled out
from the sinking weight and from the moving ship. The \textsc{xbt}
is now the most widely used instrument for measuring the thermal
structure of the upper ocean. Approximately 65,000 are used each
year.
\begin{figure}[t!]
\includegraphics{ctdnansen}
\ \ \ \ \ \
\includegraphics{ctdnansenRight}
\ \ \ \ \ \ \\
\footnotesize
Figure 6.16 \textbf{Left} A CTD \rule{0mm}{4ex}ready to\index{CTD} be lowered
over the side of a ship. From Davis (1987). \textbf{Right} Nansen water bottles
before (I), during (II), and after (III) reversing. Both instruments are shown
at close to the same scale. After Defant (1961: 33).
\label{fig:ctdnansen}
\vspace{-3ex}
\end{figure}
The streamlined weight falls through the water at a constant velocity. So
depth can be calculated from fall time with an accuracy\index{accuracy!depth from XBT} of
$\pm$2\%. Temperature accuracy\index{accuracy!temperature!XBT} is $\pm$0.1\degrees{C}.
And, vertical resolution is typically 65 cm. Probes reach to depths of 200 m to 1830 m
depending on model.
\paragraph{Nansen Bottles} (figure 6.16) were deployed from ships stopped at hydrographic stations.
\index{temperature!measurement with depth!by reversing thermometers} \index{thermometer!reversing!on Nansen bottles}\index{Nansen bottles}
\textit{Hydrographic stations} \index{hydrographic stations|textbf}are places where
oceanographers measure water properties from the surface to some depth, or to the
bottom, using instruments lowered from a ship. Usually 20 bottles were attached at
intervals of a few tens to hundreds of meters to a wire lowered over the side of the
ship. The distribution with depth was selected so that most bottles are in the upper
layers of the water column where the rate of change of temperature in the vertical
is greatest. A protected reversing thermometer\index{thermometer!reversing} for measuring
temperature was attached to each bottle along with an unprotected reversing
thermometer\index{thermometer!reversing} for measuring depth. The bottle contains a tube with
valves on each end to collect sea water at depth. Salinity was determined by laboratory
analysis of water sample collected at depth.
After bottles had been attached to the wire and all had been lowered to their
selected depths, a lead weight was dropped down the wire. The weight tripped a
mechanism on each bottle, and the bottle flipped over, reversing the
thermometers\index{thermometer!reversing}, shutting the valves and trapping water in the tube,
and releasing another weight. When all bottles had been tripped, the string of bottles was
recovered. The deployment and retrieval typically took several hours.
\paragraph{CTD} Mechanical instruments on Nansen bottles were replaced
\index{temperature!measurement with depth!by CTD}\index{CTD}beginning in the
1960s by an electronic instrument, called a \textsc{ctd}, that
measured conductivity, temperature, and depth (figure 6.16). The
measurements are recorded in digital form either within the
instrument as it is lowered from a ship or on the ship.
Temperature is usually measured by a thermistor. Conductivity is
measured by a conductivity cell. Pressure is measured by a quartz crystal.
Modern instruments have accuracy\index{accuracy!CTD} summarized in table 6.2.
\begin{table}[h!]\small \centering
\vspace{1ex}
\begin{tabular*}{95mm}{@{}lr@{}ll@{}}
\multicolumn{4}{@{}l@{}}{\bfseries Table 6.2 Summary of \rule[-1ex]{0mm}{1ex}Measurement Accuracy} \\
\hline
Variable & Range & & \rule{0ex}{2.5ex}Best Accuracy \\
\hline
Temperature & \rule{0ex}{2.5ex}42 &\ \degrees C & $\pm$ 0.001 \degrees C \\
Salinity & 1 &\ & $\pm$ 0.02 by titration \\
& & & $\pm$ 0.005 by conductivity \\
Pressure & 10,00 &\ dbar & $\pm$ 0.65 dbar \\
Density & 2 &\ kg/m$^3$ & $\pm$ 0.005 kg/m$^3$ \\
Equation of State & & & $\pm$ 0.005 kg/m$^3$ \\ [0.5ex]
\hline
\end{tabular*} \\[0.5ex]
\vspace{-3ex}
\end{table}
\paragraph{CTD on Drifters}
Perhaps\index{CTD} the most common source of temperature and salinity as a function of depth in the upper two kilometers of the ocean is the set of profiling \textsc{argo} floats\index{floats!Argo} described in \S{11.8}. The floats drift at a depth of 1 km, sink to 2 km, then rise to the surface. They profile temperature and salinity while changing depth using instruments very similar to those on a \textsc{ctd}. Data are sent to shore via the Argos system\index{Argos system} on the \textsc{noaa} polar-orbiting satellites. In 2006, nearly 2500 floats were producing one profile every 10 days throughout most of the ocean. The accuracy of data from the floats is 0.005\degrees C for temperature, 5 decibars for pressure, and 0.01 for salinity (Riser et al (2008).
\paragraph{Data Sets}
Data are in the Marine Environment and Security For European Area \textsc{mersea} Enact/Ensembles (\textsc{en}3 Quality Controlled in situ Ocean Temparature and Salinity Profiles database. As of 2008 the database contained about one million \textsc{xbt} profiles, 700,000 \textsc{ctd} profiles, 60,000 \textsc{argos} profiles, 1,100,000 Nansen bottle data of high quality in the upper 700 m of the ocean (Domingues et al, 2008).
\section{Light in the Ocean and Absorption of Light}
\index{light}\index{light!absorption of}Sunlight in the ocean is
important for many reasons: It heats sea water, warming the
surface layers; it provides energy required by phytoplankton; it
is used for navigation by animals near the surface; and reflected
subsurface light is used for mapping chlorophyll concentration
from space.
Light in the ocean travels at a velocity equal to the velocity of light in a
vacuum divided by the index of refraction ($n$), which is typically $n = 1.33$.
Hence the velocity in water is about $2.25 \times 10^8$ m/s. Because light
travels slower in water than in air, some light is reflected at the sea surface.
For light shining straight down on the sea, the reflectivity is $(n - 1)^2 /(n +
1)^2 $. For seawater, the reflectivity is $0.02 = 2$\%. Hence most sunlight
reaching the sea surface is transmitted into the sea, little is reflected. This
means that sunlight incident on the ocean in the tropics is mostly absorbed
below the sea surface.
\begin{figure}[t!]
\centering
\makebox [120mm][c]{\includegraphics{attenuation}}
\footnotesize
Figure 6.17 Absorption \rule{0mm}{3ex}coefficient for pure water as a function of wavelength
$\lambda$ of the radiation. Redrawn from Morel (1974: 18, 19). See Morel (1974) for references.
\label{fig:attenuation}
\vspace{-4ex}
\end{figure}
The rate at which sunlight is attenuated determines the depth which is
lighted and heated by the sun. Attenuation is due to absorption by pigments and
scattering by molecules and particles. Attenuation depends on wavelength.
Blue light is absorbed least, red light is absorbed most strongly.
Attenuation per unit distance is proportional to the radiance or the irradiance of
light:
\begin{equation}
\frac{dI}{dx} = -c \, I
\end{equation}
where $x$ is the distance along beam, $c$ is an attenuation coefficient (figure
6.17), and $I$ is irradiance or radiance.
\textit{Radiance} \index{radiance|textbf}is the power per unit area per solid angle. It is
useful for describing the energy in a beam of light coming from a particular direction.
Sometimes we want to know how much light reaches some depth in the ocean regardless of which
direction it is going. In this case we use
\textit{irradiance}\index{irradiance|textbf}, which is the power per unit area of surface.
If the absorption coefficient is constant, the light intensity decreases
exponentially with distance.
\begin{equation}
I_2 = I_1 \: \exp(-cx)
\end{equation}
where $I_1$ is the original radiance or irradiance of light, and $I_2$
is the radiance or irradiance of light after absorption.
\textit{Clarity of Ocean Water} \index{water!clarity of|textbf}Sea water in the middle of the
ocean is very clear---clearer than distilled water. These waters are a very deep, cobalt,
blue---almost black. Thus the strong current which flows northward offshore of Japan carrying
very clear water from the central Pacific into higher latitudes is known as the Black Current,
or Kuroshio\index{Kuroshio} in Japanese. The clearest ocean water is called Type I waters by
Jerlov (figure 6.18). The water is so clear that 10\% of the light transmitted below the sea
surface reaches a depth of 90 m.
\begin{figure}[t!]
\makebox [120mm][c]{\includegraphics{jerlov}}
\footnotesize
Figure 6.18 \textbf{Left:} Transmittance \rule{0mm}{3ex}of daylight in the ocean in \% per
meter as a function of wavelength. I: extremely pure ocean water; II: turbid
tropical-subtropical water; III: mid-latitude water; 1-9: coastal waters of increasing
turbidity. Incidence angle is 90\degrees for the first three cases, 45\degrees for the other
cases. \textbf{Right:} Percentage of 465 nm light reaching indicated depths for the same types
of water. After Jerlov (1976).
\label{fig:jerlov}
\vspace{-3ex}
\end{figure}
In the subtropics and mid-latitudes closer to the coast, sea water contains more
phytoplankton than the very clear central-ocean waters. Chlorophyll pigments in
phytoplankton absorb light, and the plants themselves scatter light. Together,
the processes change the color of the ocean as seen by observer looking
downward into the sea. Very productive waters, those with high concentrations of
phytoplankton, appear blue-green or green (figure 6.19). On clear days the color
can be observed from space. This allows ocean-color scanners, such as those on
SeaWiFS, to map the distribution of phytoplankton over large areas.
As the concentration of phytoplankton increases, the depth where sunlight is
absorbed in the ocean decreases. The more turbid tropical and mid-latitude
waters are classified as type II and III waters by Jerlov (figure 6.18). Thus
the depth where sunlight warms the water depends on the productivity of the
waters. This complicates the calculation of solar heating of the mixed layer\index{mixed
layer!solar heating and phytoplankton}.
Coastal waters are much less clear than waters offshore. These are the type
1--9 waters shown in figure 6.18. They contain pigments from land, sometimes
called gelbstoffe, which just means yellow stuff, muddy water from rivers, and
mud stirred up by waves in shallow water. Very little light penetrates more than
a few meters into these waters.
\begin{figure}[t!]
\makebox [120mm][c]{\includegraphics{reflectance}}
\footnotesize
Figure 6.19 Spectral \rule{0mm}{4ex}reflectance of sea water observed from an
aircraft flying at 305 m over waters of different colors in the Northwest
Atlantic. The numerical values are the average chlorophyll concentration in the
euphotic (sunlit) zone in units of mg/m$^3$. The reflectance is for vertically
polarized light observed at Brewster's angle of 53\degrees. This angle minimizes
reflected skylight and emphasizes the light from below the sea surface. After
Clarke, Ewing, and Lorenzen (1970).
\label{fig:reflectance}
\vspace{-3ex}
\end{figure}
\paragraph{Measurement of Chlorophyll from Space}\index{chlorophyll!measurement from space}The color of the ocean, and hence the chlorophyll concentration in the upper layers of the ocean has been measured by the Coastal Zone Color Scanner carried on the Nimbus-7 satellite launched in 1978, by the Sea-viewing Wide Field-of-view Sensor (SeaWiFS) carried on SeaStar, launched in 1997, and on the Moderate Resolution Imaging Spectrometer (\textsc{modis})\index{MODIS} carried on the Terra\index{Terra satellite} and Aqua\index{Agua satellite} satellites launched in 1999 and 2002 respectively. \textsc{modis} measures upwelling\index{upwelling!radiance} radiance in 36 wavelength bands between 405 nm and 14,385 nm.
Most of the upwelling radiance seen by the satellite comes from the
atmosphere. Only about 10\% comes from the sea surface. Both air molecules
and aerosols scatter light, and very accurate techniques have been developed to
remove the influence of the atmosphere.
The total radiance $L_t$ received by an instrument in space is:
\begin{equation}
L_t(\lambda_i) = t(\lambda_i)L_W(\lambda_i)+L_r(\lambda_i)+L_a(\lambda_i)
\end{equation}
where $\lambda_i$ is the wavelength of the radiation in the band measured by the
instrument, $L_W$ is the radiance leaving the sea surface, $L_r$ is radiance
scattered by molecules, called the Rayleigh radiance, $L_a$ is radiance
scattered from aerosols, and $t$ is the transmittance of the atmosphere. $L_r$
can be calculated from theory, and $L_a$ can be calculated from the amount of red
light received at the instrument because very little red light is reflected from
the water. Therefore $L_W$ can be calculated from the radiance measured at the
spacecraft.
Chlorophyll concentration \index{chlorophyll!calculating
concentration}in the water column is calculated from the ratio of
$L_W$ at two frequencies. Using data from the Coastal Zone Color
Scanner, Gordon et al. (1983) proposed
\begin{subequations}
\begin{align}
C_{13} = & 1.1298 \left[ \frac{L_W(443)}{L_W(550)}\right]^{-1.71}\\
C_{23} = & 3.3266 \left[ \frac{L_W(520)}{L_W(550)}\right]^{-2.40}
\end{align}
\end{subequations}
where $C$ is the chlorophyll concentration in the surface layers in mg
pigment/m$^3$, and $L_W(443), L_W(520), and L_W(550)$ is the radiance at
wavelengths of 443, 520, and 550 nm.
$C_{13}$ is used when $C_{13} \le 1.5$ mg/m$^3$, otherwise $C_{23}$ is used.
The technique is used to calculate chlorophyll concentration within a factor of
50\% over a wide range of concentrations from 0.01 to 10 mg/m$^3$.
\section{Important Concepts}
\begin{enumerate}
\item Density in the ocean is determined by temperature, salinity, and pressure.
\vitem Density changes in the ocean are very small, and studies of water masses
and currents require density with an accuracy\index{accuracy!density} of 10 parts per million.
\vitem Density is not measured, it is calculated from measurements of
temperature, salinity, and pressure using the equation of state of sea water.
\vitem Accurate calculations of density require accurate definitions of
temperature and salinity and an accurate equation of state.
\vitem Salinity is difficult to define and to measure. To avoid the difficulty,
oceanographers use conductivity instead of salinity. They measure conductivity
and calculate density from temperature, conductivity, and pressure.
\vitem A mixed layer\index{mixed layer} of constant temperature and salinity is usually found
in the top 1--100 meters of the ocean. The depth is determined by wind speed and
the flux of heat through the sea surface.
\vitem To compare temperature and density of water masses at different depths in
the ocean, oceanographers use potential temperature and potential density which
remove most of the influence of pressure on density.
\vitem Water parcels below the mixed layer\index{mixed layer} move along neutral surfaces.
\vitem Surface temperature of the ocean was usually measured at sea using bucket
or injection temperatures. Global maps of temperature combine these observations
with observations of infrared radiance from the sea surface measured by an
\textsc{avhrr} \index{Advanced Very High Resolution Radiometer (AVHRR)}in space.
\vitem Temperature and conductivity are usually measured digitally as a function
of pressure using a \textsc{ctd}\index{CTD}. Before 1960--1970 the salinity and temperature
were measured at roughly 20 depths using Nansen bottles lowered on a line from a
ship. The bottles carried reversing thermometers\index{thermometer!reversing} which recorded
temperature and depth and they returned a water sample from that depth which was used to
determine salinity on board the ship.
\vitem Light is rapidly absorbed in the ocean. 95\% of sunlight is absorbed in
the upper 100 m of the clearest sea water. Sunlight rarely penetrates deeper
than a few meters in turbid coastal waters.
\vitem Phytoplankton change the color of sea water, and the change in color can
be observed from space. Water color is used to measure phytoplankton
concentration from space.
\end{enumerate}
\chapter[The Equations of Motion]{Some Mathematics: The Equations of Motion}
In this chapter I consider the response of a fluid to internal and external forces. This leads to a derivation of some of the basic equations describing ocean dynamics. In the next chapter, we will consider the influence of viscosity, and in chapter 12 we will consider the consequences of vorticity.
Fluid mechanics used in oceanography is based on Newtonian mechanics modified by our evolving understanding of turbulence\index{turbulence}. Conservation of mass, momentum, angular momentum, and energy lead to particular equations having names that hide their origins (table 7.1).
\begin{table}[h!] \small
\vspace{-1ex}
\begin{tabular*}{121mm}{@{}ll@{}}
\multicolumn{2} {@{}l@{}}{\bfseries Table 7.1 Conservation Laws
\index{conservation laws}Leading to
Basic\rule[-1ex]{0mm}{1ex} Equations of Fluid Motion} \\
\hline
\rule{0ex}{2.5ex}Conservation of Mass: & Leads to Continuity Equation. \\
Conservation of Energy: & Conservation of heat leads to Heat Budgets. \\
& Conservation of mechanical energy leads to \\
& \hspace{1em}Wave Equation. \\
Conservation of Momentum:& Leads to Momentum (Navier-Stokes) Eq. \\
Conservation of Angular Momentum: & Leads to Conservation of Vorticity. \\
\hline
\end{tabular*} \\[0.5ex]
\vspace{-3ex}
\end{table}
\section{Dominant Forces for Ocean Dynamics}
\index{ocean!dominant forces in}Only a few forces are important in physical oceanography: gravity, friction, and Coriolis (table 7.2). Remember that forces are vectors. They have magnitude and direction.
\begin{enumerate}
\vitem \textit{Gravity} \index{gravity|textbf}is the dominant force. The weight of the water in the ocean produces pressure. Changes in gravity, due to the motion of sun\index{sun} and moon\index{moon} relative to earth produces tides, tidal currents, and tidal mixing\index{mixing!tidal} in the interior of the ocean.
\textit{Buoyancy} \index{buoyancy|textbf}is the upward or downward force due to gravity
acting on a parcel of water that is more or less dense than other water at its
level. For example, cold air blowing over the sea cools surface waters causing them
to be more dense than the water beneath. Gravity acting on the difference in density results in a
force that causes the water to sink.
\textit{Horizontal pressure gradients} \index{pressure gradient!horizontal|textbf}are due to the varying weight of water in different regions of the ocean.
\vitem \textit{Friction} \index{friction|textbf}is the force acting on a body as it moves past another body while in contact with that body. The bodies can be parcels of water or air.
\textit{Wind stress} \index{wind|textbf}is the friction due to wind blowing across the sea surface. It transfers horizontal momentum to the sea, creating currents. Wind blowing over waves on the sea surface leads to an uneven distribution of pressure over the waves. The pressure distribution transfers energy to the waves, causing them to grow into bigger waves.
\vitem\textit{Pseudo-forces} \index{pseudo-forces|textbf}are apparent forces that arise from motion in
curvilinear or rotating coordinate systems. For example, Newton's first law states that there is no change in motion of a body unless a resultant force acts on it. Yet a body moving at constant velocity
seems to change direction when viewed from a rotating coordinate system. The change
in direction is due to a pseudo-force, the Coriolis force.
\textit{Coriolis Force} \index{Coriolis force|textbf}is the dominant
pseudo-force influencing motion in a coordinate system fixed to the earth.
\end{enumerate}
\vspace{-1ex}
\begin{table}[h!] \small{{\textbf{Table 7.2 Forces in Geophysical Fluid Dynamics}}
\\[1ex]
\vspace{-6ex}
\begin{tabular*}{121mm}{@{}ll@{}}
\hline
\textbf{Dominant Forces}\rule{0pt}{2.5ex} & \\
Gravity & Gives rise to pressure gradients, buoyancy, and tides. \\
Coriolis & Results from motion in a rotating coordinate system \\
Friction & Is due to relative motion between two fluid parcels. \\
& Wind stress is an important frictional force. \\
\textbf{Other Forces} & \rule{0mm}{4.5ex}\\
Atmospheric Pressure & Results in inverted barometer effect. \\
Seismic & Results in \textit{tsunamis} driven by earthquakes. \\ [0.5ex]
\hline
\multicolumn{2}{@{}l@{}}{Note\rule{0mm}{2.5ex} that the last two forces are much less important than the first three.} \\
\end{tabular*} \\ [0.5ex]}
\rule{0ex}{2ex}
%\vspace{-3ex}
\end{table}
\section{Coordinate System}
\index{coordinate systems}Coordinate systems allow us to find
locations in theory and practice. Various systems are used
depending on the size of the features to be described or mapped. I
will refer to the simplest systems; descriptions of other systems
can be found in geography and geodesy books.
\begin{enumerate}
\vitem\textit{Cartesian Coordinate System} \index{coordinate
systems!Cartesian|textbf}is the one I will use most commonly in the following
chapters to keep the discussion as simple as possible. We can describe most
processes in Cartesian coordinates without the mathematical complexity of spherical
coordinates. The standard convention in geophysical fluid mechanics is $x$ is to the
east, $y$ is to the north, and $z$ is up.
\textit{f-Plane}
\index{coordinate systems!f-plane|textbf}\index{f-plane@\textit{f}-plane|textbf}is a Cartesian
coordinate system in which the Coriolis force is assumed constant. It is useful for describing
flow in regions small compared with the radius of the earth and larger than a few tens of
kilometers.
\textit{$\beta$-plane}
\index{coordinate
systems!B-plane@$\beta$-plane|textbf}\index{B-plane@$\beta$-plane|textbf}is a
Cartesian coordinate system in which the Coriolis force is assumed to vary linearly
with latitude. It is useful for describing flow over areas as large as ocean basins.
\vitem\textit{Spherical coordinates} \index{coordinate systems!spherical coordinates|textbf}are used to describe
flows that extend over large distances and in numerical calculations of basin and global scale flows.
\end{enumerate}
\section{Types of Flow in the ocean}
\index{flow!types of}Many terms are used for describing the ocean
circulation. Here are a few of the more commonly used terms for
describing currents and waves.
\begin{enumerate}
\vitem\textit{General Circulation} \index{general circulation|textbf}is
the permanent, time-averaged circulation.
\vitem\textit{Abyssal} \index{abyssal circulation|textbf}also called the
\textit{Deep Circulation} \index{deep circulation|textbf}is
the circulation of mass, in the meridional plane, in the deep ocean, driven by mixing.
\vitem\textit{Wind-Driven Circulation} \index{wind-driven circulation|textbf}is the
circulation in the upper kilometer of the ocean forced by the wind. The circulation
can be caused by local winds or by winds in other regions.
\vitem\textit{Gyres}\index{gyres|textbf} are wind-driven cyclonic or anticyclonic currents with
dimensions nearly that of ocean basins.
\vitem\textit{Boundary Currents} \index{boundary currents|textbf}are currents
flowing parallel to coasts. Two types of boundary currents are important:
\begin{itemize}
\vitem Western boundary currents on the western edge of the ocean tend to be
fast, narrow jets such as the Gulf Stream\index{Gulf Stream!as a western boundary current} and
Kuroshio\index{Kuroshio!as a western boundary current}.
\vitem Eastern boundary currents are weak, \textit{e.g}. the California
Current.
\end{itemize}
\vitem\textit{Squirts} \index{squirts|textbf}or \textit{Jets}
\index{jets|textbf}are long narrow currents, with dimensions of a few hundred
kilometers, that are nearly perpendicular to west coasts.
\vitem\textit{Mesoscale
Eddies}
\index{mesoscale eddies|textbf}are turbulent or spinning flows on scales of a few
hundred kilometers.
\end{enumerate}
\vspace{-0.5ex}
In addition to flow due to currents, there are many types of oscillatory flows
due to waves. Normally, when we think of waves in the ocean, we visualize waves
breaking on the beach or the surface waves influencing ships at sea. But many
other types of waves occur in the ocean.
\begin{enumerate}
\vitem\textit{Planetary Waves} \index{waves!planetary waves|textbf}depend on the rotation of the
earth for a restoring force, and they including Rossby\index{waves!Rossby},
Kelvin\index{waves!Kelvin}, Equatorial\index{waves!equatorial}, and Yanai
waves\index{waves!Yanai}.
\vitem\textit{Surface Waves} \index{waves!surface
waves|textbf}sometimes called gravity waves, are the waves that eventually break on
the beach. The restoring force is due to the large density contrast between air and
water at the sea surface.
\vitem\textit{Internal Waves} \index{waves!internal waves|textbf}are sub-sea wave
similar in some respects to surface waves. The restoring force is due to change in
density with depth.
\vitem\textit{Tsunamis}
\index{waves!tsunamis|textbf}\index{tsunamis|textbf}are surface waves with periods near 15 minutes generated by
earthquakes.
\vitem\textit{Tidal Currents} \index{waves!tidal currents|textbf}\index{tidal!
currents|textbf}are horizontal currents and currents associated with internal waves
driven by the tidal potential.
\vitem\textit{Edge Waves} \index{waves!edge
waves|textbf}are surface waves with periods of a few minutes confined to shallow
regions near shore. The amplitude of the waves drops off exponentially with
distance from shore.
\end{enumerate}
\section{Conservation of Mass and Salt}
\index{conservation of mass}Conservation of mass and salt can be
used to obtain very useful information about flows in the ocean.
For example, suppose we wish to know the net loss of fresh water,
evaporation minus precipitation, from the Mediterranean Sea. We
could carefully calculate the latent heat flux over the surface,
but there are probably too few ship reports for an accurate
application of the bulk formula. Or we could carefully measure the
mass of water flowing in and out of the sea through the Strait of
Gibraltar, but the difference is small and perhaps impossible to
measure accurately.
We can, however, calculate the net evaporation knowing the salinity of the flow
in $S_i$ and out $S_o$, together with a rough estimate of the volume of water
$V_o$ flowing out, where $V_o$ is a volume flow in units of m$^3$/s (figure 7.1).
\begin{figure}[h!]
\vspace{-1ex}
\makebox[120mm] [c]{\includegraphics{basin}}
\centering
\footnotesize
Figure 7.1 Schematic \rule{0mm}{3ex}diagram of flow into and out of a basin.\\Values from Bryden and Kinder (1991).
\label{fig:basin}
\vspace{-1ex}
\end{figure}
The mass flowing out is, by definition, $\rho_o \, V_o$. If the volume
of the sea does not change, conservation of mass requires:
\begin{equation}
\rho_i\,V_i = \rho_o\,V_o
\end{equation}
where, $\rho_i, \,\rho_o$ are the densities of the water flowing in and out. We
can usually assume, with little error, that
$\rho_i = \rho_o$.
If there is precipitation $P$ and evaporation $E$ at the
surface of the basin and river inflow $R$, conservation of mass becomes:
\begin{equation}
V_i + R + P = V_o + E
\end{equation}
Solving for ($V_o$ - $V_i$):
\begin{equation}
V_o - V_i = (R + P) - E
\end{equation}
which states that the net flow of water into the basin must balance
precipitation plus river inflow minus evaporation when averaged over a
sufficiently long time.
Because salt is not deposited or removed from the sea, conservation of salt
requires :
\begin{equation}
\rho_i\,V_i\,S_i = \rho_o\,V_o\,S_o
\end{equation}
Where $\rho_i$, $S_i$ are the density and salinity of the incoming water, and
$\rho_o$, $S_o$ are density and salinity of the outflow. With little error,
we can again assume that $\rho_i$ = $\rho_o$.
\paragraph{An Example of Conservation of Mass and Salt}
Using the values for the flow at the Strait of Gibraltar measured by Bryden and Kinder (1991) and shown in figure 7.1, solving (7.4) for $V_i$ assuming that $\rho_i = \rho_o$, and using the estimated value of $V_o$, gives $V_i = 0.836$ Sv $= 0.836 \times 10^6$ m$^3$/s, where Sv = Sverdrup\index{Sverdrup} $= 10^6$ m$^3$/s is the unit of volume transport \index{transport!volume}used in oceanography. Using $V_i$ and $V_o$ in (7.3) gives $(R + P - E) = - 4.6 \times 10^4$ m$^3$/s.
Knowing $V_i$ we can also calculate a minimum flushing time for replacing water
in the sea by incoming water. The minimum flushing time $T_m$ is the volume of the
sea divided by the volume of incoming water. The Mediterranean has a volume of around $4 \times 10^6$ km$^3$. Converting $0.836 \times 10^6$ m$^3$/s to km$^3$/yr we obtain $2.64 \times10^4$ km$^3$/yr. Then, $T_m = 4 \times 10^6$ km$^3$/$2.64 \times 10^4$ km$^3$/yr $= 151$ yr. The actual time depends on mixing\index{mixing!and flushing time} within the sea. If the waters are well mixed, the
flushing time is close to the minimum time, if they are not well mixed, the flushing time is
longer.
Our example of flow into and out of the Mediterranean Sea is an example of a \textit{box
model}\index{box model|textbf}. A box model replaces large systems, such as the Mediterranean Sea,
with boxes. Fluids or chemicals or organisms can move between boxes, and conservation equations are
used to constrain the interactions within systems.
\begin{figure}[b!]
%\vspace{-3ex}
\includegraphics{derivativesketch}
\centering
\footnotesize
Figure 7.2 Sketch of flow \rule{0mm}{4ex}used for deriving the
total derivative.
\label{fig:derivativesketch}
\vspace{-3ex}
\end{figure}
\section{The Total Derivative (D/Dt)}
\index{total derivative|textbf}If the number of boxes in a system increases to a very large number as the size of each box shrinks, we eventually approach limits used in differential calculus. For example, if we subdivide the flow of water into boxes a few meters on a side, and if we use conservation of mass, momentum, or other properties within each box, we can derive the differential equations governing fluid flow.
Consider the example of acceleration of flow in a small box of fluid. The
resulting equation is called the \textit{total derivative}. It relates the
acceleration of a particle $Du/Dt$ to derivatives of the velocity field at a
fixed point in the fluid. We will use the equation to derive the equations for
fluid motion from Newton's Second Law which requires calculating the
acceleration of a particles passing a fixed point in the fluid.
We begin by considering the flow of a quantity $q_{in}$ into and $q_{out}$ out
of the small box sketched in figure 7.2. If $q$ can change continuously in time
and space, the relationship between $q_{in}$ and $q_{out}$ is:
\begin{equation}
q_{out} = q_{in} + \frac{\partial{q}}{\partial{t}}\,\delta{t} + \frac{\partial{q}}{\partial{x}}\,\delta{x}
\end{equation}
The rate of change of the quantity $q$ within the volume is:
\begin{equation}
\frac{Dq}{Dt} = \frac{q_{out} - q_{in}}{\delta{t}}=
\frac{\partial{q}}{\partial{t}} +
\frac{\partial{q}}{\partial{x}}\frac{\delta{x}}{\delta{t}}
\end{equation}
But $\delta x /\delta t$ is the velocity $u$, and therefore:
\begin{displaymath}
\frac{Dq}{Dt} = \frac{\partial{q}}{\partial{t}} +
u\frac{\partial{q}}{\partial{x}}
\end{displaymath}
In three dimensions, the total derivative becomes:
\begin{subequations}
\begin{align}
\frac{D}{Dt} = & \:\frac{\partial}{dt} + u\frac{\partial}{\partial{x}} + v\frac{\partial}{\partial y} + w\frac{\partial
}{\partial z}
\\
\frac{D}{Dt} = & \:\frac{\partial}{dt} +
\mathbf{u}\cdot \nabla(\,)
\end{align}
\end{subequations}
where $\mathbf{u}$ is the vector velocity and $\nabla$ is the
operator \textit{del} of vector field theory (See Feynman, Leighton, and Sands
1964: 2--6).
This is an amazing result. Transforming coordinates from one
following a particle to one fixed in space converts a simple linear derivative
into a non-linear partial derivative. Now let's use the equation to calculate the
change of momentum of a parcel of fluid.
\section{Momentum Equation}
\index{momentum equation}Newton's Second Law relates the change of
the momentum of a fluid mass due to an applied force. The change
is:
\begin{equation}
\frac{D(m\textbf{v})}{Dt} = \textbf{F}
\end{equation}
where \textbf{F} is force, $m$ is mass, and \textbf{v} is velocity. I have emphasized the need to use the total derivative because we are calculating the force on a particle. We can assume that the mass is constant, and (7.8) can be written:
\begin{equation}
\frac{D\textbf{v}}{Dt} = \frac{\textbf{F}}{m} = \textbf{f$_m$}
\end{equation}
where \textbf{f$_m$} is force per unit mass.
Four forces are important: pressure gradients, Coriolis force, gravity, and
friction. Without deriving the form of these forces (the derivations are given
in the next section), we can write (7.9) in the following form.
\begin{equation}
\frac{D\mathbf{v}}{Dt} = -\,\frac{1}{\rho}\nabla\,p -
\,2\boldsymbol{\Omega}
\times \mathbf{v} + \mathbf{g} + \mathbf{F_r}
\end{equation}
Acceleration equals the negative pressure gradient minus the Coriolis force plus gravity plus other forces. Here
\textbf{g} is acceleration of gravity, $\mathbf{F_r}$ is friction, and the magnitude $\Omega$ of
$\boldsymbol{\Omega}$ is the \textit{Rotation Rate of earth}\index{earth!rotation
rate|textbf}, 2$\pi$ radians per sidereal day or
\begin{equation}
\fbox{$\D
\Omega = 7.292 \times 10^{-5} \text{ radians/s} $}
\end{equation}
\paragraph{Momentum Equation in Cartesian coordinates:}
\index{momentum equation!Cartesian coordinates|textbf}Expanding the derivative in (7.10) and writing the components
in a Cartesian coordinate system gives the \textit{Momentum Equation}:
\begin{subequations}
\begin{align}
\frac{\partial{u}}{\partial{t}} + u\,\frac{\partial{u}}{\partial{x}}
+ v\,\frac{\partial{u}}{\partial{y}} +w\,\frac{\partial{u}}{\partial{z}} &=
-\,\frac{1}{\rho}\,\frac{\partial{p}}{\partial{x}} +
2\,\Omega\,{v}\,\sin\varphi + F_x \\
\frac{\partial{v}}{\partial{t}} + u\,\frac{\partial{v}}{\partial{x}} +
v\,\frac{\partial{v}}{\partial{y}} +w\,\frac{\partial{v}}{\partial{z}} &=
-\,\frac{1}{\rho}\,\frac{\partial{p}}{\partial{y}} - 2\,\Omega\,u\,\sin\varphi
+ F_y
\\
\frac{\partial{w}}{\partial{t}} + u\,\frac{\partial{w}}{\partial{x}} +
v\,\frac{\partial{w}}{\partial{y}} +w\,\frac{\partial{w}}{\partial{z}} &=
-\,\frac{1}{\rho}\,\frac{\partial{p}}{\partial{z}} + 2\,\Omega\,{u}\,\cos\varphi
- g + F_z
\end{align}
\end{subequations}
where $F_i$ are the components of any frictional force per unit mass, and
$\varphi$ is latitude. In addition, we have assumed that $w << v$, so the
$2\,\Omega\,w \cos \varphi$ has been dropped from equation in (7.12a).
Equation (7.12) appears under various names. Leonhard Euler (1707--1783) first wrote out the general form for fluid
flow with external forces, and the equation is sometimes called the \textit{Euler equation}\index{Euler equation} or
the \textit{acceleration equation}\index{acceleration equation}. Louis Marie Henri Navier (1785--1836) added the
frictional terms, and so the equation is sometimes called the \textit{Navier-Stokes equation}\index{Navier-Stokes
equation}.
The term $2\,\Omega\,u\, \cos{\varphi}$ in (7.12c) is small compared with $g$,
and it can be ignored in ocean dynamics. It cannot be ignored, however, for
gravity surveys made with gravimeters on moving ships.
\begin{figure}[h!]
\makebox[121mm][c]{\includegraphics{pressuresketch}}
\centering
\footnotesize
Figure 7.3 Sketch of flow \rule{0mm}{3ex}used for deriving the
pressure term in the momentum equation.
\label{fig:pressuresketch}
\vspace{-3ex}
\end{figure}
\paragraph{Derivation of Pressure Term}
Consider the forces acting on the sides of a small cube of fluid (figure 7.3).
The net force $\delta F_x$ in the $x$ direction is
\begin{align}
\delta F_x &= p\,\delta{y}\,\delta{z} - (p + \delta{p})\,\delta{y}\,\delta{z}
\notag \\
\delta F_x &= - \delta{p}\,\delta{y}\,\delta{z} \notag
\end{align}
But
\begin{displaymath}
\delta{p} = \frac{\partial{p}}{\partial{x}}\,\delta{x}
\end{displaymath}
and therefore
\begin{align}
\delta F_x &= - \frac{\partial{p}}{\partial{x}}\,\delta{x}\,\delta{y}\,\delta{z}
\notag \\
\delta F_x &=-\,\frac{\partial{p}}{\partial{x}}\, \delta{V} \notag
\end{align}
Dividing by the mass of the fluid $\delta m$ in the box, the acceleration of
the fluid in the $x$ direction is:
\begin{equation}
a_x = \frac{{\delta F_x}}{\delta{m}} = - \frac{\partial{p}}{\partial{x}}\,
\frac{\delta{V}}{\delta{m}} \notag
\end{equation}
\begin{equation}
\boxed{a_x = - \frac{1}{\rho}\,\frac{\partial{p}}{\partial{x}}}
\end{equation}
The pressure forces and the acceleration due to the pressure forces in the $y$
and
$z$ directions are derived in the same way.
\paragraph{The Coriolis Term in the Momentum Equation}
\index{Coriolis force}\index{momentum equation!Coriolis term}The Coriolis term exists because we describe currents in a reference frame fixed on earth. The derivation of the Coriolis terms is not easy. Henry Stommel, the noted oceanographer at the Woods Hole Oceanographic Institution even wrote a book on the subject with Dennis Moore (Stommel \& Moore, 1989).
Usually, we just state that the force per unit mass, the acceleration of a parcel of fluid in a rotating system, can be written:
\begin{equation}
\textbf{a}_{fixed} = \left(\frac{D\textbf{v}}{Dt}\right)_{fixed} =
\left(\frac{D\textbf{v}}{Dt}\right)_{rotating} + \left( 2 \boldsymbol{\Omega}
\times \mathbf{v} \right) + \boldsymbol{\Omega} \times \left(
\boldsymbol{\Omega} \times \mathbf{R} \right)
\end{equation}
where \textbf{R} is the vector distance from the center of earth,
$\boldsymbol{\Omega}$ is the angular velocity vector of earth,
and \textbf{v} is the velocity of the fluid parcel in coordinates fixed to earth.
The term $2 \boldsymbol{\Omega} \times \mathbf{v}$ is the Coriolis force, and the
term $\boldsymbol{\Omega} \times \left( \boldsymbol{\Omega} \times \mathbf{R}
\right)$ is the centrifugal acceleration. The latter term is included in gravity
(figure 7.4).
\paragraph{The Gravity Term in the Momentum Equation}
\index{momentum equation!gravity term}The gravitational attraction
of two masses $M_1$ and $m$ is:
\begin{displaymath}
\textbf{F}_g = \frac{G\,M_1\, m}{R^2}
\end{displaymath}
where $R$ is the distance between the masses, and $G$ is the gravitational
constant. The vector force $\textbf{F}_g$ is along the line connecting the two
masses.
The force per unit mass due to gravity is:
\begin{equation}
\frac{\textbf{F}_g}{m} = \textbf{g}_f =\frac{G\,M_E}{R^2}
\end{equation}
where $M_E$ is the mass of earth. Adding the centrifugal acceleration to (7.15)
gives gravity $\textbf{g}$ (figure 7.4):
\begin{equation}
\textbf{g} = \textbf{g}_f - \boldsymbol{\Omega} \times
\left( \boldsymbol{\Omega} \times \mathbf{R}
\right)
\end{equation}
Note that gravity does not point toward earth's center of mass. The centrifugal
acceleration causes a plumb bob to point at a small angle to the line directed
to earth's center of mass. As a result, earth's surface including the ocean's
surface is not spherical but it is a oblate ellipsoid. A rotating fluid planet
has an equatorial bulge.
\begin{figure}[h!]
\makebox[120mm][c]{\includegraphics{gravitysketch}}
\footnotesize
Figure 7.4 Downward \rule{0mm}{4ex}acceleration $g$ of a body at
rest on earth's surface is the sum of gravitational acceleration between the body
and earth's mass $g_f$ and the centrifugal acceleration due to earth's
rotation $\Omega\times(\Omega\times{R})$. The surface of an ocean at
rest must be perpendicular to $g$, and such a surface is close to an
ellipsoid of rotation. earth's ellipticity is greatly exaggerated here.
\label{fig:gravitysketch}
\vspace{-2ex}
\end{figure}
\section{Conservation of Mass: The Continuity Equation}
\index{conservation of mass}\index{continuity equation}Now let's
derive the equation for the conservation of mass in a fluid. We
begin by writing down the flow of mass into and out of a small box
(figure 7.5).
\begin{figure}[h!]
\makebox[120mm][c]{\includegraphics{continuitysketch}}
\centering
\footnotesize
Figure 7.5 Sketch of flow \rule{0mm}{4ex}used for deriving the
continuity equation.
\label{fig:continuitysketchR}
\vspace{-3ex}
\end{figure}
\begin{align}
\text{Mass flow in} &= \rho \, u \, \delta z \, \delta y \notag \\
\text{Mass flow out} &= (\rho + \delta \rho ) (u + \delta u) \delta z \, \delta y \notag
\end{align}
The mass flux into the volume must be (mass flow out) $-$ (mass flow
in). Therefore,
\begin{equation}
\text{Mass flux} = ( \rho \, \delta u + u \, \delta \rho + \delta \rho \, \delta u) \delta z \, \delta y \notag
\end{equation}
But
\begin{equation}
\delta u = \frac{\partial u}{\partial x} \delta x \,;\quad \delta \rho = \frac{\partial {\rho}}{\partial x} \delta x \notag
\end{equation}
Therefore
\begin{equation}
\text{Mass flux} = \left(\rho \, \frac{\partial{u}}{\partial{x}} + u\,\frac{\partial{\rho}}{\partial{x}}
+\frac{\partial{\rho}}{\partial{x}}\,\frac{\partial{u}}{\partial{x}}\,\delta{x}\right)\delta{x}\,\delta{y}\,\delta{z}
\notag
\end{equation}
The third term inside the parentheses becomes much smaller than the first two terms
as $\delta x \rightarrow 0$, and
\begin{equation}
\text{Mass flux} =
\frac{\partial{(\rho{u})}}{\partial{x}}\,\delta{x}\,\delta{y}\,\delta{z} \notag
\end{equation}
In three dimensions:
\begin{displaymath}
\mbox{Mass flux} = \left(\frac{\partial{(\rho{u})}}{\partial{x}} +
\frac{\partial{(\rho{v})}}{\partial{y}} +
\frac{\partial{(\rho{w})}}{\partial{z}}\right)\delta{x}\,\delta{y}\,\delta{z}
\end{displaymath}
The mass flux must be balanced by a change of mass inside the volume, which is:
\begin{displaymath}
\frac{\partial\rho}{\partial{t}}\,\delta{x}\,\delta{y}\,\delta{z}
\end{displaymath}
and conservation of mass requires:
\begin{equation}
\frac{\partial\rho}{\partial{t}} + \frac{\partial{(\rho{u})}}{\partial{x}} + \frac{d(\rho{v})}{\partial{y}} + \frac{\partial{(\rho{w})}}{\partial{z}} = 0
\end{equation}
This is the \textit{continuity equation}\index{continuity equation|textbf} for compressible flow, first derived by
Leonhard Euler (1707--1783).
The equation can be put in an alternate form by expanding the derivatives of
products and rearranging terms to obtain:
\begin{displaymath}
\frac{\partial{\rho}}{\partial{t}} + u\,\frac{\partial{\rho}}{\partial{x}} + v\,\frac{\partial{\rho}}{\partial{y}} + w\,\frac{\partial{\rho}}{\partial{z}} +
\rho\,\frac{\partial{u}}{\partial{x}} + \rho\,\frac{\partial{v}}{\partial{y}} + \rho\,\frac{\partial{w}}{\partial{z}} = 0
\end{displaymath}
The first four terms constitute the total derivative of density $D\rho/Dt$ from (7.7), and we can write (7.17) as:
\begin{equation}
\fbox{$\D
\frac{\D1}{\D\rho}\frac{\D{D}\D\rho}{\D{Dt}} + \frac{\D\partial{u}}{\D\partial{x}} + \frac{\D\partial{v}}{\D\partial{y}} + \frac{\D\partial{w}}{\D\partial{z}} =
0
$}\end{equation}
This is the alternate form for the continuity equation for a compressible fluid.
\paragraph{The Boussinesq Approximation}
\index{Boussinesq approximation}Density is nearly constant in the
ocean, and Joseph Boussinesq (1842--1929) noted that we can safely
assume density is constant except when it is multiplied by $g$ in
calculations of pressure in the ocean. The assumption greatly
simplifies the equations of motion.
Boussinesq's assumption requires that:
\begin{enumerate}
\vitem Velocities in the ocean must be small compared to the speed of
sound\index{sound!speed!and Boussinesq approximation}
$c$. This ensures that velocity does not change the density. As velocity
approaches the speed of sound, the velocity field can produces large changes of
density such as shock waves.
\vitem The phase speed of waves or disturbances must be small compared with $c$.
Sound speed\index{sound!speed!in incompressible fluid} in incompressible flows is infinite, and
we must assume the fluid is compressible when discussing sound in the ocean. Thus the
approximation is not true for sound. All other waves in the ocean have speeds small compared
to sound.
\vitem The vertical scale of the motion must be small compared with $c^2$/$g$,
where $g$ is gravity. This ensures that as pressure increases with depth in the
ocean, the increase in pressure produces only small changes in density.
\end{enumerate}
The approximations are true for oceanic flows, and they ensure that oceanic
flows are incompressible. See Kundu (1990: 79 and 112), Gill (1982: 85),
Batchelor (1967: 167), or other texts on fluid dynamics for a more complete
description of the approximation.
\paragraph{Compressibility}
\index{water!compressibility coefficient|textbf}The Boussinesq approximation is equivalent to
assuming sea water is incompressible. Now let's see how the assumption simplifies the continuity
equation. We define the \textit{coefficient of compressibility}
\begin{displaymath}
\beta \equiv -\frac{1}{V}\,\frac{\partial{V}}{\partial{p}} =
-\frac{1}{V}\,\frac{dV}{dt}\Big/\frac{dp}{dt}
\end{displaymath}
where $V$ is volume, and $p$ is pressure. For incompressible flows, $\beta$ = 0,
and:
\begin{displaymath}
\frac{1}{V}\,\frac{dV}{dt} = 0
\end{displaymath}
because $dp$/$dt$ $\not=$ 0. Remembering that density is mass
$m$ per unit volume $V$, and that mass is constant:
\begin{displaymath}
\frac{1}{V}\frac{dV}{dt} = -\,V\frac{d}{dt}\left(\frac{1}{V}\right) =
- \frac{V}{m}\,\frac{d}{dt}\left(\frac{m}{V}\right)
=\,-\,\frac{1}{\rho}\,\frac{d\rho}{dt} =\,-\,\frac{1}{\rho}\, \frac{D\rho}{Dt} = 0
\end{displaymath}
If the flow is incompressible, (7.18) becomes:
\begin{equation}
\fbox{$ \D
\frac{\D\partial{u}}{\D\partial{x}} + \frac{\D\partial{v}}{\D\partial{y}} + \frac{\D\partial{w}}{\D\partial{z}} = 0 $}
\end{equation}
This is the \index{continuity equation} \textit{Continuity Equation for
Incompressible Flows}.
\section{Solutions to the Equations of Motion}
Equations (7.12) and (7.19) are four equations, the three components of
the momentum equation plus the continuity equation, with four unknowns: $u$, $v$,
$w$, $p$. Note, however, that these are non-linear partial differential equations.
Conservation of momentum, when applied to a fluid, converted a simple,
first-order, ordinary, differential equation for velocity (Newton's Second Law),
which is usually easy to solve, into a non-linear partial differential equation,
which is almost impossible to solve.
\paragraph{Boundary Conditions:}
In fluid mechanics, we generally assume:
\begin{enumerate}
\vitem No velocity normal to a boundary, which means there is no flow through the
boundary; and
\vitem No flow parallel to a solid boundary, which means no slip at the solid
boundary.
\end{enumerate}
\paragraph{Solutions}
We expect that four equations in four unknowns plus boundary conditions give a
system of equations that can be solved in principle. In practice, solutions are
difficult to find even for the simplest flows. First, as far as I know, there
are no exact solutions for the equations with friction. There are very few exact
solutions for the equations without friction. Those who are interested in ocean
waves might note that one such exact solution is Gerstner's solution for water
waves (Lamb, 1945: 251). Because the equations are almost impossible to solve, we
will look for ways to greatly simplify the equations. Later, we will find
that even numerical calculations are difficult.
Analytical solutions can be obtained for much simplified forms of the equations
of motion. Such solutions are used to study processes in the ocean, including
waves. Solutions for oceanic flows with realistic coasts and bathymetric
features must be obtained from numerical solutions. In the next few
chapters we seek solutions to simplified forms of the equations. In Chapter 15 we
will consider numerical solutions.
\section{Important Concepts}
\begin{enumerate}
\item Gravity, buoyancy\index{buoyancy}, and wind are the dominant forces acting on
the ocean.
\vitem Earth's rotation produces a pseudo force, the Coriolis force.
\vitem Conservation laws applied to flow in the ocean lead to equations of
motion. Conservation of salt, volume and other quantities can lead to
deep insights into oceanic flow.
\vitem The transformation from equations of motion applied to fluid parcels to
equations applied at a fixed point in space greatly complicates the
equations of motion. The linear, first-order, ordinary differential equations
describing Newtonian dynamics of a mass accelerated by a force become
nonlinear, partial differential equations of fluid mechanics.
\vitem Flow in the ocean can be assumed to be incompressible except when
describing sound. Density can be assumed to be constant except when density is
multiplied by gravity $g$. The assumption is called the Boussinesq
approximation\index{Boussinesq approximation}.
\vitem Conservation of mass leads to the continuity equation, which has an
especially simple form for an incompressible fluid.
\end{enumerate}
\chapter{Equations of Motion With Viscosity}
Throughout most of the interior of the ocean and atmosphere friction is relatively
small, and we can safely assume that the flow is frictionless. At the boundaries,
friction, in the form of viscosity, becomes important. This thin, viscous layer is
called a \textit{boundary layer}\index{boundary layer|textbf}. Within the layer,
the velocity slows from values typical of the interior to zero at a
solid boundary. If the boundary is not solid, then the boundary layer is a thin
layer of rapidly changing velocity whereby velocity on one side of the boundary
changes to match the velocity on the other side of the boundary. For example, there
is a boundary layer at the bottom of the atmosphere, the planetary boundary layer
I described in Chapter 3. In the planetary boundary layer, velocity goes from many
meters per second in the free atmosphere to tens of centimeters per second at the
sea surface. Below the sea surface, another boundary layer, the \index{Ekman
layer}Ekman layer described in Chapter 9, matches the flow at the sea surface to the
deeper flow.
In this chapter I consider the role of friction in fluid flows, and the stability
of the flows to small changes in velocity or density.
\section{The Influence of Viscosity}
\index{viscosity}Viscosity is the tendency of a fluid to resist shear. In the last chapter I wrote the $x$--component of the
momentum equation for a fluid in the form (7:12a):
\begin{equation}
\frac{\partial{u}}{\partial{t}}+u\,\frac{\partial{u}}{\partial{x}}+v\,
\frac{\partial{u}}{\partial{y}}+w\,\frac{\partial{u}}{\partial{z}}=-\,
\frac{1}{\rho}\frac{\partial{p}}{\partial{x}} + 2\,\Omega\,v\,\sin\vartheta + F_x
\end{equation}
where $F_x$ was a frictional force per unit mass. Now we can consider the form
of this term if it is due to viscosity.
Molecules in a fluid close to a solid boundary sometime strike the boundary and
transfer momentum to it (figure 8.1). Molecules further from the boundary
collide with molecules that have struck the boundary, further transferring the
change in momentum into the interior of the fluid. This transfer of momentum is
\textit{molecular viscosity}\index{molecular
viscosity|textbf}\index{viscosity!molecular|textbf}. Molecules, however, travel
only micrometers between collisions, and the process is very inefficient for
transferring momentum even a few centimeters. Molecular viscosity is important only
within a few millimeters of a boundary.
Molecular viscosity $\rho \nu$ is the ratio of the stress $T$ tangential to the
boundary of a fluid and the velocity shear at the boundary. So the
stress has the form:
\begin{equation}
T_{xz} = \rho \nu \,\frac{\partial{u}}{\partial{z}}
\end{equation}
for flow in the $(x, z)$ plane within a few millimetres of the surface, where $\nu$
is the kinematic molecular viscosity. Typically $\nu = 10^{-6}$ m$^2$/s for water at
20\degrees{C}.
Generalizing (8.2) to three dimensions leads to a stress tensor giving
the nine components of stress at a point in the fluid, including
pressure, which is a normal stress, and shear stresses. A derivation of the
stress tensor is beyond the scope of this book, but you can find the details in
Lamb (1945: \S 328) or Kundu (1990: p. 93). For an incompressible fluid, the
frictional force per unit mass in (8.1) takes the form:
\begin{equation}
F_x= \frac{\partial }{\partial x} \left[ \nu \frac{\partial u}{\partial x} \right]
+ \frac{\partial }{\partial y} \left[ \nu \frac{\partial u}{\partial y} \right]
+ \frac{\partial }{\partial z} \left[ \nu \frac{\partial u}{\partial z} \right]
= \frac{1}{\rho} \left[ \frac{\partial T_{xx}}{\partial x} +
\frac{\partial T_{xy}}{\partial y} +
\frac{\partial T_{xz}}{\partial z} \right]
\end{equation}
\begin{figure}[t!]
\makebox[120mm] [c]{\includegraphics{viscositysketch}}
\centering
\footnotesize
Figure 8.1 Molecules \rule{0mm}{4ex}colliding with the wall and
with each other transfer\\momentum from the fluid to the wall, slowing the fluid
velocity.
\label{fig:viscositysketch}
\vspace{-3ex}
\end{figure}
\begin{figure}[t!]
\makebox[120mm] [c]{\includegraphics{reynoldsexp}}
\footnotesize
Figure 8.2 Reynolds \rule{0mm}{4ex}apparatus for investigating the transition to
turbulence\index{turbulence!transition to} in pipe flow, with photographs of near-laminar flow
(left) and turbulent flow (right) in a clear pipe much like the one used by Reynolds. After
Binder (1949: 88-89).
\label{fig:reynoldsexp}
\vspace{-4ex}
\end{figure}
\section{Turbulence}
\index{viscosity!turbulent}If molecular viscosity is important only over distances
of a few millimeters, and if it is not important for most oceanic flows, unless of
course you are a zooplankter trying to swim in the ocean, how then is the influence
of a boundary transferred into the interior of the flow? The answer is: through
turbulence.
Turbulence arises from the non-linear terms in the momentum equation
$(u\,\partial{u}/\partial{x}$, \textit{etc}.). The importance of these terms is
given by a non-dimensional number, the Reynolds Number $Re$, which is the ratio of
the non-linear terms to the viscous terms:
\begin{equation}
Re = \frac{\text{Non-linear Terms}}{\text{Viscous
Terms}} =
\cfrac{\left( \displaystyle u\,\cfrac{\partial{u}}{\partial{x}}\right) }{\left( \displaystyle \nu\,\cfrac{\partial^2{u}}{\partial{x^2}}\right)
} \approx \cfrac{\D U\,\cfrac{U}{L\vphantom{y}}}{\D \nu\,\cfrac{U}{L^2}}
= \frac{UL}{\nu}
\end{equation}
where, $U$ is a typical velocity of the flow and $L$ is a typical length
describing the flow. You are free to pick whatever $U,L$ might be typical of
the flow. For example $L$ can be either a typical cross-stream distance, or an
along-stream distance. Typical values in the open ocean are $U = 0.1$ m/s and
$L = 1$ megameter, so $Re = 10^{11}$. Because non-linear terms are important if
Re $>$ 10 -- 1000, they are certainly important in the ocean. The ocean is
turbulent.
The Reynolds number is named after Osborne Reynolds (1842--1912) who conducted
experiments in the late 19th century to understand turbulence\index{turbulence!measurement of}.
In one famous experiment (Reynolds 1883), he injected dye into water flowing at various speeds
through a tube (figure 8.2). If the speed was small, the flow was smooth. This is
called
\textit{laminar flow}. At higher speeds, the flow became irregular and
turbulent. The transition occurred at Re $ = VD/\nu \approx 2000$, where $V$ is
the average speed in the pipe, and $D$ is the diameter of the pipe.
As Reynolds number increases above some critical value, the flow becomes more and
more turbulent. Note that flow pattern is a function of Reynold's number. All
flows with the same geometry and the same Reynolds number have the same flow
pattern. Thus flow around all circular cylinders, whether 1 mm or 1 m in
diameter, look the same as the flow at the top of figure 8.3 if the Reynolds
number is 20. Furthermore, the boundary layer is confined to a very thin layer
close to the cylinder, in a layer too thin to show in the figure.
\begin{figure}[t!]
\makebox[120mm][c]{\includegraphics{turbsketch}}
\footnotesize
Figure 8.3 Flow past a circular \rule{0mm}{3ex}cylinder
as a function of Reynolds number between one and a million. From Richardson
(1961). The appropriate flows are: A---a toothpick moving at 1 mm/s; B---finger
moving at 2 cm/s; F---hand out a car window at 60 mph. All flow at the same
Reynolds number has the same streamlines. Flow past a 10 cm diameter cylinder at
1 cm/s looks the same as 10 cm/s flow past a cylinder 1 cm in diameter because in
both cases Re $= 1000$.
\label{fig:turbsketch}
\vspace{-2ex}
\end{figure}
\paragraph{Turbulent Stresses: The Reynolds Stress}
\index{turbulent!stress}\index{Reynolds Stress}Prandtl, Karman and others who
studied fluid mechanics in the early 20th century, hypothesized that parcels of
fluid in a turbulent flow played the same role in transferring momentum within the
flow that molecules played in laminar flow. The work led to the idea of turbulent
stresses.
To see how these stresses might arise, consider the momentum equation for a flow
with mean $(U, V, W)$ and turbulent $(u', v', w')$ components:
\begin{equation}
u=U+u' \,;\quad v = V+v' \,;\quad w=W+w' \, ;\quad p=P+p'
\end{equation}
where the mean value $U$ is calculated from a time or space average:
\begin{equation}
U = \langle u \rangle =\frac{1}{T}\int^T_0\,u(t)\,dt \quad \text{or}\quad
U = \langle u \rangle =\frac{1}{X}\int^X_0\,u(x)\,dx
\end{equation}
The non-linear terms in the momentum equation can be written:
\begin{align}
\left< (U+u')\frac{\partial{(U+u')}}{\partial{x}} \right> &= \left<
U\,\frac{\D \partial{U}}{\partial{x}}\right> +
\left< U\,\frac {\partial{u'}}{\partial{x}}\right> +
\left< u' \,\frac {\partial{U}}{\partial{x}}\right> + \left< u' \,\frac
{\partial{u'}}{\partial{x}}\right> \notag \\
\left< (U+u')\frac{\partial{(U+u')}}{\partial{x}} \right> &= \left<
U\,\frac{\D \partial{U}}{\partial{x}}\right> +
\left__
\end{align}
The second equation follows from the first since $\langle
U\,\partial{u'}/\partial{x}\rangle = 0$ and $\langle
u'\,\partial{U}/\partial{x}\rangle$ $= 0$, which follow from the definition of
$U$: $\langle U \partial{u'}/\partial{x}\rangle = U \partial{\langle u'
\rangle }/\partial{x}$ $ = 0$.
Using (8.5) in (7.19) gives:
\begin{equation}
\frac{\partial{U }}{\partial{x}} + \frac{\partial{V }}{\partial{y}} +\frac{\partial{W }}{\partial{z}} +
\frac{\partial{u'}}{\partial{x}} + \frac{\partial{v'}}{\partial{y}}
+\frac{\partial{w'}}{\partial{z}} =0
\end{equation}
Subtracting the mean of (8.8) from (8.8) splits the continuity equation into two equations:
\begin{subequations}
\begin{align}
\frac{\partial{U }}{\partial{x}} + \frac{\partial{V }}{\partial{y}} +\frac{\partial{W }}{\partial{z}} &= 0 \\
\frac{\partial{u'}}{\partial{x}} + \frac{\partial{v'}}{\partial{y}} +\frac{\partial{w'}}{\partial{z}} &=0
\end{align}
\end{subequations}
Using (8.5) in (8.1) taking the mean value of the resulting equation, then simplifying using
(8.7), the x-component of the momentum equation for the mean flow becomes:
\begin{equation}
\begin{split}
\frac{DU}{Dt} & = -\frac{1}{\rho}\,\frac{\partial{P}}{\partial{x}} + 2\Omega V\sin\varphi \\
& + \frac{\partial }{\partial x} \left[ \nu \frac{\partial U}{\partial x} - \langle u'u'\rangle \right]
+ \frac{\partial }{\partial y} \left[ \nu \frac{\partial U}{\partial y} - \langle u'v'\rangle \right] +
\frac{\partial }{\partial z} \left[ \nu \frac{\partial U}{\partial z} - \langle u'w'\rangle \right]
\end{split}
\end{equation}
The derivation is not as simple as it seems. See Hinze (1975: 22) for details. Thus the additional force per unit mass due to the turbulence\index{turbulent!stress} is:
\begin{equation}
F_x=-\frac{\partial}{\partial{x}}\langle u'u' \rangle
-\frac{\partial}{\partial{y}}\langle u'v' \rangle
-\frac{\partial}{\partial{z}}\langle u'w'\rangle
\end{equation}
The terms $\rho{\langle u' u' \rangle}$, $\rho{\langle u' v' \rangle}$, and $\rho{\langle u' w' \rangle}$ transfer
eastward momentum ($\rho u' $) in the $x$, $y$, and $z$ directions. For example, the term $\rho{\langle u' w'
\rangle}$ gives the downward transport \index{transport!momentum}of eastward momentum across a
horizontal plane. Because they transfer momentum, and because they were first derived by
Osborne Reynolds, they are called \textit{Reynolds Stresses}\index{Reynolds Stress|textbf}.
\section{Calculation of Reynolds Stress:}
\index{Reynolds Stress!calculation of}The Reynolds stresses such as $\partial{\langle u'w'\rangle}/\partial{z}$ are called virtual stresses (cf. Goldstein, 1965: \S 69 \& \S 81) because we assume that they play the same role as the viscous terms in the equation of motion. To proceed further, we need values or functional form for the Reynolds stress. Several approaches are used.
\paragraph{From Experiments}
We can calculate Reynolds stresses from direct measurements of ($u', v', w'$) made
in the laboratory or ocean. This is accurate, but hard to generalize to other
flows. So we seek more general approaches.
\paragraph{By Analogy with Molecular Viscosity}
Let's return to the example in figure 8.1, which shows flow above a surface in the $x$, $y$ plane. Prandtl, in a revolutionary paper published in 1904, stated that turbulent viscous effects are only important in a very thin layer close to the surface, the boundary layer. Prandtl's invention of the boundary layer allows us to describe very accurately turbulent flow of wind above the sea surface, or flow at the bottom boundary layer in the ocean, or flow in the mixed layer\index{mixed layer} at the sea surface. See the box \textit{Turbulent Boundary Layer Over a Flat Plate}.
To calculate flow in a boundary layer, we assume that flow is constant in the $x$, $y$ direction, that the statistical properties of the flow vary only in the $z$ direction, and that the mean flow is steady. Therefore $\partial
/\partial t = \partial /\partial x = \partial /\partial y = 0$, and (8.10) can be written:
\begin{equation}
2 \Omega V \sin \varphi + \frac{\partial }{\partial z} \left[\nu \frac{\partial
U}{\partial z} - \langle u'w'\rangle \right] = 0
\end{equation}
We now assume, in analogy with (8.2)
\begin{equation}
- \rho \langle u'w'\rangle = T_{xz} = \rho A_z \frac{\partial U}{\partial z}
\end{equation}
where $A_z$ is an \textit{eddy viscosity}\index{eddy
viscosity|textbf}\index{viscosity!eddy|textbf} or \textit{eddy
diffusivity}\index{eddy diffusivity|textbf} which replaces the molecular viscosity
$\nu$ in (8.2). With this assumption,
\begin{equation}
\frac{\partial{T_{xz}}}{\partial{z}} =
\frac{\partial}{\partial{z}}\left(A_z\frac{\partial{U}}{\partial{z}}\right)
\approx A_z \frac{\partial^2 U}{\partial z^2}
\end{equation}
where I have assumed that $A_z$ is either constant or that it varies more slowly in
the $z$ direction than $\partial U / \partial z$. Later, I will assume that $A_z
\approx z$.
Because eddies can mix heat, salt, or other properties as well
as momentum, I will use the term eddy diffusivity. It is more general than eddy
viscosity, which is the mixing\index{mixing!of momentum} of momentum.
The $x$ and $y$ momentum equations for a homogeneous, steady-state, turbulent
boundary layer above or below a horizontal surface are:
\begin{subequations}
\begin{align}
\rho fV + \frac{\partial {T_{xz}}}{\partial z} & = 0 \\
\rho fU - \frac{\partial {T_{yz}}}{\partial z} & = 0
\end{align}
\end{subequations}
where $f=2\omega \sin \varphi$ is the Coriolis parameter\index{Coriolis parameter}, and I have
dropped the molecular viscosity term because it is much smaller than the turbulent eddy
viscosity. Note, (8.15b) follows from a similar derivation from the $y$-component of
the momentum equation. We will need (8.15) when we describe flow near the surface.
\begin{table}[h!]
\fbox{\parbox{119mm}{
\centering \small
\begin{minipage}{11.5cm}
\begin{center}\textbf{The Turbulent Boundary Layer \rule{0mm}{3ex}Over a Flat
Plate}\\
\end{center}
\vspace{-1.5 ex} \hspace*{1 em}\index{turbulent!boundary layer}The revolutionary concept of a boundary layer was invented by Prandtl in 1904 (Anderson, 2005). Later, the concept was applied to flow over a flat plate by G.I. Taylor (1886--1975), L. Prandtl (1875--1953), and T. von Karman (1881--1963) who worked independently on the theory from 1915 to 1935. Their empirical theory, sometimes called the \textit{mixing-length theory}\index{mixing-length theory|textbf} predicts well the mean velocity profile close to the boundary. Of interest to us, it predicts the mean flow of air above the sea. Here's a simplified version of the theory applied to a smooth surface.
\hspace*{1 em} We begin by assuming that the mean flow in the boundary layer is
steady and that it varies only in the $z$ direction. Within a few millimeters of
the boundary, friction is important and (8.2) has the solution
\begin{equation}
U = \frac{T_x}{\rho \nu} \,z
\end{equation}
and the mean velocity varies linearly with distance above the boundary. Usually
(8.16) is written in dimensionless form:
\begin{equation}
\frac{U}{u^*} = \frac{u^* z}{\nu}
\end{equation}
where $u^{*2} \equiv T_x/\rho$ is the \textit{friction velocity}\index{friction velocity|textbf}.
\hspace*{1 em}Further from the boundary, the flow is turbulent, and molecular
friction is not important. In this regime, we can use (8.13), and
\begin{equation}
A_z \frac{\partial U}{\partial z} = u^{*2}
\end{equation}
\hspace*{1 em}Prandtl and Taylor assumed that large eddies are more effective
in mixing\index{mixing!of momentum} momentum than small eddies, and therefore $A_z$ ought to
vary with distance from the wall. Karman assumed that it had the particular functional
form $A_z =
\kappa z u^*$, where $\kappa$ is a dimensionless constant. With this
assumption, the equation for the mean velocity profile becomes
\begin{equation}
\kappa z u^* \frac{\partial U}{\partial z} = u^{*2}
\end{equation}
\hspace*{1 em}Because $U$ is a function only of $z$, we can write $dU =
u^*/(\kappa z) \, dz$, which has the solution
\begin{equation}
U = \frac{u^*}{\kappa} \ln \left(\frac{z}{z_0}\right)
\end{equation}
where $z_0$ is distance from the boundary at which velocity goes to zero.
\hspace*{1 em}For airflow over the sea, $\kappa = 0.4$ and $z_o$ is given by
Charnock's (1955) relation $z_0 = 0.0156 \, u^{*2}/g$. The mean velocity in the boundary layer just above the sea surface described in \S 4.3 fits well the logarithmic profile of (8.20), as does the mean velocity in the upper few meters of the sea just below the sea surface. Furthermore, using (4.2) in the
definition of the friction velocity\index{friction velocity!and wind stress}, then using (8.20)
gives Charnock's form of the drag coefficient\index{drag!coefficient} as a function of wind
speed.\rule[-1ex]{0mm}{1ex}
\vspace{0.5ex}
\end{minipage}
}}
\vspace{-3ex}
\end{table}
The assumption that an eddy viscosity $A_z$ can be used to relate the Reynolds stress to the mean flow works well in turbulent boundary layers. However $A_z$ cannot be obtained from theory. It must be calculated from data collected in wind tunnels or measured in the surface boundary layer at sea. See Hinze (1975, \S5--2 and\S7--5) and Goldstein (1965: \S80) for more on the theory of turbulence\index{turbulence!theory of} flow near a flat plate.
Prandtl's theory based on assumption (8.13) works well only where friction is much larger than the Coriolis force. This is true for air flow within tens of meters of the sea surface and for water flow within a few meters of the surface. The application of the technique to other flows in the ocean is less clear. For example, the flow in the mixed layer\index{mixed layer!theory} at depths below about ten meters is less well described by the classical turbulent theory. Tennekes and Lumley (1990: 57) write:
\begin{quotation} \small
Mixing-length and eddy viscosity models should be used only to generate
analytical expressions for the Reynolds stress and mean-velocity profile if
those are desired for curve fitting purposes in turbulent flows characterized by
a single length scale and a single velocity scale. The use of mixing-length
theory\index{mixing-length theory} in turbulent flows whose scaling laws are not known
beforehand should be avoided.
\end{quotation}
Problems with the eddy-viscosity approach:
\begin{enumerate}
\vitem Except in boundary layers a few meters thick, geophysical flows may be
influenced by several characteristic scales. For example, in the atmospheric
boundary layer above the sea, at least three scales may be important: i) the
height above the sea $z$, ii) the Monin-Obukhov scale $L$ discussed in
\S4.3, and iii) the typical velocity $U$ divided by the Coriolis parameter\index{Coriolis
parameter} $U/f$.
\vitem The velocities $u',\,w'$ are a property of the \textit{fluid}, while $A_z$ is a
property of the \textit{flow};
\vitem Eddy viscosity terms are not symmetric:
\begin{align}
\langle u'v' \rangle &= \langle v'u' \rangle\,;\quad \text{but} \notag \\
A_x \frac{\partial{V}}{\partial{x}} &\neq A_y \frac{\partial{U}}{\partial{y}}
\notag
\end{align}
\end{enumerate}
\paragraph{From a Statistical Theory of Turbulence}
The Reynolds stresses can be calculated from various theories which relate $\langle u'u'
\rangle$ to higher order correlations of the form $\langle u'u'u' \rangle$. The problem then
becomes: How to calculate the higher order terms? This is the \textit{closure
problem}\index{closure problem|textbf}\index{turbulence!closure problem|textbf} in turbulence.
There is no general solution, but the approach leads to useful understanding of some forms of
turbulence such as isotropic turbulence downstream of a grid in a wind tunnel (Batchelor
1967). \textit{Isotropic turbulence}\index{isotropic
turbulence|textbf}\index{turbulence!isotropic|textbf} is turbulence with statistical
properties that are independent of direction.
The approach can be modified somewhat for flow in the ocean. In the idealized case
of high Reynolds flow, we can calculate the statistical properties of a flow in
thermodynamic equilibrium. Because the actual flow in the ocean is far from
equilibrium, we assume it will evolve towards equilibrium. Holloway
(1986) provides a good review of this approach, showing how it can be used to
derive the influence of turbulence\index{turbulent!mixing} on mixing and heat
transports\index{transport!heat}. One interesting result of the work is that zonal
mixing\index{mixing!zonal} ought to be larger than meridional mixing\index{mixing!meridional}.
\paragraph{Summary} The turbulent eddy viscosities $A_x$, $A_y$, and $A_z$ cannot
be calculated accurately for most oceanic flows.
\begin{enumerate}
\vitem They can be estimated from measurements of turbulent flows. Measurements
in the ocean, however, are difficult. Measurements in the lab, although
accurate, cannot reach Reynolds numbers of $10^{11}$ typical of the ocean.
\vitem The statistical theory of turbulence\index{turbulence!theory of} gives useful insight
into the role of turbulence in the ocean, and this is an area of active research.
\end{enumerate}
\begin{table}[h!]\centering \small
\begin{tabular*}{62mm}{@{}rcl@{}}
\multicolumn{3}{@{}l@{}}{\bfseries Table 8.1 Some \rule[-1ex]{0mm}{1ex}Values for Viscosity}
\\
\hline
$\nu_{water}$ &$=$&\rule{0mm}{3ex}$10^{-6}$ m$^2$/s \\
$\nu_{tar\,at\,15^\circ{C}}$ &$=$&\rule{0mm}{3ex}$10^6$ m$^2$/s \\
$\nu_{glacier\,ice}$ &$=$&\rule{0mm}{3ex}$10^{10}$ m$^2$/s \\
$A_{y ocean}$ &$=$&\rule{0mm}{3ex}$10^4$ m$^2$/s \\
$A_{z ocean}$ &$=$&\rule{0mm}{3ex}$(10^{-5} - 10^{-3})$ m$^2$/s \\
[0.5ex]
\hline
\end{tabular*} \\[0.5ex]
\vspace{-3ex}
\end{table}
\section{Mixing in the Ocean}
\index{mixing!oceanic}
\index{mixing} Turbulence in the ocean leads to mixing. Because the ocean
has stable stratification, vertical displacement must work against the buoyancy\index{buoyancy}
force. Vertical mixing requires more energy than horizontal mixing. As a result,
horizontal mixing along surfaces of constant density is much larger than vertical
mixing across surfaces of constant density. The latter, however, usually called
\textit{diapycnal mixing}\index{mixing!diapycnal|textbf}\index{diapycnal
mixing|textbf}, is very important because it changes the vertical structure of the
ocean, and it controls to a large extent the rate at which deep water eventually
reaches the surface in mid and low latitudes.
The equations describing mixing depend on many processes. See Garrett (2006) for a good overview of the subject. Here I consider some simple flows. A simple equation for vertical mixing\index{mixing!vertical} by eddies of a tracer $\Theta$ such as salt or temperature is:
\begin{equation}
\frac{\partial \Theta}{\partial t} + W\,\frac{\partial \Theta}{\partial z} =
\frac{\partial }{\partial z} \left( A_z \frac{\partial \Theta}{\partial z}
\right) + S
\end{equation}
where $A_z$ is the vertical eddy diffusivity, $W$ is a mean vertical velocity,
and $S$ is a source term.
\paragraph{Average Vertical Mixing}
\index{mixing!average vertical}
Walter Munk (1966) used a very simple observation to calculate
vertical mixing in the ocean. He observed that the ocean has a thermocline\index{thermocline}
almost everywhere, and the deeper part of the thermocline\index{thermocline} does not change
even over decades (figure 8.4). This was a remarkable observation because we expect downward
mixing would continuously deepen the thermocline. But it doesn't. Therefore, a steady-state
thermocline requires that the downward mixing of heat by turbulence\index{turbulent!mixing}
must be balanced by an upward transport of heat\index{transport!heat upward} by a mean vertical
current $W$. This follows from (8.21) for steady state with no sources or sinks of heat:
\begin{equation}
W \frac{\partial T}{\partial z} = A_z \frac{\partial^2 T}{\partial z^2}
\end{equation}
where $T$ is temperature as a function of depth in the thermocline.
The equation has the solution:
\begin{equation}
T \approx T_0 \exp (z/H)
\end{equation}
where $H=A_z/W$ is the scale depth of the thermocline\index{thermocline}, and $T_0$ is the
temperature near the top of the thermocline. Observations of the shape of the
deep thermocline are indeed very close to a exponential function.
Munk used an exponential function fit through the observations of $T(z)$ to get
$H$.
\begin{figure}[t!]
\makebox[121mm] [c]{\includegraphics{mixing}}
\footnotesize
Figure 8.4 Potential \rule{0mm}{3ex}temperature measured as a
function of depth (pressure) near 24.7\degrees N, 161.4\degrees W
in the central North Pacific by the \textit{Yaquina} in 1966 ($\bullet$), and by
the \textit{Thompson} in 1985 $\left( _\square \;\right)$. Data from
\textit{Atlas of Ocean Sections} produced by Swift, Rhines, and Schlitzer.
\label{fig:mixing}
\vspace{-3ex}
\end{figure}
Munk calculated $W$ from the observed vertical distribution of $^{14}$C, a
radioactive isotope of carbon, to obtain a vertical time scale. In this case,
$S=-1.24 \times 10^{-4}$ years$^{-1}$. The length and time scales gave $W=1.2$
cm/day and
\begin{equation}
\left< A_z \right> = 1.3 \times 10^{-4} \text{ m$^2$/s} \qquad \text{Average
Vertical Eddy Diffusivity}
\end{equation}
where the brackets denote average eddy diffusivity in the thermocline\index{thermocline!eddy
diffusivity in}.
Munk also used $W$ to calculate the average vertical flux of water through
the thermocline in the Pacific, and the flux agreed well with the rate of
formation of bottom water assuming that bottom water upwells almost everywhere
at a constant rate in the Pacific. Globally, his theory requires upward mixing of 25
to 30 Sverdrups of water, where one Sverdrup is $10^6$ cubic meters per second.
\paragraph{Measured Vertical Mixing}
\index{mixing!measured vertical|(}
\index{mixing!vertical!measured}Direct observations of vertical mixing required the
development of techniques for measuring: i) the fine structure of
turbulence\index{turbulent!fine structure}, including probes able to measure temperature and
salinity with a spatial resolution of a few centimeters (Gregg 1991), and ii) the distribution
of tracers such as sulphur hexafluoride (SF$_6$) which can be easily detected at
concentrations as small as one gram in a cubic kilometer of seawater.
Direct measurements of open-ocean turbulence\index{turbulence!measurement of} and the diffusion
of SF$_6$ yield an eddy diffusivity:
\begin{equation}
A_z \approx 1 \times 10^{-5} \text{ m$^2$/s} \qquad \text{Open-Ocean Vertical
Eddy Diffusivity}
\end{equation}
For example, Ledwell, Watson, and Law (1998) injected 139 kg of SF$_6$ in the Atlantic near 26\degrees N, 29\degrees W 1200 km west of the Canary Islands at a depth of 310 m. They then measured the concentration for five months as it mixed over hundreds of kilometers to obtain a diapycnal eddy diffusivity\index{mixing!diapycnal}\index{diapycnal mixing} of $A_z = 1.2 \pm 0.2 \times 10^{-5}$ m$^2$/s.
The large discrepancy between Munk's calculation of the average eddy diffusivity for vertical mixing and the small values observed in the open ocean has been resolved by recent studies that show:
\begin{equation}
A_z \approx 10^{-3} \to 10^{-1} \text{ m$^2$/s} \qquad \text{Local Vertical Eddy Diffusivity}
\end{equation}
Polzin et al. (1997) measured the vertical structure of temperature in the Brazil Basin in the south Atlantic. They found $A_z > 10^{-3}$ m$^2$/s close to the bottom when the water flowed over the western flank of the mid-Atlantic ridge at the eastern edge of the basin. Kunze and Toole (1997) calculated enhanced eddy diffusivity as large as $A= 10^{-3}$ m$^2$/s above Fieberling Guyot in the Northwest Pacific and smaller diffusivity along the flank of the seamount. And, Garbato et al (2004) calculated even stronger mixing in the Scotia Sea where the Antarctic Circumpolar Current flows between Antarctica and South America.
The results of these and other experiments show that mixing occurs mostly by breaking internal waves and by shear at oceanic boundaries: along continental slopes, above seamounts and mid-ocean ridges, at fronts, and in the mixed layer\index{mixed layer!mixing in} at the sea surface. To a large extent, the mixing is driven by deep-ocean tidal currents\index{mixing!tidal}\index{mixing!of deep waters}, which become turbulent when they flow past obstacles on the sea floor, including seamounts and mid-ocean ridges (Jayne et al, 2004).
Because water is mixed along boundaries or in other regions (Gnadadesikan, 1999), we must take care in interpreting temperature profiles such as that in figure 8.4. For example, water at 1200 m in the
central north Atlantic could move horizontally to the Gulf Stream\index{Gulf Stream!and mixing}, where it mixes with water from 1000 m. The mixed water may then move horizontally back into the central north Atlantic at a depth of 1100 m. Thus parcels of water at 1200 m and at 1100 m at some location may reach their position along entirely different paths.
\paragraph{Measured Horizontal Mixing}
\index{mixing!average horizontal|(}
Eddies mix fluid in the horizontal, and large eddies mix
more fluid than small eddies. Eddies range in size from a few meters due to
turbulence\index{turbulent!mixing} in the thermocline\index{thermocline} up to several hundred
kilometers for geostrophic\index{geostrophic currents!eddies} eddies discussed in Chapter 10.
In general, mixing depends on Reynolds number $R$ (Tennekes 1990: p. 11)
\begin{equation}
\frac{A}{\gamma} \approx \frac{A}{\nu} \sim \frac{UL}{\nu} = R
\end{equation}
where $\gamma$ is the molecular diffusivity of heat, $U$ is a typical velocity in an eddy, and
$L$ is the typical size of an eddy. Furthermore, horizontal eddy diffusivity are ten thousand
to ten million times larger than the average vertical eddy diffusivity.
Equation (8.27) implies $A_x\sim UL$. This functional form agrees well with
Joseph and Sender's (1958) analysis, as reported in (Bowden 1962) of spreading
of radioactive tracers, optical turbidity, and Mediterranean Sea water in the
north Atlantic. They report
\begin{gather}
A_x = P L \\
10 \text{ km} < L < 1500 \text{ km} \notag \\
P = 0.01 \pm 0.005 \text{ m/s} \notag
\end{gather}
where $L$ is the distance from the source, and $P$ is a constant.
The horizontal eddy diffusivity (8.28) also agrees well with more recent reports
of horizontal diffusivity. Work by Holloway (1986) who used satellite
altimeter observations of geostrophic currents\index{geostrophic currents!altimeter
observations of}, Freeland who tracked \textsc{sofar} underwater floats, and Ledwell Watson,
and Law (1998) who used observations of currents and tracers to find
\begin{equation}
A_x \approx 8 \times 10^2 \text{ m$^2$/s} \qquad \text{Geostrophic Horizontal
Eddy Diffusivity}
\end{equation}
Using (8.28) and the measured $A_x$ implies eddies with typical scales of 80 km, a value near the size of geostrophic eddies\index{geostrophic currents!eddies} responsible for the mixing.
Ledwell, Watson, and Law (1998) also measured a horizontal eddy diffusivity. They found
\begin{equation}
A_x \approx 1 \text{ -- } 3 \text{ m$^2$/s} \qquad \text{Open-Ocean Horizontal
Eddy Diffusivity}
\end{equation}
over scales of meters due to turbulence\index{turbulent!mixing} in the thermocline\index{thermocline!mixing in} probably driven by breaking internal waves. This value, when used in (8.28) implies typical lengths of 100 m for the small eddies responsible for mixing in this experiment.\index{mixing!average horizontal|)}
\paragraph{Comments on horizontal mixing}
\begin{enumerate}
\vitem Horizontal eddy diffusivity is $10^5 - 10^8$ times larger than vertical eddy diffusivity.
\vitem \index{mixing!horizontal}Water in the interior of the ocean moves along sloping surfaces of constant density with little local mixing until it reaches a boundary where it is mixed vertically. The mixed water then moves back into the open ocean again along surfaces of constant density (Gregg 1987).
One particular case is particularly noteworthy. When water, mixing downward through the base of the mixed layer,\index{mixed layer!mixing through base of} flows out into the thermocline along surfaces of constant density, the mixing leads to the \textit{ventilated thermocline}\index{thermocline!ventilated|textbf} model of oceanic density distributions.
\vitem Observations of mixing in the ocean imply that numerical models of the oceanic circulation should use mixing schemes that have different eddy diffusivity parallel and perpendicular to surfaces of constant density, not parallel and perpendicular to level surfaces\index{level surface} of constant $z$ as I used above. Horizontal mixing along surfaces of constant $z$ leads to mixing across layers of
constant density because layers of constant density are inclined to the horizontal by about
$10^{-3}$ radians (see \S10.7 and figure 10.13).
Studies by Danabasoglu, McWilliams, and Gent (1994) show that numerical models using isopycnal and diapycnal mixing\index{mixing!diapycnal}\index{diapycnal mixing} leads to much more realistic
simulations of the oceanic circulation.
\vitem Mixing is horizontal and two dimensional for horizontal scales greater than $NH/(2f)$
where $H$ is the water depth, $N$ is the stability frequency\index{stability!frequency} (8.36),
and
$f$ is the Coriolis parameter (Dritschel, Juarez, and Ambaum (1999).
\end{enumerate}
\vspace{-2ex}
\section{Stability}
We saw in section 8.2 that fluid flow with a sufficiently large Reynolds number
is turbulent. This is one form of instability. Many other types of instability
occur in the in the ocean. Here, let's consider three of the more important ones:
i) \textit{static stability}\index{stability!static|textbf} associated with change
of density with depth, ii) \textit{dynamic
stability}\index{stability!dynamic|textbf} associated with velocity shear, and iii)
\textit{double-diffusion}\index{double diffusion} associated with salinity and
temperature gradients in the ocean.
\paragraph{Static Stability and the Stability Frequency} Consider first static
stability. If more dense water lies above less dense water, the fluid is
unstable. The more dense water will sink beneath the less dense. Conversely, if less
dense water lies above more dense water, the interface between the two is stable.
But how stable? We might guess that the larger the density contrast across the
interface, the more stable the interface. This is an example of static stability.
Static stability is important in any \textit{stratified} flow
where density increases with depth, and we need some criterion for determining the
importance of the stability.
\begin{figure}[b!]
\vspace{1ex}
\makebox[120mm] [c]{\includegraphics{stabilitysketch}}
\centering
\footnotesize
Figure 8.5 Sketch for \rule{0mm}{4ex}calculating static stability
and stability frequency\index{stability!frequency!sketch of}.
\label{fig:stabilitysketch}
%\vspace{-2ex}
\end{figure}
Consider a parcel of water that is displaced vertically and adiabatically in a
stratified fluid (figure 8.5). The buoyancy\index{buoyancy} force $F$ acting on the displaced
parcel is the difference between its weight $V g \rho '$ and the weight of the surrounding
water
$V g \rho_2$, where $V$ is the volume of the parcel:
\begin{displaymath}
F=V\,g\,(\rho_2-\rho{'})
\end{displaymath}
The acceleration of the displaced parcel is:
\begin{equation}
a=\frac{F}{m}=\frac{g\,(\rho_2-\rho{'})}{\rho{'}}
\end{equation}
but
\begin{align}
\rho_2 &= \rho + \left( \frac{d {\rho}}{d {z}}\right)_{water}
\delta z \\
\rho{'} &= \rho + \left( \frac{d {\rho}}{d {z}}\right)_{parcel}
\delta z
\end{align}
Using (8.32) and (8.33) in (8.31), ignoring terms proportional to
$\delta{z^2}$, we obtain:
\begin{equation}
E = -\frac{1}{\rho}\,\Biggl[\left(\frac{d \rho}{d {z}}\right)_{water}
- \,\left(\frac{d \rho}{d {z}}\right)_{parcel}\Biggr]
\end{equation}
where $E \equiv -a/(g \, \delta z)$ is the
\textit{stability}\index{stability|textbf} of the water column (McDougall, 1987;
Sverdrup, Johnson, and Fleming, 1942: 416; or Gill, 1982: 50).
In the upper kilometer of the ocean stability is large, and the first term in (8.34)
is much larger than the second. The first term is proportional to the rate of
change of density of the water column. The second term is proportional to the
compressibility of sea water, which is very small. Neglecting the second term, we
can write the
\textit{stability equation}\index{stability!equation|textbf}:
\begin{equation}
\boxed{E \approx -\frac{1}{\rho}\,\frac{d{\rho}}{d{z}} }
\end{equation}
The approximation used to derive (8.35) is valid for $E > 50 \times
10^{-8}$/m.
Below about a kilometer in the ocean, the change in density with depth is so small
that we must consider the small change in density of the parcel due to changes in
pressure as it is moved vertically.
Stability is defined such that
\begin{align}
E>0 & \quad \text{Stable} \notag \\
E=0 & \quad \text{Neutral Stability} \notag \\
E<0 & \quad \text{Unstable} \notag
\end{align}
In the upper kilometer of the ocean, $z < 1,000$ m, $E = (50$---$1000) \times
10^{-8}$/m, and in deep trenches where $z > 7,000$ m, $E = 1 \times 10^{-8}$/m.
\begin{figure}[t!]
%\vspace{-1ex}
\makebox[120mm] [c]{\includegraphics{stabilityfreq}}
\footnotesize
Figure 8.6. Observed \rule{0pt}{3ex} stability frequency\index{stability!frequency!in Pacific}
in the Pacific.
\textbf{Left:} Stability of the deep thermocline\index{thermocline!stability of} east of the
Kuroshio\index{Kuroshio!thermocline}.
\textbf{Right:} Stability of a shallow thermocline typical of the tropics. Note
the change of scales.
\label{fig:stabilityfreq}
\vspace{-3ex}
\end{figure}
The influence of stability is usually expressed by a
\textit{stability frequency}\index{stability!frequency|textbf}
$N$:
\begin{equation}
N^2 \equiv -g E
\end{equation}
The stability frequency\index{stability!frequency} is often called the \textit{buoyancy
frequency}\index{buoyancy!frequency|textbf} or the \textit{Brunt-Vaisala
frequency}\index{Brunt-Vaisala frequency|textbf}. The frequency quantifies the importance of
stability, and it is a fundamental variable in the dynamics of stratified flow. In simplest
terms, the frequency can be interpreted as the vertical frequency excited by a vertical
displacement of a fluid parcel. Thus, it is the maximum frequency of internal waves in the
ocean. Typical values of $N$ are a few cycles per hour (figure 8.6).
\paragraph{Dynamic Stability and Richardson's Number}
If velocity changes with depth in a stable, stratified flow, then the flow may
become unstable if the change in velocity with depth, the \textit{current
shear}\index{current shear|textbf}, is large enough. The simplest example is wind
blowing over the ocean. In this case, stability is very large across the sea
surface. We might say it is infinite because there is a step discontinuity in
$\rho$, and (8.36) is infinite. Yet, wind blowing on the ocean creates waves, and
if the wind is strong enough, the surface becomes unstable and the waves break.
This is an example of
\textit{dynamic instability}\index{instability!dynamic|textbf}\index{dynamic
instability|textbf} in which a stable fluid is made unstable by velocity shear.
Another example of dynamic instability, the Kelvin-Helmholtz instability, occurs
when the density contrast in a sheared flow is much less than at the sea surface,
such as in the thermocline\index{thermocline} or at the top of a stable, atmospheric boundary
layer (figure 8.7).
\begin{figure}[t!]
\makebox[120mm] [c]{\includegraphics{helmholtz}}
\footnotesize
Figure 8.7
Billow clouds showing a Kelvin-Helmholtz \rule{0mm}{4ex}instability at the top of
a stable atmospheric layer. Some billows can become large enough
that more dense air overlies less dense air, and then the billows collapse into
turbulence\index{turbulent!mixing}. Photography copyright Brooks Martner, \textsc{noaa}
Environmental Technology Laboratory.
\label{fig:helmholtz}
\vspace{-3ex}
\end{figure}
The relative importance of static stability and dynamic instability is expressed by
the \textit{Richardson Number}\index{Richardson Number|textbf}:
\begin{equation}
\boxed{R_i\equiv\frac{g\,E}{(\partial{U}/\partial{z})^2} }
\end{equation}
where the numerator is the strength of the static stability, and the denominator
is the strength of the velocity shear.
\begin{align}
R_i &>0.25 \quad \text{Stable} \notag \\
R_i &<0.25 \quad \text{Velocity Shear Enhances Turbulence} \notag
\end{align}
Note that a small Richardson number is not the only criterion for instability.
The Reynolds number must be large and the Richardson number must be less than
0.25 for turbulence. These criteria are met in some oceanic flows. The
turbulence mixes fluid in the vertical, leading to a vertical eddy viscosity and
eddy diffusivity. Because the ocean tends to be strongly stratified and currents
tend to be weak, turbulent mixing is intermittent and rare. Measurements of
density as a function of depth rarely show more dense fluid over less dense
fluid as seen in the breaking waves in figure 8.7 (Moum and Caldwell 1985).
\paragraph{Double Diffusion and Salt Fingers}
\index{double diffusion!salt fingers}In some regions of the ocean, less dense water
overlies more dense water, yet the water column is unstable even if there are no
currents. The instability occurs because the molecular diffusion of heat is about
100 times faster than the molecular diffusion of salt. The instability was first
discovered by Melvin Stern in 1960 who quickly realized its importance in
oceanography.
\begin{figure}[h!]
\makebox[120mm] [c]{\includegraphics{saltfingers}}
\footnotesize
Figure 8.8 \textbf{Left:} Initial \rule{0mm}{4ex}distribution of
density in the vertical. \textbf{Right:} After some time, the diffusion of heat
leads to a thin unstable layer between the two initially stable layers. The thin
unstable layer sinks into the lower layer as salty fingers. The vertical scale
in the figures is a few centimeters.
\label{fig:saltfingers}
\vspace{-2ex}
\end{figure}
Consider two thin layers a few meters thick separated by a sharp interface (figure
8.8). If the upper layer is warm and salty, and if the lower is colder and less
salty than the upper layer, the interface becomes unstable even if the upper layer
is less dense than the lower.
Here's what happens. Heat diffuses across the interface faster than salt, leading to a thin, cold, salty layer
between the two initial layers. The cold salty layer is more dense than the cold, less-salty layer below, and the
water in the layer sinks. Because the layer is thin, the fluid sinks in fingers 1--5 cm in diameter and 10s of
centimeters long, not much different in size and shape from our fingers. This is \textit{salt fingering}\index{salt
fingering|textbf}. Because two constituents diffuse across the interface, the process is called \textit{double
diffusion}\index{double diffusion|textbf}.
There are four variations on this theme. Two variables taken two at a time
leads to four possible combinations:
\begin{enumerate}
\vitem \textit{Warm salty over colder less salty}. This process is called
\textit{salt fingering}. It occurs in the thermocline below the surface waters of sub-tropical
gyres and the western tropical north Atlantic, and in the North-east Atlantic beneath the outflow from
the Mediterranean Sea. Salt fingering eventually leads to density increasing with depth
in a series of steps. Layers of constant-density are separated by thin layers with
large changes in density, and the profile of density as a function of depth looks
like stair steps. Schmitt et al (1987) observed 5--30 m thick steps in the western,
tropical north Atlantic that were coherent over 200--400 km and that lasted for at
least eight months. Kerr (2002) reports a recent experiment by Raymond Schmitt,
James Leswell, John Toole, and Kurt Polzin showed salt fingering off Barbados
mixed water 10 times faster than turbulence\index{turbulent!mixing}.
\vitem \textit{Colder less salty over warm salty}. This process is called
\textit{diffusive convection}\index{diffusive convection|textbf}. It is much less
common than salt fingering, and it us mostly found at high latitudes. Diffusive
convection also leads to a stair step of density as a function of depth. Here's
what happens in this case. Double diffusion leads to a thin, warm, less-salty layer
at the base of the upper, colder, less-salty layer. The thin layer of water rises
and mixes with water in the upper layer. A similar processes occurs in the lower
layer where a colder, salty layer forms at the interface. As a result of the
convection in the upper and lower layers, the interface is sharpened. Any small
gradients of density in either layer are reduced. Neal et al (1969) observed 2--10
m thick layers in the sea beneath the Arctic ice. \vitem \textit{Cold salty over
warmer less salty}. Always statically unstable. \vitem \textit{Warmer less salty
over cold salty}. Always stable and double diffusion diffuses the interface between
the two layers.
\end{enumerate}
Double diffusion mixes ocean water, and it cannot be ignored. Merryfield et al
(1999), using a numerical model of the ocean circulation that included double
diffusion, found that double-diffusive mixing changed the
regional distributions of temperature and salinity although it had little
influence on large-scale circulation of the ocean.
\section{Important Concepts}
\begin{enumerate}
\item
Friction in the ocean is important only over distances of a few
millimeters. For most flows, friction can be ignored.
\vitem The ocean is turbulent for all flows whose typical dimension exceeds a few
centimeters, yet the theory for turbulent flow in the ocean is poorly understood.
\vitem
The influence of turbulence\index{turbulence!Reynolds number} is a function of the Reynolds
number of the flow. Flows with the same geometry and Reynolds number have the same
streamlines.
\vitem
Oceanographers assume that turbulence influences flows over
distances greater than a few centimeters in the same way that molecular
viscosity influences flow over much smaller distances.
\vitem
The influence of turbulence leads to Reynolds stress terms
in the momentum equation.
\vitem
The influence of static stability in the ocean is expressed as a
frequency, the stability frequency\index{stability!frequency}. The larger the frequency, the
more stable the water column.
\vitem
The influence of shear stability is expressed through the Richardson
number. The greater the velocity shear, and the weaker the static stability, the
more likely the flow will become turbulent.
\vitem
Molecular diffusion of heat is much faster than the diffusion of salt.
This leads to a double-diffusion instability which modifies the density
distribution in the water column in many regions of the ocean.
\vitem
Instability in the ocean leads to mixing. Mixing across surfaces of
constant density is much smaller than mixing along such surfaces.
\vitem
Horizontal eddy diffusivity in the ocean is much greater than vertical eddy
diffusivity.
\vitem
Measurements of eddy diffusivity indicate water is mixed vertically near
oceanic boundaries such as above seamounts and mid-ocean ridges.
\end{enumerate}
\chapter{Response of the Upper Ocean to Winds}
If you have had a chance to travel around the United States, you may have noticed
that the climate of the east coast differs considerably from that on the west
coast. Why? Why is the climate of Charleston, South Carolina so different from
that of San Diego, although both are near 32\degrees N, and both are on or near
the ocean? Charleston has 125--150 cm of rain a year, San Diego has 25--50 cm,
Charleston has hot summers, San Diego has cool summers. Or why is the climate of
San Francisco so different from that of Norfolk, Virginia?
If we look closely at the characteristics of the atmosphere along the two coasts
near 32\degrees N, we find more differences that may explain the
climate. For example, when the wind blows onshore toward San Diego, it brings a
cool, moist, marine, boundary layer a few hundred meters thick
capped by much warmer, dry air. On the east coast, when the wind blows onshore, it
brings a warm, moist, marine, boundary layer that is much thicker. Convection, which
produces rain, is much easier on the east coast than on the west coast. Why then is
the atmospheric boundary layer over the water so different on the two coasts? The
answer can be found in part by studying the ocean's response to local winds, the
subject of this chapter.
\section{Inertial Motion}
\index{inertial!motion}To begin our study of currents near the sea surface, let's
consider first a very simple solution to the equations of motion, the response of
the ocean to an impulse that sets the water in motion. For example, the impulse
can be a strong wind blowing for a few hours. The water then moves only under the
influence of Coriolis force. No other force acts on the water.
Such motion is said to be inertial. The mass of water continues to move due to its
inertia. If the water were in space, it would move in a straight line according
to Newton's second law. But on a rotating earth, the motion is much
different.
From (7.12) the equations of motion for a parcel of water moving in the ocean without friction are:
\begin{subequations}
\begin{align}
\frac{du}{dt}&=-\frac{1}{\rho}\,\frac{\partial{p}}{\partial{x}} + 2\Omega v\,\sin\varphi \\
\frac{dv}{dt}&=-\frac{1}{\rho}\,\frac{\partial{p}}{\partial{y}} - 2\Omega u\,\sin\varphi \\
\frac{dw}{dt}&=-\frac{1}{\rho}\,\frac{\partial{p}}{\partial{z}} + 2\Omega u\,\cos\varphi - g
\end{align}
\end{subequations}
where $p$ is pressure, $\Omega = 2\,\pi$/(sidereal day) $= 7.292 \times 10^{-5}$
rad/s is the rotation of the earth in fixed coordinates, and $\varphi$ is
latitude.
Let's now look for simple solutions to these equations. To do this we must
simplify the momentum equations. First, if only the Coriolis force acts on
the water, there must be no horizontal pressure gradient:
\begin{displaymath}
\frac{\partial{p}}{\partial{x}} = \frac{\partial{p}}{\partial{y}} = 0
\end{displaymath}
Furthermore, we can assume that the flow is horizontal, and (9.1) becomes:
\begin{subequations}
\begin{align}
\frac{du}{dt}&=2\Omega \,v \sin\varphi =fv \\
\frac{dv}{dt}&=-2\Omega \,u \sin\varphi = -fu
\end{align}
\end{subequations}
where:
\begin{equation}
\boxed{f = 2\,\Omega\,\sin\varphi }
\end{equation}
is the \textit{Coriolis Parameter}\index{Coriolis parameter|textbf} and $\Omega = 7.292 \times
10^{-5}$/s is the rotation rate of earth.
Equations (9.2) are two coupled, first-order, linear, differential equations which
can be solved with standard techniques. If we solve the second equation for $u$,
and insert it into the first equation we obtain:
\begin{displaymath}
\frac{du}{dt}=-\frac{1}{f}\,\frac{d^2v}{dt^2}=fv
\end{displaymath}
Rearranging the equation puts it into a standard form we should recognize, the
equation for the harmonic oscillator:
\begin{equation}
\frac{d^2v}{dt^2} + f^2v = 0
\end{equation}
which has the solution (9.5). This current is called an \textit{inertial
current}\index{inertial!current|textbf} or
\textit{inertial oscillation}\index{inertial!oscillation|textbf}:
\begin{equation}
\fbox{$ \D \begin{aligned}%{cc}
u &= V\, \sin ft \\
v &= V\, \cos ft \\
V^2 &= u^2+v^2 \end{aligned}$}
\end{equation}
Note that (9.5) are the parametric equations for a circle with diameter
$D_i = 2V/f$ and period $T_i = (2\pi)/f= T_{sd}/(2\sin\varphi)$ where $T_{sd}$ is
a sidereal day.
\begin{figure}[t]
\makebox[120mm] [c]{\includegraphics{inertialcur}}
\footnotesize
Figure 9.1 Inertial currents \rule{0mm}{3ex}in the North Pacific in October 1987
(days 275--300) measured by holey-sock drifting buoys drogued at a depth of 15
meters. Positions were observed 10--12 times per day by the Argos system\index{Argos system} on
\textsc{noaa} polar-orbiting weather satellites and interpolated to positions every
three hours. The largest currents were generated by a storm on day 277. Note
these are not individual eddies. The entire surface is rotating. A drogue placed
anywhere in the region would have the same circular motion. After van Meurs (1998).
\label{fig:inertialcur}
\vspace{-3ex}
\end{figure}
$T_i$ is the \textit{inertial period}\index{inertial!period|textbf}. It is one
half the time required for the rotation of a local plane on earth's surface (Table
9.1). The rotation is \textit{anti-cyclonic}\index{anti-cyclonic|textbf}: clockwise in the
northern hemisphere, counterclockwise in the southern. Inertial currents are the free motion
of parcels of water on a rotating plane.
\begin{table}[h!]\centering \small
\vspace{-1ex}
\begin{tabular*}{50mm}{@{}ccc@{}}
\multicolumn{3}{@{}l@{}} {\bfseries Table 9.1 Inertial Oscillations} \\
\hline
Latitude $(\varphi)$ & $T_i$ (hr) & D\rule{0ex}{2.5ex} (km) \\
& \multicolumn{2}{c}{for V = 20 cm/s} \\
\hline
90\degrees & 11.97\rule{0ex}{2.5ex} & 2.7 \\
35\degrees & 20.87 & 4.8 \\
10\degrees & 68.93 & 15.8 \\ [0.5ex]
\hline
\end{tabular*} \\[0.5ex]
\vspace{-3ex}
\end{table}
Inertial currents are the most common currents in the ocean (figure 9.1). Webster
(1968) reviewed many published reports of inertial currents\index{inertial!current} and
found that currents have been observed at all depths in the ocean and at all latitudes. The
motions are transient and decay in a few days. Oscillations at different depths or at different
nearby sites are usually incoherent.
Inertial currents are caused by rapid changes of wind at the sea surface, with
rapid changes of strong winds producing the largest oscillations. Although we
have derived the equations for the oscillation assuming frictionless flow, friction
cannot be completely neglected. With time, the oscillations decay into other surface
currents. (See, for example, Apel, 1987: \S6.3 for more information.)
\section{Ekman Layer at the Sea Surface}
Steady winds \index{Ekman layer|(}blowing on \index{Ekman layer!sea surface|(}(the sea
surface produce a thin, horizontal boundary layer, the
\textit{Ekman layer}. \index{Ekman layer!defined|textbf} By thin, I mean a layer
that is at most a few-hundred meters thick, which is thin compared with the depth of
the water in the deep ocean. A similar boundary layer exists at the bottom of the
ocean, the
\textit{bottom Ekman layer}\index{Ekman layer!bottom|textbf}, and at the bottom of the
atmosphere just above the sea surface, the planetary boundary layer or frictional
layer described in \S 4.3. The Ekman layer is named after Professor Walfrid Ekman, who
worked out its dynamics for his doctoral thesis.
Ekman's work was the first of a remarkable series of studies conducted during the
first half of the twentieth century that led to an understanding of how winds
drive the ocean's circulation (Table 9.1). In this chapter we consider Nansen and
Ekman's work. The rest of the story is given in chapters 11 and 13.
\begin{table}[t!]\small
\begin{tabular*}{120mm}{@{}lcl@{}}
\multicolumn{3}{@{}l@{}}{\bfseries Table 9.2 Contributions to the Theory of the
\rule[-1ex]{0mm}{1ex}Wind-Driven Circulation} \\
\hline
Fridtjof Nansen & (1898) & Qualitative \rule{0ex}{2.5ex}theory, currents transport water at an \\
&&\hspace{1em}angle to the wind. \\
Vagn Walfrid Ekman & (1902) & Quantitative theory for wind-driven
transport\index{transport!wind-driven} at \\ &&\hspace{1em}the sea surface. \\
Harald Sverdrup & (1947) & Theory for wind-driven circulation in the eastern \\
&&\hspace{1em}Pacific. \\
Henry Stommel & (1948) & Theory for westward intensification of wind-driven \\
&&\hspace{1em}circulation (western boundary currents). \\
Walter Munk & (1950) & Quantitative theory for main features of the wind- \\
&&\hspace{1em}driven circulation. \\
Kirk Bryan & (1963) & Numerical models of the oceanic circulation. \\
Bert Semtner & (1988) & Global, eddy-resolving, realistic model of the \\
\ \ and Robert Chervin &&\hspace{1em}ocean's circulation. \\
\hline
\end{tabular*} \\[0.5ex]
\vspace{-3ex}
\end{table}
\paragraph{Nansen's Qualitative Arguments}
Fridtjof Nansen noticed that wind tended to blow ice at an angle of
20\degrees--40\degrees\ to the right of the wind in the Arctic, by which he meant
that the track of the iceberg was to the right of the wind looking downwind (See
figure 9.2). He later worked out the balance of forces that must exist
when wind tried to push icebergs downwind on a rotating earth.
Nansen argued that three forces must be important:
\begin{enumerate}
\vitem Wind Stress, \textbf{W};
\vitem Friction \textbf{F} (otherwise the iceberg would move as fast as the wind);
\vitem Coriolis Force\index{Coriolis force}, \textbf{C}.
\end{enumerate}
Nansen argued further that the forces must have the following attributes:
\begin{enumerate}
\vitem Drag must be opposite the direction of the ice's velocity;
\vitem Coriolis force must be perpendicular to the velocity;
\vitem The forces must balance for steady flow.
\end{enumerate}
\begin{center}
\textbf{W} + \textbf{F} + \textbf{C} = 0
\end{center}
\paragraph{Ekman's Solution}
Nansen \index{Ekman layer!theory of}asked Vilhelm Bjerknes to let one of Bjerknes'
students make a theoretical study of the influence of earth's rotation on wind-driven
currents. Walfrid Ekman was chosen, and he presented the results in his thesis at
Uppsala (Kullenberg, 1954). Ekman later expanded the study to include the influence of continents and
differences of density of water (Ekman, 1905). The following follows Ekman's line of
reasoning in that paper.
\begin{figure}[t!]
\centering
\makebox[120mm] [c]{\includegraphics{forcesketch}}
\footnotesize
Figure 9.2 The balance of forces \rule{0mm}{3ex}acting on an iceberg
in a wind on a rotating earth.
\label{fig:forcesketch}
\vspace{-3ex}
\end{figure}
Ekman assumed\index{Ekman layer!Ekman's assumptions} a steady, homogeneous, horizontal
flow with friction on a rotating earth. Thus horizontal and temporal derivatives are
zero:
\begin{equation}
\frac{\partial}{\partial{t}}=\frac{\partial}{\partial{x}}=\frac{\partial}{\partial{y}}=0
\end{equation}
For flow on a rotating earth, this leaves a balance between frictional and Coriolis forces (8.15).
Ekman further assumed a constant vertical eddy viscosity of the form (8:13):
\begin{equation}
T_{xz} \,=\,\rho\, A_z \,\frac{\partial{u}}{\partial{z}}\: , \qquad T_{yz}\,=\,\rho\, A_z \,\frac{\partial{v}}{\partial{z}}
\end{equation}
where $T_{xz}$, $T_{yz}$ are the components of the wind stress\index{wind stress!components} in
the
$x$,
$y$ directions, and $\rho$ is the density of sea water.
Using (9.7) in (8.15), the $x$ and $y$ momentum equations are:
\begin{subequations}
\begin{align}
fv + A_z \, \frac{\partial{^2 u}}{\partial{z^2}} &= 0 \\
-fu + A_z \, \frac{\partial{^2 v}}{\partial{z^2}} &= 0
\end{align}
\end{subequations}
where $f$ is the Coriolis parameter\index{Coriolis parameter}.
It is easy to verify that the equations (9.9) have solutions:
\begin{subequations}
\begin{align}
u &= V_0\,\exp(az)\,\cos(\pi/4 + az) \\
v &= V_0\,\exp(az)\,\sin(\pi/4 + az)
\end{align}
\end{subequations}
when the wind is blowing to the north $(T = T_{yz})$. The constants are
\begin{equation}
V_0 = \frac{T}{\sqrt{\rho^2_w\,f\,A_z}} \qquad \text{and} \qquad
a=\sqrt{\frac{f}{2A_z}}
\end{equation}
and $V_0$ is the velocity of the current at the sea surface.
Now let's look at the form of the solutions. At the sea surface $z = 0$,
$\exp(z=0) = 1$, and
\begin{subequations}
\begin{align}
u(0) &= V_0\, \cos(\pi/4) \\
v(0) &= V_0\, \sin(\pi/4)
\end{align}
\end{subequations}
The current has a speed of $V_0$ to the northeast. In general, the surface
current is 45\degrees\ to the right of the wind when looking downwind in the
northern hemisphere. The current is 45\degrees\ to the left of the wind in the
southern hemisphere. Below the surface, the velocity decays exponentially with
depth (figure 9.3):
\begin{equation}
\left[u^2(z) + v^2(z) \right]^{1/2} =V_0\,\exp(az)
\end{equation}
\begin{figure}[h!]
\makebox[121mm] [c]{\includegraphics{ekmancurrent}}
\centering
\footnotesize
Figure 9.3. Ekman current \rule{0mm}{4ex}generated by a 10 m/s wind at 35\degrees\ N.
\label{fig:ekmancurrent}
\end{figure}
\paragraph{Values for Ekman's Constants}
To proceed further, \index{Ekman layer!surface-layer constants}we need
values for any two of the free parameters: the velocity at the surface, $V_0$; the
coefficient of eddy viscosity, $A_z$; or the wind stress\index{wind stress!and Ekman layer} $T$.
The wind stress\index{wind stress!and Ekman layer} is well known, and Ekman used the bulk
formula (4.2):
\begin{equation}
T_{yz} = T = \rho_{air}\, C_D \,U_{10}^2
\end{equation}
where $\rho_{air}$ is the density of air, $C_D$ is the drag
coefficient\index{drag!coefficient}, and
$U_{10}$ is the wind speed at 10 m above the sea. Ekman turned to the literature
to obtain values for $V_0$ as a function of wind speed. He found:
\begin{equation}
V_0 = \frac{0.0127}{\sqrt{\sin|\varphi|}}\, U_{10}, \qquad \qquad
|\varphi|\ge 10
\end{equation}
With this information, he could then calculate the velocity as a function of
depth knowing the wind speed $U_{10}$ and wind direction.
\paragraph{Ekman Layer Depth}
\index{Ekman layer!depth|textbf}The thickness of the Ekman layer is arbitrary
because the Ekman currents decrease exponentially with depth. Ekman proposed that
the thickness be the depth $D_E$ at which the current velocity is opposite the
velocity at the surface, which occurs at a depth $D_E = \pi/a$, and the
\textit{Ekman layer depth} is:
\begin{equation}
\boxed{D_E = \sqrt{\frac{2\pi^2\,A_z}{f}}}
\end{equation}
Using (9.13) in (9.10), dividing by $U_{10}$, and using (9.14) and (9.15) gives:
\begin{equation}
D_E = \frac{7.6}{\sqrt{\sin|\varphi|}}\, U_{10}
\end{equation}
in SI units. Wind in meters per second gives depth in meters. The constant
in (9.16) is based on $\rho = 1027 $ kg/m$^3$, $\rho_{air} = 1.25 $
kg/m$^3$, and Ekman's value of $C_D = 2.6 \times 10^{-3}$ for the drag
coefficient\index{drag!coefficient}.
Using (9.16) with typical winds, the depth of the Ekman layer varies from about
45 to 300 meters (Table 9.3), and the velocity of the surface current varies from
2.5\% to 1.1\% of the wind speed depending on latitude\index{Ekman layer!sea
surface|)}.
\begin{table}[h!]\small \centering
\vspace{-1ex}
\begin{tabular*}{60mm}{@{}c|cc}
\multicolumn{3}{@{}c@{}}{\bfseries Table 9.3 Typical Ekman Depths \rule[-1ex]{0mm}{1ex}} \\
\hline
&\multicolumn{2}{c}{Latitude} \\
U$_{10}$ [m/s] & 15\degrees & 45\degrees \\
\hline
5 & \rule{0ex}{3ex}75 m & 45 m \\
10 & 150 m & 90 m \\
20 & 300 m & 180 m \\
\hline
\end{tabular*} \\[0.5ex]
\vspace{-3ex}
\end{table}
\paragraph{The Ekman Number: Coriolis and Frictional Forces}
The depth of the Ekman layer is closely related to the depth at
which frictional force is equal to the Coriolis force\index{Coriolis force} in the momentum
equation (9.9). The Coriolis force is
$f u$, and the frictional force is $A_z \partial^2 U/\partial z^2$. The ratio of the
forces, which is non dimensional, is called the \textit{Ekman Number}\index{Ekman
Number|textbf} $E_z$:
\begin{displaymath}
E_z=\frac{\text{Friction Force}}{\text{Coriolis Force}} = \frac{\D A_z\frac{\D\partial^2u}{\partial{z^2}}}{fu} = \frac{\D A_z\frac{\D
u}{\D d^2}}{\D fu}
\end{displaymath}
\begin{equation}
\boxed{E_z = \frac{A_z}{f\,d^2}}
\end{equation}
where we have approximated the terms using typical velocities $u$, and typical
depths $d$. The subscript $z$ is needed because the ocean is stratified and mixing
in the vertical is much less than mixing in the horizontal. Note that as depth
increases, friction becomes small, and eventually, only the Coriolis force
remains.
Solving (9.17) for $d$ gives
\begin{equation}
d = \sqrt{\frac{A_z}{fE_z}}
\end{equation}
which agrees with the functional form (9.15) proposed by Ekman. Equating (9.18)
and (9.15) requires $E_z = 1/(2\pi^2) = 0.05$ at the Ekman depth. Thus Ekman chose
a depth at which frictional forces are much smaller than the Coriolis force.
\paragraph{Bottom Ekman Layer}
\index{Ekman layer!bottom}The Ekman layer at the bottom of the ocean and the atmosphere differs from the layer at the
ocean surface. The solution for a bottom layer below a fluid with velocity $U$ in the $x$-direction is:
\begin{subequations}
\begin{align}
u&=U[1 - \exp(-az)\,\cos\,az] \\
v&=U\,\exp(-az)\,\sin\,az
\end{align}
\end{subequations}
The velocity goes to zero at the boundary, $u = v = 0$ at $z = 0$. The direction
of the flow close to the boundary is 45\degrees\ to the left of the flow $U$
outside the boundary layer in the northern hemisphere, and the direction of the
flow rotates with distance above the boundary (figure 9.4). The direction of
rotation is anti\-cyclonic with distance above the bottom.
\begin{figure}[b!]
\makebox[121mm] [c]{\includegraphics{bottomekman}}
\footnotesize
Figure 9.4 Ekman \rule{0mm}{3ex}layer in the lowest kilometer of
the atmosphere (solid line), with wind velocity measured by Dobson
(1914) -\ -\ - \ . The numbers give height above the surface in meters. The
Ekman layer at the sea floor has a similar shape. After Houghton
(1977: 107).
\label{fig:bottomekman}
%\vspace{-3ex}
\end{figure}
Winds above the planetary boundary layer are perpendicular to the pressure
gradient in the atmosphere and parallel to lines of constant surface pressure.
Winds at the surface are 45\degrees\ to the left of the winds aloft, and surface
currents are 45\degrees\ to the right of the wind at the surface. Therefore we
expect currents at the sea surface to be nearly in the direction of winds
above the planetary boundary layer and parallel to lines of constant
pressure. Observations of surface drifters\index{drifters!in Pacific} in the Pacific tend to
confirm the hypothesis (figure 9.5).
\begin{figure}[t!]
%\vspace{-2ex}
\makebox[121mm] [c]{\includegraphics{drifterplot}}
\footnotesize
Figure 9.5 Trajectories \rule{0mm}{3ex}of surface drifters\index{drifters!in Pacific} in
April 1978 together with surface pressure in the atmosphere averaged for the
month. Note that drifters tend to follow lines of constant pressure except in the
Kuroshio\index{Kuroshio!observed by drifters} where ocean currents are fast compared with
velocities in the Ekman layer in the ocean. After McNally et al. (1983).
\label{drifterplot}
\vspace{-3ex}
\end{figure}
\paragraph{Examining Ekman's Assumptions}
Before \index{Ekman layer!Ekman's assumptions|(}considering the validity of Ekman's
theory for describing flow in the surface boundary layer of the ocean, let's first
examine the validity of Ekman's assumptions. He assumed:
\begin{enumerate}
\vitem No boundaries. This is valid away from coasts.
\vitem Deep water. This is valid if depth $\gg 200$ m.
\vitem $f$-plane. This is valid.
\vitem Steady state. This is valid if wind blows for longer than a pendulum day.
Note however that Ekman also calculated a time-dependent solution, as did
Hasselmann (1970).
\vitem $A_z$ is a function of $U^2_{10}$ only. It is assumed to be
independent of depth. This is not a good assumption. The mixed layer\index{mixed layer!and
Ekman layer} may be thinner than the Ekman depth, and $A_z$ will change rapidly at the bottom
of the mixed layer\index{mixed layer!and Ekman layer} because mixing is a function of
stability. Mixing across a stable layer is much less than mixing through a layer of a neutral
stability. More realistic profiles for the coefficient of eddy viscosity as a function of
depth change the shape of the calculated velocity profile. I reconsider this problem below.
\vitem Homogeneous density. This is probably good, except as it effects stability.
\end{enumerate}\index{Ekman layer!Ekman's assumptions|)}
\paragraph{Observations of Flow Near the Sea Surface}
Does \index{Ekman layer!observations of}the flow close to the sea surface agree with
Ekman's theory? Measurements of currents made during several, very careful experiments
indicate that Ekman's theory is remarkably good. The theory accurately describes the
flow averaged over many days.
Weller and Plueddmann (1996) measured currents from 2 m to 132 m using 14
vector-measuring current meters deployed from the Floating Instrument Platform
\textsc{flip} in February and March 1990 500 km west of point Conception,
California. This was the last of a remarkable series of experiments
coordinated by Weller using instruments on \textsc{flip}.
Davis, DeSzoeke, and Niiler (1981) measured currents from 2 m to 175 m using 19
vector-measuring current meters deployed from a mooring for 19 days in August and
September 1977 at 50\degrees N, 145\degrees W in the northeast Pacific.
Ralph and Niiler (2000) tracked 1503 drifters\index{drifters!measurement of Ekman currents}
drogued to 15 m depth in the Pacific from March 1987 to December 1994. Wind velocity was
obtained every 6 hours from the European Centre for Medium-Range Weather Forecasts
\textsc{ecmwf}.
The results of the experiments indicate that:
\begin{enumerate}
\vitem
Inertial currents are the largest component of the flow.
\vitem
The flow is nearly independent of depth within the mixed layer\index{mixed layer!velocity
within} for periods near the inertial period\index{inertial!period}. Thus the mixed layer moves
like a slab at the inertial period. Current shear is concentrated at the top of the
thermocline\index{thermocline!and current shear}.
\vitem
The flow averaged over many inertial periods is almost exactly that calculated
from Ekman's theory. The shear of the Ekman currents extends through the averaged
mixed layer\index{mixed layer!and Ekman layer} and into the thermocline. Ralph and Niiler
found (using SI units, $U$ in m/s):
\begin{align}
D_E =& \frac{7.12}{\sqrt{\sin|\varphi|}}\, U_{10}\\
V_0 =& \frac{0.0068}{\sqrt{\sin|\varphi|}}\, U_{10}
\end{align}
The Ekman-layer depth $D_E$ is almost exactly that proposed
by Ekman (9.16), but the surface current $V_0$ is half his value (9.14).
\vitem
The angle between the wind and the flow at the surface depends on latitude, and it is near 45\degrees\ at mid latitudes (figure 9.6).
\vitem
The transport is 90\degrees\ to the right \index{transport!Ekman}of the wind in the northern
hemisphere. The transport direction agrees well with Ekman's theory.
\end{enumerate}
\begin{figure}[t!]
%\vspace{-2ex}
\makebox[120mm] [c]{\includegraphics{ekmanangle}}
\footnotesize
Figure 9.6 Angle \rule{0mm}{3ex}between the wind and flow at the surface calculated by Maximenko and Niiler using positions from drifters drogued at 15 m with satellite-altimeter, gravity, and \textsc{grace} \index{GRACE}data and winds from the \textsc{ncar/ncep} reanalysis.
\label{fig:ekmanangle}
\vspace{-3ex}
\end{figure}
\paragraph{Influence of Stability in the Ekman Layer}
\index{Ekman layer!influence of stability}Ralph and Niiler (2000) point out that
Ekman's choice of an equation for surface currents (9.14), which leads to (9.16), is
consistent with theories that include the influence of stability in the upper ocean.
Currents with periods near the inertial period\index{inertial!period} produce shear in
the thermocline\index{thermocline!and current shear}. The shear mixes the surface layers when
the Richardson number falls below the critical value (Pollard et al. 1973). This idea, when
included in mixed-layer theories, leads to a surface current $V_0$ that is proportional to
$\sqrt{N/f}$
\begin{equation}
V_0 \sim U_{10} \sqrt{N/f}
\end{equation}
where $N$ is the stability frequency defined by (8.36). Furthermore
\begin{equation}
A_z \sim U_{10}^2 / N \qquad \text{and} \qquad D_E \sim U_{10} / \sqrt{Nf}
\end{equation}
Notice that (9.22) and (9.23) are now dimensionally correct. The equations used
earlier, (9.14), (9.16), (9.20), and (9.21) all required a dimensional
coefficient\index{Ekman layer|)}.
\section{Ekman Mass Transport}
\index{Ekman transport!mass transport defined|textbf} \index{transport!Ekman mass} \index{Ekman
transport|(}Flow in the Ekman layer at the sea surface carries mass. For many reasons we may
want to know the total mass transported in the layer. The \textit{Ekman mass transport} $M_E$ is
defined as the integral of the Ekman velocity $U_E, V_E$ from the surface to a depth
$d$ below the Ekman layer. The two components of the transport are $M_{Ex}$, $M_{Ey}$ :
\begin{equation}
M_{Ex} = \int^0_{-d} \rho U_E \, dz, \qquad
M_{Ey} = \int^0_{-d} \rho V_E \, dz
\end{equation}
The transport has units kg/(m$\cdot$s). It is the mass of water passing
through a vertical plane one meter wide that is perpendicular to the
transport and extending from the surface to depth $-d$ (figure 9.7).
\begin{figure}[h!]
\vspace{-2ex}
\makebox[120mm] [c]{\includegraphics{transportsketch}}
\centering
\footnotesize
Figure 9.7 Sketch for \rule{0mm}{3ex}defining \textbf{Left:} mass
transports\index{transport!Ekman mass}, and
\textbf{Right:} volume transports\index{transport!Ekman volume}.
\label{fig:transportsketch}
%\vspace{-1ex}
\end{figure}
We calculate the Ekman mass transports\index{transport!Ekman} by integrating (8.15) in (9.24):
\begin{align}
f\int_{-d}^0\rho\,V_E\,dz =f\,M_{Ey} &=-\int_{-d}^0 \,d T_{xz} \notag \\
f\,M_{Ey} &=-T_{xz}\big|_{z=0} + T_{xz}\big|_{z=-d}
\end{align}
A few hundred meters below the surface the Ekman velocities approach zero, and the
last term of (9.25) is zero. Thus mass transport\index{transport!Ekman mass} is due only to
wind stress\index{wind stress!and mass transport in ocean} at the sea surface $(z = 0)$. In a
similar way, we can calculate the transport in the $x$ direction to obtain the two components
of the
\textit{Ekman transport}:
\begin{subequations}
\begin{align}
f\,M_{Ey} &= -T_{xz}(0) \\
f\,M_{Ex} &= \;\;\; T_{yz}(0)
\end{align}
\end{subequations}
where $T_{xz}(0), T_{yz}(0)$ are the two components of the stress at the sea
surface.
Notice that the transport is perpendicular to the wind stress\index{wind stress!and mass
transport in ocean}, and to the right of the wind in the northern hemisphere. If the wind is to
the north in the positive $y$ direction (a south wind), then $T_{xz}(0) = 0$, $M_{Ey} =
0$, and $M_{Ex} = T_{yz}(0)/f$. In the northern hemisphere, $f$ is positive,
and the mass transport is in the $x$ direction, to the east.
It may seem strange that the drag of the wind on the water leads to a current at
right angles to the drag. The result follows from the assumption that friction is
confined to a thin surface boundary layer, that it is zero in the interior of the
ocean, and that the current is in equilibrium with the wind so that it is no
longer accelerating.
\textit{Volume transport}\index{Ekman transport!volume transport
defined|textbf}\index{transport!Ekman volume} $Q$ is the mass transport divided by the density
of water and multiplied by the width perpendicular to the transport.
\begin{equation}
Q_x=\frac{Y M_x}{\rho}, \qquad Q_y=\frac{X M_y}{\rho}
\end{equation}
where $Y$ is the north-south distance across which the eastward
transport\index{transport!eastward} $Q_x$ is calculated, and $X$ in the east-west distance
across which the northward transport $Q_y$ is calculated. Volume transport has dimensions of
cubic meters per second. A convenient unit for volume transport in the ocean is a million cubic
meters per second. This unit is called a \textit{Sverdrup}\index{Sverdrup|textbf}, and it is
abbreviated Sv.
Recent observations of Ekman transport \index{transport!Ekman, observations of}in the ocean
agree with the theoretical values (9.26). Chereskin and Roemmich (1991) measured the Ekman
volume transport across 11\degrees N in the Atlantic using an acoustic Doppler current profiler
described in Chapter 10. They calculated a transport of $Q_y = 12.0
\pm 5.5$ Sv (northward) from direct measurements of current, $Q_y = 8.8 \pm 1.9$ Sv from
measured winds using (9.26) and (9.27), and $Q_y = 13.5 \pm 0.3$ Sv from mean
winds averaged over many years at 11\degrees N.\index{Ekman transport|)}
\paragraph{Use of Transports}
Mass \index{Ekman transport!uses}transports\index{transport!Ekman} are widely used for two
important reasons. First, the calculation is much more robust than calculations of velocities
in the Ekman layer. By robust, I mean that the calculation is based on fewer assumptions,
and that the results are more likely to be correct. Thus the calculated mass
transport does not depend on knowing the distribution of velocity in the Ekman
layer or the eddy viscosity.
Second, the variability of transport in space has important consequences.
Let's look at a few applications.
\section{Application of Ekman Theory}
Because steady winds blowing on the sea surface produce an Ekman layer that
transports \index{transport!and Ekman pumping}water at right angles to the wind direction, any
spatial variability of the wind, or winds blowing along some coasts, can lead to
upwelling\index{upwelling!due to Ekman pumping}. And upwelling\index{upwelling!importance of}
is important:
\begin{enumerate}
\vitem Upwelling enhances biological productivity, which feeds fisheries.
\vitem Cold upwelled water alters local weather. Weather onshore of regions of
upwelling\index{upwelling!and water temperature} tend to have fog, low stratus clouds, a stable
stratified atmosphere, little convection, and little rain.
\vitem Spatial variability of transports in the open ocean leads to
upwelling\index{upwelling!due to Ekman pumping} and downwelling, which leads to redistribution
of mass in the ocean, which leads to wind-driven geostrophic currents\index{geostrophic
currents!and Ekman pumping} via Ekman pumping\index{Ekman pumping}.
\end{enumerate}
\paragraph{Coastal Upwelling}
To \index{Ekman layer!coastal upwelling}see how winds lead to
upwelling\index{upwelling!coastal}, consider north winds blowing parallel to the California
Coast (figure 9.8 left). The winds produce a mass transport\index{transport!and upwelling} away
from the shore everywhere along the shore. The water pushed offshore can be replaced only by
water from below the Ekman layer. This is
\textit{upwelling}\index{upwelling|textbf} (figure 9.8 right). Because the upwelled
water is cold, the upwelling leads to a region of cold water at the surface along the
coast. Figure 10.16 shows the distribution of cold water off the coast of California.
\begin{figure}[t!]
\makebox[120mm] [c]{\includegraphics{upwelling}}
\footnotesize
Figure 9.8 Sketch of \rule{0mm}{5ex}Ekman transport along a coast
leading to upwelling\index{upwelling!coastal} of cold water along the coast. \textbf{Left:} Plan view. North winds along a west coast in the northern hemisphere cause Ekman transports away from the shore. \textbf{Right:} Cross section. The water transported offshore must be replaced by water upwelling from below the mixed layer\index{mixed layer!upwelling through}.
\label{fig:upwelling}
\vspace{-3ex}
\end{figure}
Upwelled water is colder than water normally found on the surface, and it is
richer in nutrients. The nutrients fertilize phytoplankton in the mixed layer\index{mixed
layer!and phytoplankton}, which are eaten by zooplankton, which are eaten by small fish, which
are eaten by larger fish and so on to infinity. As a result, upwelling\index{upwelling!and
fisheries} regions are productive waters supporting the world's major fisheries. The important
regions are offshore of Peru, California, Somalia, Morocco, and Namibia.
Now I can answer the question I asked at the beginning of the chapter: Why is
the climate of San Francisco so different from that of Norfolk, Virginia?
Figures 4.2 or 9.8 show that wind along the California and Oregon coasts has a
strong southward component. The wind causes upwelling\index{upwelling!coastal} along the coast,
which leads to cold water close to shore. The shoreward component of the wind brings
warmer air from far offshore over the colder water, which cools the incoming air
close to the sea, leading to a thin, cool atmospheric boundary layer. As the air
cools, fog forms along the coast. Finally, the cool layer of air is blown over
San Francisco, cooling the city. The warmer air above the boundary layer, due to
downward velocity of the Hadley circulation in the atmosphere (see figure 4.3),
inhibits vertical convection, and rain is rare. Rain forms only when winter
storms coming ashore bring strong convection higher up in the atmosphere.
In addition to upwelling\index{upwelling!coastal}, other processes influence weather in
California and Virginia.
\begin{enumerate}
\vitem
The oceanic mixed layer\index{mixed layer!in eastern basins} tends to be thin on the eastern
side of ocean, and upwelling can easily bring up cold water.
\vitem
Currents along the eastern side of the ocean at mid-latitudes tend to bring
colder water from higher latitudes.
\end{enumerate}
All these processes are reversed offshore of east coasts, leading to warm
water close to shore, thick atmospheric boundary layers, and frequent convective
rain. Thus Norfolk is much different that San Francisco due to
upwelling\index{upwelling!coastal} and the direction of the coastal currents.
\paragraph{Ekman Pumping}
The \index{Ekman pumping|(}horizontal variability of the wind blowing on the sea surface leads to horizontal variability of the Ekman transports. Because mass must be conserved, the spatial variability of the transports must lead to vertical velocities at the top of the Ekman layer. To calculate this velocity, we first integrate the continuity equation (7.19) in the vertical:
\begin{equation}
\begin{aligned}
\rho\int_{-d}^0 \left( \frac{\partial{u}}{\partial{x}}+\frac{\partial{v}}{\partial{y}}+\frac{\partial{w}}{\partial{z}} \right) dz &=0
\notag
\\
\frac{\partial}{\partial{x}}\int_{-d}^0 \rho\,u\,dz +\frac{\partial}{\partial{y}}\int_{-d}^0\rho\,v\,dz
&=- \rho \int_{-d}^0 \frac{\partial{w}}{\partial{z}}\, dz
\notag \\
\frac{\partial{M_{Ex}}}{\partial{x}}+\frac{\partial{M_{Ey}}}{\partial{y}} &=-\rho \left[ w(0)-w(-d)\right]
\end{aligned}
\end{equation}
By definition, the Ekman velocities approach zero at the base of the Ekman layer,
and the vertical velocity at the base of the layer $w_E(-d)$ due to divergence of
the Ekman flow must be zero. Therefore:
\begin{subequations}
\begin{equation}
\frac{\partial M_{Ex}}{\partial{x}}+\frac{\partial M_{Ey}}{\partial{y}} = -
\rho\,w_E(0)
\end{equation}
\vspace{-3ex}
\begin{equation}
\boxed{\nabla_H \cdot \mathbf{M}_E = -\rho\,w_E(0)}
\end{equation}
\end{subequations}
Where $\mathbf{M}_E$ is the vector mass transport\index{transport!Ekman mass} due to Ekman flow
in the upper boundary layer of the ocean, and
$\nabla_H$ is the horizontal divergence operator. (9.28) states that the horizontal
divergence of the Ekman transports leads to a vertical velocity in the upper boundary
layer of the ocean, a process called \textit{Ekman Pumping}\index{Ekman
pumping!defined|textbf}.
If we use the Ekman mass transports\index{transport!and Ekman pumping} (9.26) in (9.28) we can
relate Ekman pumping\index{Ekman pumping} to the wind stress\index{wind stress!and Ekman pumping}.
\begin{subequations}
\begin{align}
w_E(0)
&=-\frac{1}{\rho}\left[ \frac{\partial}{\partial{x}} \left( \frac{T_{yz}(0)}{f}
\right) -\frac{\partial}{\partial{y}} \left( \frac{T_{xz}(0)}{f} \right) \right]
\\ w_E(0) &=-\text{curl}_z \left( \frac{\mathbf{T}}{\rho\,f} \right)
\end{align}
\end{subequations}
where $\mathbf{T}$ is the vector wind stress and the subscript $z$ indicates the vertical component of the curl.
The vertical velocity at the sea surface $w(0)$ must be zero because the surface
cannot rise into the air, so $w_E(0)$ must be balanced by another vertical
velocity. We will see in Chapter 12 that it is balanced by a geostrophic\index{geostrophic
currents!velocity of} velocity $w_G(0)$ at the top of the interior flow in the ocean.
Note that the derivation above follows Pedlosky (1996: 13), and it differs from the
traditional approach that leads to a vertical velocity at the base of the Ekman
layer. Pedlosky points out that if the Ekman layer is very thin compared with the
depth of the ocean, it makes no difference whether the velocity is calculated at
the top or bottom of the Ekman layer, but this is usually not true for the ocean.
Hence, we must compute vertical velocity at the top of the layer\index{Ekman
pumping|)}.
\begin{figure}[t!]
%\vspace{-2ex}
\makebox[120mm] [c]{\includegraphics{langmuir}}
\footnotesize
Figure 9.9 A three-dimensional \rule{0mm}{3ex}view of the Langmuir
circulation at the surface of the Pacific observed from the Floating Instrument
Platform
\textsc{flip}. The heavy dashed line on the sea surface indicate lines of
convergence marked by cards on the surface. Vertical arrows are individual
values of vertical velocity measured every 14 seconds at 23 m depth as the
platform drifted through the Langmuir currents. Horizontal arrows, which are
drawn on the surface for clarity, are values of horizontal velocity at 23 m. The
broad arrow gives the direction of the wind. After Weller et al. (1985).
\label{fig:langmuir}
\vspace{-2ex}
\end{figure}
\section{Langmuir Circulation}
\index{Langmuir circulation}Measurements of surface currents show that winds
generate more than Ekman and inertial currents\index{inertial!current} at the sea surface.
They also generate a Langmuir circulation (Langmuir, 1938), a current that spiral around an
axis parallel to the wind direction. Weller et al. (1985) observed such a flow during an
experiment to measure the wind-driven circulation in the upper 50 meters of the sea.
They found that during a period when the wind speed was 14 m/s, surface currents
were organized into Langmuir cells spaced 20 m apart, the cells were aligned at an
angle of 15\degrees\ to the right of the wind, and vertical velocity at 23 m depth
was concentrated in narrow jets under the areas of surface convergence (figure 9.9).
Maximum vertical velocity was $-0.18$ m/s. The seasonal thermocline\index{thermocline!seasonal}
was at 50 m, and no downward velocity was observed in or below the
thermocline\index{thermocline}.
\section{Important Concepts}
\begin{enumerate}
\item Changes in wind stress\index{wind stress!and surface currents} produce transient
oscillations in the ocean called inertial currents\index{inertial!current}
\begin{enumerate}
\vitem Inertial currents are very common in the ocean.
\vitem The period of the current is $(2 \pi)/f$.
\end{enumerate}
\vitem Steady winds produce a thin boundary layer, the Ekman layer, at the top of the
ocean. Ekman boundary layers also exist at the bottom of the ocean and the atmosphere.
The Ekman layer in the atmosphere above the sea surface is called the planetary
boundary layer. \vitem The Ekman layer\index{Ekman layer!characteristics} at the sea
surface has the following characteristics:
\begin{enumerate}
\vitem \textit{Direction}: 45\degrees to the right of the wind looking downwind
in the Northern Hemisphere.
\vitem \textit{Surface Speed}: 1--2.5\% of wind speed depending on latitude.
\vitem \textit{Depth}: approximately 40--300 m depending on latitude and wind
velocity.
\end{enumerate}
\vitem Careful measurements of currents near the sea surface show that:
\begin{enumerate}
\vitem Inertial oscillations are the largest component of the current in the mixed
layer\index{mixed layer!and inertial oscillations}.
\vitem The flow is nearly independent of depth within the mixed layer\index{mixed
layer!currents within} for periods near the inertial period\index{inertial!period}. Thus the
mixed layer moves like a slab at the inertial period.
\vitem An Ekman layer exists in the atmosphere just above the sea (and land)
surface.
\vitem Surface drifters\index{drifters} tend to drift parallel to lines of constant atmospheric
pressure at the sea surface. This is consistent with Ekman's theory.
\vitem The flow averaged over many inertial periods is almost exactly that
calculated from Ekman's theory.
\end{enumerate}
\vitem Transport is 90\degrees\ to the right of the wind in the northern
hemisphere.
\vitem Spatial variability of Ekman transport, due to spatial variability of winds
over distances of hundreds of kilometers and days, leads to convergence and
divergence of the transport.
\begin{enumerate}
\vitem Winds blowing toward the equator along west coasts of continents produces
upwelling\index{upwelling!coastal} along the coast. This leads to cold, productive waters
within about 100 km of the shore.
\vitem Upwelled water along west coasts of continents modifies the weather along
the west coasts.
\end{enumerate}
\vitem Ekman pumping\index{Ekman
pumping}, which is driven by spatial variability of winds, drives a
vertical current, which drives the interior geostrophic\index{geostrophic currents}
circulation of the ocean.
\end{enumerate}
\chapter{Geostrophic Currents}
Within the ocean's interior away from the top and bottom Ekman layers\index{Ekman
layer}, for horizontal distances exceeding a few tens of kilometers, and for times
exceeding a few days, horizontal pressure gradients in the ocean almost exactly
balance the Coriolis force resulting from horizontal currents. This balance is known
as the \textit{geostrophic balance}\index{geostrophic balance|textbf}.
The dominant forces acting in the vertical are the vertical pressure gradient and
the weight of the water. The two balance within a few parts per million. Thus
pressure at any point in the water column is due almost entirely to the weight of
the water in the column above the point. The dominant forces in the horizontal
are the pressure gradient and the Coriolis force. They balance within a few parts
per thousand over large distances and times (See Box).
Both balances require that viscosity and nonlinear terms in the equations of
motion be negligible. Is this reasonable? Consider viscosity. We know that a
rowboat weighing a hundred kilograms will coast for maybe ten meters after the
rower stops. A super tanker moving at the speed of a rowboat may coast for
kilometers. It seems reasonable, therefore that a cubic kilometer of water
weighing $10^{15}$ kg would coast for perhaps a day before slowing to a stop. And
oceanic mesoscale eddies\index{mesoscale eddies} contain perhaps 1000 cubic kilometers of
water. Hence, our intuition may lead us to conclude that neglect of viscosity is reasonable.
Of course, intuition can be wrong, and we need to refer back to scaling
arguments.
\begin{figure} [t!]
\fbox{\parbox{12cm}{
\centering \small
\begin{minipage}{11.5cm}
\begin{center}\textbf{Scaling the Equations: The \rule{0mm}{3ex} Geostrophic
Approximation}\\
\end{center}
\vspace{-1em}
\hspace*{1em}\index{geostrophic approximation}We wish to simplify the
\rule{0mm}{3ex}equations of motion to obtain solutions that describe the deep-sea
conditions well away from coasts and below the Ekman boundary layer at the
surface. To begin, let's examine the typical size of each term in the equations in
the expectation that some will be so small that they can be dropped without
changing the dominant characteristics of the solutions. For interior, deep-sea
conditions, typical values for distance $L$, horizontal velocity $U$, depth $H$,
Coriolis parameter\index{Coriolis parameter} $f$, gravity $g$, and density $\rho$ are:
\vspace{-2ex}
\begin{align}
L\, &\approx\, 10^6 \text{ m} & H_1\, &\approx\,10^3 \text{ m} & f\, &\approx\,10^{-4}\,\text{ s}^{-1} & \rho\, &\approx\,10^3 \text{ kg/m}^3 \notag \\
U\, &\approx\,10^{-1} \text{ m/s} & H_2\, &\approx\, 1 \text{ m} & \rho\, &\approx\,10^3 \text{ kg/m}^3 & g\, &\approx\,10 \text{ m/s}^2 \notag
\end{align}
where $H_1$ and $H_2$ are typical depths for pressure in the vertical and horizontal.
\hspace*{1em}From these variables we can calculate typical values for vertical
velocity $W$, pressure $P$, and time $T$:
\begin{align} \notag
\frac{\partial W}{\partial z} & =-\,\left(\frac{\partial U}{\partial x}\,
+\,\frac{\partial v}{\partial y}\right) \notag \\
\frac{W}{H_1} & =\frac{U}{L}; \quad W =\frac{UH_1}{L}\,=\,
\frac{10^{-1}\,10^3}{10^6}\text{ m/s} =10^{-4} \text{m/s} \notag \\
P & =\rho g H_1 =10^3\,10^1\,10^3=10^7 \text{ Pa;} \quad \partial{p} / \partial{x} = \rho g H_2 / L = 10^{-2} \text{Pa/m} \notag
\\ T & = L/U =10^7 \text{ s} \notag
\end{align}
The momentum equation for vertical velocity is therefore:
\begin{align}
\frac{\partial w}{\partial t}\,+\,u\,\frac{\partial w}{\partial x}\,+\,v\,
\frac{\partial w}{\partial y}\,+\,w\,\frac{\partial w}{\partial z}&
=-\frac{1}{\rho}\frac{\partial p}{\partial z}\,+\,2\Omega \, u \cos\varphi -g
\notag\\ \notag
\frac{W}{T}\,+\,\frac{UW}{L}\;+\;\frac{UW}{L}\,+\;\;\;\frac{W^2}{H}&=\frac{P}{\rho\,H_1}\,+\,f\,U\,-\,g
\notag\\ \notag 10^{-11} + 10^{-11} + 10^{-11} + 10^{-11} & =10\quad + 10^{-5} - 10
\notag
\end{align}
and the only important balance in the vertical is hydrostatic:
\begin{displaymath}
\frac{\partial p}{\partial z}=-\rho g \quad \text{Correct to}\quad 1:10^6.
\end{displaymath}
The momentum equation for horizontal velocity in the $x$ direction is:
\begin{align}
\frac{\partial u}{\partial t}\,+\,u\,\frac{\partial u}{\partial
x}\,+\,v\,\frac{\partial u}{\partial y}\,+\,w\,\frac{\partial u} {\partial z} &
=-\frac{1}{\rho}\frac{\partial p}{\partial x}\,+fv \notag\\ \notag 10^{-8} +\;\;
10^{-8} + \:\;10^{-8} + \;10^{-8} & =\quad 10^{-5}\; + 10^{-5}
\notag
\end{align}
Thus the Coriolis force balances the pressure gradient within one part per thousand. This is called the \textit{geostrophic balance}\index{geostrophic balance|textbf}, and the \textit{geostrophic equations}\index{geostrophic equations|textbf} are:
\begin{displaymath}
\frac{1}{\rho}\,\frac{\partial p}{\partial x} =f v; \quad
\frac{1}{\rho}\,\frac{\partial p}{\partial y} =-f u; \quad
\frac{1}{\rho}\,\frac{\partial p}{\partial z} = -g
\end{displaymath} \notag
This balance applies to oceanic flows with horizontal dimensions larger than
roughly 50 km and times greater than a few days.
\vspace{0.7ex}
\end{minipage}}}
\vspace{-5ex}
\end{figure}
\section{Hydrostatic Equilibrium}
\index{hydrostatic equilibrium}Before describing the geostrophic balance, let's first
consider the simplest solution of the momentum equation, the solution for an ocean at
rest. It gives the hydrostatic pressure within the ocean. To obtain the solution, we
assume the fluid is stationary:
\begin{equation}
u=v=w=0;
\end{equation}
the fluid remains stationary:
\begin{equation}
\frac{du}{dt}=\frac{dv}{dt}=\frac{dw}{dt} = 0;
\end{equation}
and, there is no friction:
\begin{equation}
f_x=f_y=f_z=0.
\end{equation}
With these assumptions the momentum equation (7.12) becomes:
\begin{equation}
\frac{1}{\rho}\frac{\partial p}{\partial x}=0; \qquad \qquad
\frac{1}{\rho}\frac{\partial p}{\partial y}=0; \qquad \qquad
\frac{1}{\rho}\frac{\partial p}{\partial z}=-\,g(\varphi,z)
\end{equation}
where I have explicitly noted that gravity $g$ is a function of latitude
$\varphi$ and height $z$. I will show later why I have kept this explicit.
Equations (10.4) require surfaces of constant pressure to be level surface\index{level surface} (see page 30). A surface of constant pressure is an
\textit{isobaric surface}\index{isobaric surface|textbf}. The last equation can be integrated to obtain the pressure
at any depth $h$. Recalling that $\rho$ is a function of depth for an ocean at rest.
\begin{equation}
p=\int_{-h}^0\,g(\varphi,z)\,\rho(z)\,dz
\end{equation}
For many purposes, $g$ and $\rho$ are constant, and $p = \rho \,g\,h$.
Later, I will show that (10.5) applies with an accuracy\index{accuracy!equation!momentum} of
about one part per million even if the ocean is not at rest.
The\index{pressure!units of} SI unit for pressure is the pascal (Pa). A bar is another
unit of pressure. One bar is exactly $10^5$ Pa (table 10.1). Because the depth in meters
and pressure in decibars are almost the same numerically, oceanographers prefer to state
pressure in decibars.
\begin{table}[h!]\small \centering
\vspace{-1ex}
\begin{tabular*}{70mm}{lcl}
\multicolumn{3}{@{}l@{}}{\bfseries Table\rule[-1ex]{0mm}{1ex} 10.1 Units of
Pressure} \\
\hline
1 \rule{0mm}{2.5ex}pascal (Pa) &=& 1 N/m$^2$ = 1 kg$\cdot$s$^{-2}\cdot $m$^{-1}$
\\ 1 bar &=& 10$^5$ Pa \\
1 decibar &=& 10$^4$ Pa \\
1 millibar &=& 100 Pa \\
\hline
\end{tabular*} \\[0.5ex]
\vspace{-2ex}
\end{table}
\section{Geostrophic Equations}
The geostrophic \index{geostrophic currents!equations for|(}balance requires that the Coriolis
force \index{Coriolis force}balance the horizontal pressure gradient. The equations for geostrophic balance are derived from the equations of motion assuming the flow has no acceleration, $du/dt
= dv/dt = dw/dt = 0$; that horizontal velocities are much larger than vertical, $w \ll u,v$;
that the only external force is gravity; and that friction is small. With these assumptions
(7.12) become
%\begin{subequations}
\begin{equation}
\frac{\partial p}{\partial x}= \rho fv; \quad
\frac{\partial p}{\partial y}= - \rho f u; \quad
\frac{\partial p}{\partial z}= - \rho g
\end{equation}
where $f = 2 \Omega \sin \varphi$ is the Coriolis parameter\index{Coriolis parameter}. These
are the \textit{geostrophic equations}\index{geostrophic equations|textbf}.
The equations can be written:
\begin{subequations}
\begin{equation}
u= -\frac{1}{f\rho}\frac{\partial p}{\partial y}; \qquad
v= \frac{1}{f\rho}\frac{\partial p}{\partial x}
\end{equation}
\begin{equation}
p=p_0+\int_{-h}^{\,\zeta}\,g(\varphi,z)\rho(z)dz
\end{equation}
\end{subequations}
where $p_0$ is atmospheric pressure at $z = 0$, and $\zeta$ is the height of the
sea surface. Note that I have allowed for the sea surface to be above or below
the surface $z = 0$; and the pressure gradient at the sea surface is balanced by
a surface current $u_s$.
Substituting (10.7b) into (10.7a) gives:
\begin{subequations}
\begin{align}
u\,&= -\frac{1}{f\rho}\,\frac{\partial}{\partial
y}\int_{-h}^{0}\,g(\varphi,z)\,\rho(z)\,dz -
\frac{g}{f}\,\frac{\partial \zeta}{\partial y} \notag \\
u &= -\frac{1}{f\rho}\,\frac{\partial}{\partial
y}\int_{-h}^0\,g(\varphi,z)\,\rho(z)\,dz - u_s
\end{align}
where I have used the Boussinesq approximation\index{Boussinesq approximation}, retaining full
accuracy\index{accuracy!Boussinesq approximation} for
$\rho$ only when calculating pressure.
In a similar way, we can derive the equation for $v$.
\begin{align}
v&= \frac{1}{f\rho}\,\frac{\partial}{\partial
x}\int_{-h}^0\,g(\varphi,z)\,\rho(z)\,dz + \frac{g}{f}\,\frac{\partial
\zeta}{\partial x} \notag \\
v&= \frac{1}{f\rho}\,\frac{\partial}{\partial x}\int_{-h}^0
\,g(\varphi,z)\,\rho(z)\,dz + v_s
\end{align}
\end{subequations}
If the ocean is homogeneous and density and gravity are constant, the first term
on the right-hand side of (10.8) is equal to zero; and the horizontal pressure
gradients within the ocean are the same as the gradient at $z = 0$. This is
barotropic flow described in \S10.4.
If the ocean is stratified, the horizontal pressure gradient has two terms,
one due to the slope at the sea surface, and an additional term due to
horizontal density differences. These equations include baroclinic flow also
discussed in \S10.4. The first term on the right-hand side of (10.8) is due to
variations in density $\rho (z)$, and it is called the relative velocity. Thus
calculation of geostrophic currents from the density distribution requires the
velocity
$\left(u_0, v_0\right)$ at the sea surface or at some other depth.
\begin{figure}[h!]
\makebox[120mm][c]{\includegraphics{surfacesketch}}
\centering
\footnotesize
Figure 10.1 Sketch \rule{0mm}{3ex}defining $\zeta$ and $r$, used for
calculating pressure just below the sea surface.
\label{fig:surfacesketch}
\vspace{-3ex}
\end{figure}
\section{Surface Geostrophic Currents From Altimetry}
\index{geostrophic currents!surface}\index{geostrophic currents!from altimetry|(}The
geostrophic approximation applied at $z = 0$ leads to a very simple relation: surface
geostrophic currents are proportional to surface slope. Consider a level surface\index{level surface} slightly
below the sea surface, say two meters below the sea surface, at
$z = -r$ (figure 10.1).
The pressure on the level surface\index{level surface} is:
\begin{equation}
p = \rho\,g\,\left(\zeta + r\right)
\end{equation}
assuming $\rho$ and $g$ are essentially constant in the upper few meters of the
ocean.
Substituting this into (10.7a), gives the two components ($u_s, v_s$)
of the surface geostrophic current:
\begin{equation}
u_s =-\frac{g}{f}\frac{\partial\zeta}{\partial y}; \qquad \qquad
v_s =\frac{g}{f}\frac{\partial\zeta}{\partial x}
\end{equation}
where $g$ is gravity, $f$ is the Coriolis parameter\index{Coriolis parameter}, and $\zeta$ is
the height of the sea surface above a level surface\index{geostrophic currents!from
altimetry|)}\index{geostrophic currents!equations for|)}.
\paragraph{The Oceanic Topography}
\index{topography!oceanic|textbf}In \S 3.4 we define the topography of the sea surface $\zeta$
to be the height of the sea surface relative to a particular level surface\index{level surface}, the geoid\index{geoid}; and we
defined the geoid\index{geoid} to be the level surface\index{level surface} that coincided with the surface of the ocean at
rest. Thus, according to (10.10) the surface geostrophic currents are proportional to the
slope of the topography (figure 10.2), a quantity that can be measured by satellite altimeters
if the geoid\index{geoid} is known.
\begin{figure}[h!]
\vspace{-2ex}
\makebox[120mm][c]{\includegraphics{geostrophicsketch}}
\footnotesize
Figure 10.2 The \rule{0mm}{3ex}slope of the sea surface relative to the geoid\index{geoid}
$(\partial\zeta/\partial x)$ is directly related to surface geostrophic currents $v_s$.
The slope of 1 meter per 100 kilometers (10 $\mu$rad) is typical of strong currents.
$V_s$ is into the paper in the northern hemisphere.
\label{fig:geostrophicsketch}
\vspace{-2ex}
\end{figure}
Because the geoid\index{geoid} is a level surface\index{level surface}, it is a surface of constant geopotential. To see this, consider the work done in moving a mass $m$ by a distance $h$ perpendicular to a level surface\index{level surface}. The work is $W=mgh$, and the change of potential energy per unit mass is $gh$. Thus level surfaces\index{level surface} are surfaces of constant geopotential, where the \textit{geopotential}\index{geopotential} $\Phi = gh$.
Topography is due to processes that cause the ocean to move: tides, ocean currents, and
the changes in barometric pressure that produce the inverted barometer effect. Because
the ocean's topography is due to dynamical processes, it is usually called
\textit{dynamic topography}\index{dynamic
topography|textbf}\index{topography!dynamic|textbf}. The topography is approximately one
hundredth of the geoid undulations\index{geoid!undulations}. Thus the shape of the sea surface is
dominated by local variations of gravity. The influence of currents is much smaller.
Typically, sea-surface topography has amplitude of $\pm$1m (figure 10.3). Typical slopes are
$\partial\zeta/\partial x \approx $ 1--10 microradians for $v = $ 0.1--1.0 m/s at mid latitude.
The height of the geoid\index{geoid}, smoothed over horizontal distances greater than roughly 400 km, is known with an accuracy\index{accuracy!geoid} of $\pm$1mm from data collected by the Gravity Recovery and Climate Experiment \textsc{grace}\index{GRACE|textbf} satellite mission.
\begin{figure}[t!]
%\vspace{-2ex}
\makebox[120mm][c]{\includegraphics{sshprofile}}
\footnotesize
Figure 10.3 Topex/Poseidon\index{Topex/Poseidon!observations of Gulf Stream}
\rule{0mm}{3ex}altimeter observations of the Gulf Stream\index{Gulf Stream!mapped by
Topex/Poseidon}\index{geostrophic currents!from altimetry}. When the altimeter observations
are subtracted from the local geoid\index{geoid}, they yield the oceanic topography, which is due primarily
to ocean currents in this example. The gravimetric geoid\index{geoid} was determined by the Ohio State
University from ship and other surveys of gravity in the region. From Center for Space
Research, University of Texas.
\label{sshprofile}
\vspace{-5ex}
\end{figure}
\paragraph{Satellite Altimetry}
\index{satellite altimetry}Very accurate, satellite-altimeter systems are needed for
measuring the oceanic topography. The first systems, carried on Seasat, Geosat\index{Geosat},
\textsc{ers}--1, and \textsc{ers}--2\index{ERS satellites} were designed to measure
week-to-week variability of currents. Topex/Poseidon\index{Topex/Poseidon}, launched in 1992,
was the first satellite designed to make the much more accurate measurements necessary for
observing the permanent (time-averaged) surface circulation of the ocean, tides, and the
variability of gyre-scale currents. It was followed in 2001 by Jason\index{Jason} and in 2008 by Jason-2\index{Jason-2}.
Because the geoid\index{geoid} was not well known locally before about 2004, altimeters were usually flown in orbits that have an exactly repeating ground track. Thus Topex/Poseidon\index{Topex/Poseidon} and Jason\index{Jason} fly over the same ground track every 9.9156 days. By subtracting
sea-surface height from one traverse of the ground track from height measured on a later
traverse, changes in topography can be observed without knowing the geoid\index{geoid}. The
geoid\index{geoid} is constant in time, and the subtraction removes the geoid, revealing changes
due to changing currents, such as mesoscale eddies\index{mesoscale eddies}, assuming tides
have been removed from the data (figure 10.4). Mesoscale variability includes eddies with
diameters between roughly 20 and 500 km.
\begin{figure}[t!]
\makebox[120mm][c]{\includegraphics{sshvariability}}
\footnotesize
Figure 10.4 Global distribution of \rule{0mm}{3ex}standard deviation of topography from
Topex/Poseidon\index{Topex/Poseidon!observations of topography} altimeter data from 10/3/92 to
10/6/94. The height variance is an indicator of variability of surface geostrophic
currents\index{geostrophic currents!from altimetry}. From Center for Space Research,
University of Texas.
\label{fig:sshvariability}
\vspace{-4ex}
\end{figure}
The great accuracy\index{accuracy!altimeter} and precision of the Topex/Poseidon\index{Topex/Poseidon!accuracy of} and Jason\index{Jason!accuracy of} altimeter systems allow them to measure the oceanic topography over ocean basins with an accuracy of $\pm$(2--5) cm (Chelton et al, 2001). This allows them to measure:
\begin{enumerate}
\vitem Changes in the mean volume of the ocean and sea-level rise with an accuracy of $\pm 0.4$ mm/yr since 1993 (Nerem et al, 2006);
\vitem Seasonal heating and cooling of the ocean (Chambers et al 1998);
\vitem Open ocean tides with an accuracy of $\pm$(1--2) cm (Shum et al, 1997);
\vitem Tidal dissipation (Egbert and Ray, 1999; Rudnick et al, 2003);
\vitem The permanent surface geostrophic current system (figure 10.5);
\vitem Changes in surface geostrophic currents on all scales (figure 10.4); and
\vitem Variations in topography of equatorial current systems such as those associated with El Ni\~{n}o (figure 10.6).
\end{enumerate}
\begin{figure}[t!]
\makebox[120mm][c]{\includegraphics{sshmean}}
\footnotesize
Figure 10.5 Global distribution of \rule{0pt}{3ex} time-averaged topography of the
ocean from Topex/Pos\-eidon altimeter data from 10/3/92 to 10/6/99 relative to the
\textsc{jgm}--3 geoid\index{geoid}. Geostrophic cur\-rents at the ocean surface\index{geostrophic
currents!from altimetry} are parallel to the contours. Compare with figure 2.8 calculated from
hydrographic data\index{hydrographic data}. From Center for Space Research, University of
Texas.
\label{fig:sshmean}
\vspace{-4ex}
\end{figure}
\begin{figure}[t!]
\makebox[120mm][c]{\includegraphics{texas-may01}}
\footnotesize
Figure 10.6 Time-longitude \rule{0pt}{3ex}plot of sea-level
anomalies\index{anomalies!sealevel} in the Equatorial Pacific observed by
Topex/Poseidon during the 1997--1998 El Ni\~{n}o\index{Topex/Poseidon!observations of El Ni\~{n}o}. Warm anomalies\index{anomalies!sea-surface temperature} are light gray, cold anomalies are dark
gray. The anomalies are computed from 10-day deviations from a three-year mean surface from 3
Oct 1992 to 8 Oct 1995. The data are smoothed with a Gaussian weighted filter with a
longitudinal span of 5\degrees and a latitudinal span of 2\degrees. The annotations on the
left are cycles of satellite data. From Center for Space Research, University of Texas.
\label{texas-may01}
\vspace{-3ex}
\end{figure}
\vspace{-1ex}
\paragraph{Altimeter Errors (Topex/Poseidon and Jason)}
\index{satellite altimetry!errors in}\index{Topex/Poseidon}The most accurate observations of the sea-surface topography are from Topex/Poseidon and Jason\index{Jason}. Errors for these satellite altimeter system are due to (Chelton et al, 2001):
\begin{enumerate}
\vitem Instrument noise, ocean waves, water vapor, free electrons in the ionosphere, and mass of the atmosphere. Both satellites carried a precise altimeter system able to observe the height of the satellite above the sea surface between $\pm$66\degrees\ latitude with a precision of $\pm$(1--2) cm and an accuracy\index{accuracy!altimeter} of $\pm$(2--5) cm. The systems consist of a two-frequency radar altimeter to measure height above the sea, the influence of the ionosphere, and wave height, and a three-frequency microwave radiometer able to measure water vapor in the troposphere.
\vitem Tracking errors. The satellites carried three tracking systems that enable their position in space, the ephemeris, to be determined with an accuracy\index{accuracy!satellite tracking systems} of $\pm$(1--3.5) cm.
\vitem Sampling error. The satellites measure height along a ground track that repeats within $\pm$1 km every 9.9156 days. Each repeat is a cycle. Because currents are measured only
along the sub-satellite track, there is a sampling error\index{sampling error}. The satellite
cannot map the topography between ground tracks, nor can they observe changes with
periods less than $2 \times 9.9156$ d (see \S 16.3).
\vitem Geoid error\index{geoid!errors}. The permanent topography is not well known over distances
shorter than a hundred kilometers because geoid errors dominate for short distances. Maps of
topography smoothed over greater distances are used to study the dominant features of the permanent geostroph\-ic currents at the sea surface (figure 10.5). New satellite systems \textsc{grace}\index{GRACE} and \textsc{champ} are measuring earth's gravity accurately enough \index{accuracy!geoid} that the geoid error is now small enough to ignore over distances greater than 100 km.
\end{enumerate}
Taken together, the measurements of height above the sea and the satellite position give sea-surface height in geocentric coordinates within $\pm$(2--5) cm. Geoid error adds further errors that depend on the size of the area being measured.
\section{Geostrophic Currents From Hydrography}
\index{geostrophic currents!from hydrographic data|(}The geostrophic equations are widely used
in oceanography to calculate currents at depth. The basic idea is to use
hydrographic\index{hydrographic data} measurements of temperature,
salinity or conductivity, and pressure to calculate the density field of the ocean using the
equation of state of sea water. Density is used in (10.7b) to calculate the internal pressure
field, from which the geostrophic currents are calculated using (10.8a, b). Usually, however,
the constant of integration in (10.8) is not known, and only the relative velocity field can
be calculated.
At this point, you may ask, why not just measure pressure directly as is done in
meteorology, where direct measurements of pressure are used to calculate winds.
And, aren't pressure measurements needed to calculate density from the
equation of state? The answer is that very small changes in depth make large
changes in pressure because water is so heavy. Errors in pressure caused by errors in
determining the depth of a pressure gauge are much larger than the pressure due to
currents. For example, using (10.7a), we calculate that the pressure gradient due to
a 10 cm/s current at 30\degrees latitude is
$7.5 \times 10^{-3}$ Pa/m, which is 750 Pa in 100 km. From the hydrostatic
equation (10.5), 750 Pa is equivalent to a change of depth of 7.4 cm. Therefore,
for this example, we must know the depth of a pressure gauge with an
accuracy\index{accuracy!pressure} of much better than 7.4 cm. This is not possible.
\paragraph{Geopotential Surfaces Within the Ocean}
\index{geopotential!surface}Calculation of pressure gradients within the ocean must be
done along surfaces of constant geopotential just as we calculated surface pressure
gradients relative to the geoid\index{geoid} when we calculated surface geostrophic currents. As
long ago as 1910, Vilhelm Bjerknes (Bjerknes and Sandstrom, 1910) realized that such surfaces
are not at fixed heights in the atmosphere because $g$ is not constant, and (10.4) must
include the variability of gravity in both the horizontal and vertical directions (Saunders
and Fofonoff, 1976) when calculating pressure in the ocean.
The \textit{geopotential $\Phi$}\index{geopotential|textbf} is:
\begin{equation}
\Phi =\int_0^z\,g dz
\end{equation}
Because $\Phi/9.8$ in SI units has almost the same numerical value as height in meters,
the meteorological community accepted Bjerknes' proposal that height be replaced by
\textit{dynamic meters}\index{dynamic meter|textbf} $D =
\Phi/10$ to obtain a natural vertical coordinate. Later, this was replaced by the
\textit{geopotential meter}\index{geopotential!meter|textbf} (gpm) $Z = \Phi/9.80$. The
geopotential meter is a measure of the work required to lift a unit mass from sea level
to a height $z$ against the force of gravity. Harald Sverdrup, Bjerknes' student,
carried the concept to oceanography, and depths in the ocean are often quoted in
geopotential meters. The difference between depths of constant vertical distance and
constant potential can be relatively large. For example, the geometric depth of the 1000
dynamic meter surface is 1017.40 m at the north pole and 1022.78 m at the equator, a
difference of 5.38 m.
Note that depth in geopotential meters, depth in meters, and pressure in decibars
are almost the same numerically. At a depth of 1 meter the pressure is
approximately 1.007 decibars and the depth is 1.00 geopotential meters.
\paragraph{Equations for Geostrophic Currents Within the Ocean}
\index{geostrophic currents!equations for}To calculate geo\-strophic currents, we need to calculate the horizontal
pressure gradient with\-in the ocean. This can be done using either of two approaches:
\begin{enumerate}
\vitem Calculate the slope of a constant pressure surface relative to a surface of
constant geopotential. We used this approach when we used sea-surface slope from
altimetry to calculate surface geostrophic currents. The sea surface is a
constant-pressure surface. The constant geopotential surface was the geoid\index{geoid}.
\vitem Calculate the change in pressure on a
surface of constant geopotential. Such a surface is called a \textit{geopotential
surface}\index{geopotential!surface|textbf}.
\end{enumerate}
\begin{figure}[h!]
\vspace{-2ex}
\makebox[120mm][c]{\includegraphics{hydrosketch}}
\centering
\footnotesize
Figure 10.7. Sketch of \rule{0mm}{3ex}geometry used for calculating
geostrophic current from hydrography.
\label{fig:hydrosketch}
\vspace{-2ex}
\end{figure}
Oceanographers usually calculate the slope of constant-pressure surfaces.
The important steps are:
\begin{enumerate}
\vitem Calculate differences in geopotential $\left( \Phi_A - \Phi_B \right)$
between two constant-pressure surfaces $\left( P_1 , P_2 \right)$ at hydrographic
stations\index{hydrographic stations} A and B (figure 10.7). This is similar
to the calculation of $\zeta$ of the surface layer.
\vitem Calculate the slope of the upper pressure surface relative to the
lower.
\vitem Calculate the geostrophic current at the upper surface relative to the
current at the lower. This is the current shear.
\vitem Integrate the current shear from some depth where currents are known to
obtain currents as a function of depth. For example, from the surface downward,
using surface geostrophic currents observed by satellite altimetry, or upward
from an assumed level of no motion.
\end{enumerate}
To calculate geostrophic currents oceanographers use a modified form of the
hydrostatic equation. The vertical pressure gradient (10.6) is written
\begin{subequations}
\begin{align}
\frac{\delta p}{\rho}=\alpha\,\delta p &=-g\,\delta z \\
\alpha\,\delta p&=\delta\Phi
\end{align}
\end{subequations}
where $\alpha = \alpha(S,t,p)$ is the \textit{specific volume}\index{specific
volume|textbf}, and (10.12b) follows from (10.11). Differentiating (10.12b) with
respect to horizontal distance $x$ allows the geo\-stro\-phic balance to be written
in terms of the slope of the constant-pressure surface using (10.6) with $f = 2 \Omega \sin \phi $:
\begin{subequations}
\begin{align}
\alpha\,\frac{\partial p}{\partial x} =\frac{1}{\rho}\,\frac{\partial
p}{\partial x} &=-2\,\Omega \,v\sin \varphi \\
\frac{\partial \Phi \left( p=p_0 \right)} {\partial x}
&= - 2 \, \Omega \,v \, \sin{\varphi}
\end{align}
\end{subequations}
where $\Phi$ is the geopotential at the constant-pressure surface.
Now let's see how hydrographic data\index{hydrographic data} are used
for evaluating $\partial
\Phi/\partial x$ on a constant-pressure surface. Integrating (10.12b) between two constant-pressure surfaces $\left( P_1 , P_2
\right)$ in the ocean as shown in figure 10.7 gives the geopotential difference
between two constant-pressure surfaces. At station A the integration gives:
\begin{equation}
\Phi\left(P_{1A}\right)-\Phi\left(P_{2A}\right)=\int_{P_{1A}}^{P_{2A}}
\alpha\left(S,t,p\right)dp
\end{equation}
The specific volume anomaly is written as the sum of two parts:
\begin{equation}
\alpha(S,t,p)=\alpha(35,0,p)+\delta
\end{equation}
where $\alpha (35,0,p)$ is the specific volume of sea water with salinity of 35,
temperature of 0\degrees C, and pressure $p$. The second term $\delta$ is the
\textit{specific volume anomaly}\index{specific volume!anomaly|textbf}. Using (10.15) in
(10.14) gives:
\begin{align}
\Phi(P_{1A})-\Phi(P_{2A})&=\int_{P_{1A}}^{P_{2A}}\,\alpha(35,0,p)\, dp +\int_{P_{1A}}^{P_{2A}}
\delta \,dp \notag \\
\Phi(P_{1A})-\Phi(P_{2A})&=\left(\Phi_1-\Phi_2 \right)_{std}
+\Delta\Phi_A \notag
\end{align}
where ($\Phi_1-\Phi_2 )_{std}$ is the \textit{standard
geopotential distance}\index{standard geopotential distance|textbf} between two
constant-pressure surfaces $P_1$ and $P_2$, and
\begin{equation}
\Delta\Phi_A =\int_{P_{1A}}^{P_{2A}} \,\delta\, dp
\end{equation}
is the anomaly of the geopotential distance between the surfaces. It is called the
\textit{geopotential anomaly}\index{geopotential!anomaly|textbf}. The geometric distance
between $\Phi_2$ and $\Phi_1$ is numerically approximately $(\Phi_2 - \Phi_1) /g$ where
$g= 9.8$m/s$^2$ is the approximate value of gravity. The geopotential anomaly is much
smaller, being approximately 0.1\% of the standard geopotential distance.
Consider now the geopotential anomaly between two pressure surfaces $P_1$ and
$P_2$ calculated at two hydrographic stations\index{hydrographic stations} A
and B a distance $L$ meters apart (figure 10.7). For simplicity we assume the lower
constant-pressure surface is a level surface\index{level surface}. Hence the constant-pressure and geopotential
surfaces coincide, and there is no geostrophic velocity at this depth. The slope of the
upper surface is
\begin{displaymath}
\frac{\Delta\Phi_B - \Delta\Phi_A}{L} =\text{slope of constant-pressure
surface $P_2$}
\end{displaymath}
because the standard geopotential distance is the same at stations A and B. The
geostrophic velocity\index{geostrophic currents} at the upper surface calculated from
(10.13b) is:
\begin{equation}
V =\frac{\left(\Delta\Phi_B - \Delta\Phi_A\right)}{2\Omega\,L\, \sin\varphi }
\end{equation}
where $V$ is the velocity at the upper geopotential surface. The velocity $V$ is
perpendicular to the plane of the two hydrographic stations\index{hydrographic stations} and
directed into the plane of figure 10.7 if the flow is in the northern hemisphere. \textit{A
useful rule of thumb is that the flow is such that warmer, lighter water is to the right
looking downstream in the northern hemisphere.}
Note that I could have calculated the slope of the constant-pressure surfaces using density $\rho$ instead of specific volume $\alpha$. I used $\alpha$ because it is the common practice in oceanography, and tables of specific volume anomalies\index{anomalies!specific volume} and computer code to calculate the anomalies are widely available. The common practice follows from numerical methods developed before calculators and computers were available, when all calculations were done by hand or by mechanical calculators with the help of tables and nomograms. Because the computation must be done with an accuracy\index{accuracy!density} of a few parts per million, and because all scientific fields tend to be conservative, the common practice has continued to use specific volume anomalies rather than density anomalies\index{anomalies!density}.
\paragraph{Barotropic and Baroclinic Flow:}
If the ocean were homogeneous with constant density, then constant-pressure surfaces
would always be parallel to the sea surface, and the geostrophic velocity would be
independent of depth. In this case the relative velocity is zero, and hydrographic
data\index{hydrographic data!and geostrophic currents} cannot be used to measure the
geostrophic current. If density varies with depth, but not with horizontal distance, the
constant-pressure surfaces are always parallel to the sea surface and the levels of constant
density, the
\textit{isopycnal surfaces}\index{isopycnal surfaces|textbf}. In this case, the relative flow
is also zero. Both cases are examples of \textit{barotropic flow}.
\textit{Barotropic flow}\index{barotropic flow|textbf} occurs when levels of constant
pressure in the ocean are always parallel to the surfaces of constant density. Note,
some authors call the vertically averaged flow the barotropic component of the flow.
Wunsch (1996: 74) points out that barotropic is used in so many different ways
that the term is meaningless and should not be used.
\textit{Baroclinic flow}\index{baroclinic flow|textbf} occurs when levels of constant
pressure are inclined to surfaces of constant density. In this case, density varies with
depth and horizontal position. A good example is seen in figure 10.8 which shows levels
of constant density changing depth by more than 1 km over horizontal distances of 100 km
at the Gulf Stream\index{Gulf Stream!is baroclinic}. Baroclinic flow varies with depth, and the
relative current can be calculated from hydrographic data\index{hydrographic data!and
geostrophic currents}. Note, constant-density surfaces cannot be inclined to constant-pressure
surfaces for a fluid at rest.
In general, the variation of flow in the vertical can be decomposed into a
barotropic component which is independent of depth, and a baroclinic component
which varies with depth.
\section{An Example Using Hydrographic Data} Let's now consider a specific
\index{geostrophic currents!from hydrographic data}numerical calculation of
geostrophic velocity using generally accepted procedures from \textit{Processing of
Oceanographic Station Data} (\textsc{jpots} Editorial Panel, 1991). The book has worked
examples using hydrographic data\index{hydrographic data!from Endeavor} collected
by the \textsc{r/v} \textit{Endeavor} in the north Atlantic. Data were collected on Cruise 88
along 71\degrees W across the Gulf Stream\index{Gulf Stream!south of Cape Cod} south of Cape
Cod, Massachusetts at stations 61 and 64. Station 61 is on the Sargasso Sea side of the Gulf
Stream in water 4260 m deep. Station 64 is north of the Gulf Stream in water 3892 m deep. The
measurements were made by a Conductivity-Temp\-erature-Depth-Oxygen Profiler, Mark III CTD/02\index{CTD}, made by Neil Brown Ins\-truments Systems.
The \textsc{ctd} sampled temperature, salinity, and pressure 22 times per second,
and the digital data were averaged over 2 dbar intervals as the \textsc{ctd} was
lowered in the water. Data were tabulated at 2 dbar pressure intervals centered
on odd values of pressure because the first observation is at the surface, and
the first averaging interval extends to 2 dbar, and the center of the first
interval is at 1 dbar. Data were further smoothed with a binomial filter and
linearly interpolated to standard levels reported in the first three columns of
tables 10.2 and 10.3. All processing was done by computer.
$\delta (S, t, p)$ in the fifth column of tables 10.2 and 10.3 is calculated from
the values of $t, S, p$ in the layer. $<\delta >$ is the average value of specific
volume anomaly for the layer between standard pressure levels. It is the average
of the values of $\delta (S, t, p)$ at the top and bottom of the layer (\textit{cf.} the mean-value theorem of calculus). The last
column $(10^{-5} \Delta\Phi)$ is the product of the average specific volume anomaly
of the layer times the thickness of the layer in decibars. Therefore, the last column
is the geopotential anomaly $\Delta \Phi$ calculated by integrating (10.16)
between $P_1$ at the bottom of each layer and $P_2$ at the top of each layer.
The distance between the stations is $L = 110,935$ m; the average Coriolis
parameter\index{Coriolis parameter} is $f = 0.88104 \times 10^{-4}$; and the denominator in
(10.17) is 0.10231 s/m. This was used to calculate the geostrophic currents relative to 2000 decibars
reported in table 10.4 and plotted in figure 10.8.
Notice that there are no Ekman\index{Ekman layer} currents in figure 10.8.
Ekman currents are not geostrophic, so they don't contribute directly to the topography. They contribute only indirectly through Ekman pumping (see figure 12.7).
\begin{table}[t!]\centering \renewcommand{\baselinestretch}{0.0} \small
\begin{tabular*}{108mm}{@{}rrrrrrl}
\multicolumn{7}{@{}l@{}}{\bfseries Table 10.2 Computation of Relative Geostrophic Currents.} \\
& \multicolumn{6}{@{}l@{}}{\bfseries \rule{0mm}{2.4ex}Data from Endeavor Cruise 88, Station 61} \\
& \multicolumn{6}{@{}l@{}}{\bfseries (36\degrees 40.03'N, 70\degrees 59.59'W; \rule[-1ex]{0mm}{3.5ex}23 August 1982; 1102Z)} \\
\hline
Pressure &t & S &$\sigma (\theta)$\ \ \ &$\delta(S,t,p)$ &$<\delta >$
&$10^{-5}\Delta\Phi$\rule{0mm}{2.5ex} \\ decibar&\degrees C & &kg/m$^3$
&$10^{-8}$m$^3$/kg &$10^{-8}$m$^3$/kg &m$^2$/s$^2$\rule[-1ex]{0mm}{3.5ex} \\
\hline
0& 25.698& 35.221& 23.296& 457.24\rule{0mm}{2.5ex} & \\
& & & & & 457.26& 0.046\\
1& 25.698& 35.221& 23.296& 457.28& \\
& & & & & 440.22& 0.396\\
10& 26.763& 36.106& 23.658& 423.15& \\
& & & & & 423.41& 0.423\\
20& 26.678& 36.106& 23.658& 423.66& \\
& & & & & 423.82& 0.424\\
30& 26.676& 36.107& 23.659& 423.98& \\
& & & & & 376.23& 0.752\\
50& 24.528& 36.561& 24.670& 328.48& \\
& & & & & 302.07& 0.755\\
75& 22.753& 36.614& 25.236& 275.66& \\
& & & & & 257.41& 0.644\\
100& 21.427& 36.637& 25.630& 239.15& \\
& & & & & 229.61& 0.574\\
125& 20.633& 36.627& 25.841& 220.06& \\
& & & & & 208.84& 0.522\\
150& 19.522& 36.558& 26.086& 197.62& \\
& & & & & 189.65& 0.948\\
200& 18.798& 36.555& 26.273& 181.67& \\
& & & & & 178.72& 0.894\\
250& 18.431& 36.537& 26.354& 175.77& \\
& & & & & 174.12& 0.871\\
300& 18.189& 36.526& 26.408& 172.46& \\
& & & & & 170.38& 1.704\\
400& 17.726& 36.477& 26.489& 168.30& \\
& & & & & 166.76& 1.668\\
500& 17.165& 36.381& 26.557& 165.22& \\
& & & & & 158.78& 1.588\\
600& 15.952& 36.105& 26.714& 152.33& \\
& & & & & 143.18& 1.432\\
700& 13.458& 35.776& 26.914& 134.03& \\
& & & & & 124.20& 1.242\\
800& 11.109& 35.437& 27.115& 114.36& \\
& & & & & 104.48& 1.045\\
900& 8.798& 35.178& 27.306& 94.60& \\
& & & & & 80.84& 0.808\\
1000& 6.292& 35.044& 27.562& 67.07& \\
& & & & & 61.89& 0.619\\
1100& 5.249& 35.004& 27.660& 56.70& \\
& & & & & 54.64& 0.546\\
1200& 4.813& 34.995& 27.705& 52.58& \\
& & & & & 51.74& 0.517\\
1300& 4.554& 34.986& 27.727& 50.90& \\
& & & & & 50.40& 0.504\\
1400& 4.357& 34.977& 27.743& 49.89& \\
& & & & & 49.73& 0.497\\
1500& 4.245& 34.975& 27.753& 49.56& \\
& & & & & 49.30& 1.232\\
1750& 4.028& 34.973& 27.777& 49.03& \\
& & & & & 48.83& 1.221\\
2000& 3.852& 34.975& 27.799& 48.62& \\
& & & & & 47.77& 2.389\\
2500& 3.424& 34.968& 27.839& 46.92& \\
& & & & & 45.94& 2.297\\
3000& 2.963& 34.946& 27.868& 44.96& \\
& & & & & 43.40& 2.170\\
3500& 2.462& 34.920& 27.894& 41.84& \\
& & & & & 41.93& 2.097\\
4000& 2.259& 34.904& 27.901& 42.02\rule[-1ex]{0mm}{1ex}& \\
\hline
\end{tabular*} \\[0.5ex]
\vspace{-3.ex}
\end{table}
\begin{table}[t!]\centering \renewcommand{\baselinestretch}{0.0} \small
\begin{tabular*}{108mm}{@{}rrrrrrl}
\multicolumn{7}{@{}l@{}}{\bfseries Table 10.3 Computation of Relative Geostrophic Currents.} \\
& \multicolumn{6}{@{}l@{}}{\bfseries \rule{0mm}{2.4ex}Data from Endeavor Cruise 88, Station 64} \\
& \multicolumn{6}{@{}l@{}}{\bfseries (37\degrees 39.93'N, 71\degrees 0.00'W; \rule[-1ex]{0mm}{3.5ex}24 August 1982; 0203Z)} \\
\hline
Pressure &t & S &$\sigma (\theta )$\ \ \ &$\delta(S,t,p)$ &$<\delta >$
&$10^{-5}\Delta\Phi$\rule{0mm}{2.5ex} \\ decibar&\degrees C & &kg/m$^3$
&$10^{-8}$m$^3$/kg &$10^{-8}$m$^3$/kg &m$^2$/s$^2$\rule[-1ex]{0mm}{3.5ex} \\
\hline
0& 26.148& 34.646& 22.722& 512.09\rule{0mm}{2.5ex}&\\
& & & & & 512.15& 0.051\\
1& 26.148& 34.646& 22.722& 512.21&\\
& & & & & 512.61& 0.461\\
10& 26.163& 34.645& 22.717& 513.01&\\
& & & & & 512.89& 0.513\\
20& 26.167& 34.655& 22.724& 512.76&\\
& & & & & 466.29& 0.466\\
30& 25.640& 35.733& 23.703& 419.82&\\
& & & & & 322.38& 0.645\\
50& 18.967& 35.944& 25.755& 224.93&\\
& & & & & 185.56& 0.464\\
75& 15.371& 35.904& 26.590& 146.19&\\
& & & & & 136.18& 0.340\\
100& 14.356& 35.897& 26.809& 126.16&\\
& & & & & 120.91& 0.302\\
125& 13.059& 35.696& 26.925& 115.66&\\
& & & & & 111.93& 0.280\\
150& 12.134& 35.567& 27.008& 108.20&\\
& & & & & 100.19& 0.501\\
200& 10.307& 35.360& 27.185& 92.17&\\
& & & & & 87.41& 0.437\\
250& 8.783& 35.168& 27.290& 82.64&\\
& & & & & 79.40& 0.397\\
300& 8.046& 35.117& 27.364& 76.16&\\
& & & & & 66.68& 0.667\\
400& 6.235& 35.052& 27.568& 57.19&\\
& & & & & 52.71& 0.527\\
500& 5.230& 35.018& 27.667& 48.23&\\
& & & & & 46.76& 0.468\\
600& 5.005& 35.044& 27.710& 45.29&\\
& & & & & 44.67& 0.447\\
700& 4.756& 35.027& 27.731& 44.04&\\
& & & & & 43.69& 0.437\\
800& 4.399& 34.992& 27.744& 43.33&\\
& & & & & 43.22& 0.432\\
900& 4.291& 34.991& 27.756& 43.11&\\
& & & & & 43.12& 0.431\\
1000& 4.179& 34.986& 27.764& 43.12&\\
& & & & & 43.10& 0.431\\
1100& 4.077& 34.982& 27.773& 43.07&\\
& & & & & 43.12& 0.431\\
1200& 3.969& 34.975& 27.779& 43.17&\\
& & & & & 43.28& 0.433\\
1300& 3.909& 34.974& 27.786& 43.39&\\
& & & & & 43.38& 0.434\\
1400& 3.831& 34.973& 27.793& 43.36&\\
& & & & & 43.31& 0.433\\
1500& 3.767& 34.975& 27.802& 43.26&\\
& & & & & 43.20& 1.080\\
1750& 3.600& 34.975& 27.821& 43.13&\\
& & & & & 43.00& 1.075\\
2000& 3.401& 34.968& 27.837& 42.86&\\
& & & & & 42.13& 2.106\\
2500& 2.942& 34.948& 27.867& 41.39&\\
& & & & & 40.33& 2.016\\
3000& 2.475& 34.923& 27.891& 39.26&\\
& & & & & 39.22& 1.961\\
3500& 2.219& 34.904& 27.900& 39.17&\\
& & & & & 40.08& 2.004\\
4000& 2.177& 34.896& 27.901& 40.98\rule[-1ex]{0mm}{1ex} \\
\hline
\end{tabular*} \\[0.5ex]
\vspace{-3ex}
\end{table}
\begin{table}[t!]\centering \renewcommand{\baselinestretch}{0.0} \small
\begin{tabular*}{97mm}{@{}rrrrrrl}
\multicolumn{6}{@{}l@{}}{\bfseries Table 10.4 Computation of Relative Geostrophic Currents.} \\
& \multicolumn{5}{@{}l@{}}{\bfseries Data from Endeavor Cruise 88, Station 61 and 64}\rule[-1ex]{0mm}{3.5ex}\\
\hline
Pressure &$10^{-5}\Delta\Phi_{61}$ & $\rule{0mm}{2.5ex}\Sigma\Delta\Phi $ &$10^{-5}\Delta\Phi_{64}$
&$\Sigma\Delta\Phi $ & V \\
decibar\rule[-1ex]{0mm}{4.5ex}&m$^2$/s$^2$ &at 61$^\ast$ &m$^2$/s$^2$ &at 64$^\ast$&(m/s) \\
\hline
0& &2.1872 & &1.2583 &0.95\rule{0mm}{2.5ex}\\
& 0.046 & & 0.051 \\
1& &2.1826 & &1.2532 &0.95\\
& 0.396 & & 0.461 \\
10& &2.1430 & &1.2070& 0.96\\
& 0.423 & & 0.513 \\
20& &2.1006 & &1.1557& 0.97\\
& 0.424 & & 0.466 \\
30& & 2.0583& &1.1091& 0.97\\
& 0.752 & & 0.645 \\
50& & 1.9830& & 1.0446 &0.96\\
& 0.755 & & 0.464 \\
75& & 1.9075& & 0.9982 &0.93\\
& 0.644 & & 0.340 \\
100& & 1.8431& & 0.9642& 0.90\\
& 0.574 & & 0.302 \\
125& & 1.7857& & 0.9340& 0.87\\
& 0.522 & & 0.280 \\
150& & 1.7335& & 0.9060& 0.85\\
& 0.948 & & 0.501 \\
200& & 1.6387& & 0.8559& 0.80\\
& 0.894 & & 0.437 \\
250& & 1.5493& & 0.8122& 0.75\\
& 0.871 & & 0.397 \\
300& & 1.4623& & 0.7725& 0.71\\
& 1.704 & & 0.667 \\
400& & 1.2919& & 0.7058& 0.60\\
& 1.668 & & 0.527 \\
500& & 1.1252& & 0.6531& 0.48\\
& 1.588 & & 0.468 \\
600& & 0.9664& & 0.6063& 0.37\\
& 1.432 & & 0.447 \\
700& & 0.8232& & 0.5617& 0.27\\
& 1.242 & & 0.437 \\
800& & 0.6990& & 0.5180& 0.19\\
& 1.045 & & 0.432 \\
900& & 0.5945& & 0.4748& 0.12\\
& 0.808 & & 0.431 \\
1000& & 0.5137& & 0.4317& 0.08\\
& 0.619 & & 0.431 \\
1100& & 0.4518& & 0.3886& 0.06\\
& 0.546 & & 0.431 \\
1200& & 0.3972& & 0.3454& 0.05\\
& 0.517 & & 0.433 \\
1300& & 0.3454& & 0.3022& 0.04\\
& 0.504 & & 0.434 \\
1400& & 0.2950& & 0.2588& 0.04\\
& 0.497 & & 0.433 \\
1500& & 0.2453& & 0.2155& 0.03\\
& 1.232 & & 1.080 \\
1750& & 0.1221& & 0.1075& 0.01\\
& 1.221 & & 1.075 \\
2000& & 0.0000& & 0.0000& 0.00\\
& 2.389 & & 2.106 \\
2500& & -0.2389& & -0.2106& -0.03\\
& 2.297 & & 2.016 \\
3000& & -0.4686& & -0.4123& -0.06\\
& 2.170 & & 1.961 \\
3500& & -0.6856& & -0.6083& -0.08\\
& 2.097 & & 2.004 \\
4000& & -0.8952& & -0.8087& -0.09\rule[-1ex]{0mm}{1ex}\\
\hline
\multicolumn{6}{@{}l@{}}{$\ast$ Geopotential anomaly integrated from 2000 dbar level.\rule{0mm}{2.5ex}}
\\
\multicolumn{6}{@{}l@{}}{\ \ \ Velocity \rule{0mm}{2.5ex}is calculated from
(10.17)}\\
\end{tabular*} \\[0.5ex]
\vspace{-5ex}
\end{table}
\begin{figure}[t!]
%\centering
\makebox[120mm][c]{\includegraphics{profileandsection}}
\footnotesize
Figure 10.8 \textbf{Left} Relative current as a function of depth\rule{0mm}{4ex}
calculated from hydrographic\index{hydrographic data!from Endeavor} data collected
by the \textit{Endeavor} cruise south of Cape Cod in August 1982. The Gulf Stream\index{Gulf
Stream!cross section of} is the fast current shallower than 1000 decibars. The assumed depth of
no motion is at 2000 decibars.
\textbf{Right} Cross section of potential density $\sigma_{\theta}$ across the
Gulf Stream along 63.66\degrees W calculated from \textsc{ctd}\index{CTD} data collected from
\textit{Endeavor} on 25--28 April 1986. The Gulf Stream is centered on the
steeply sloping contours shallower than 1000m between 40\degrees\ and 41\degrees.
Notice that the vertical scale is 425 times the horizontal scale. (Data contoured
by Lynn Talley, Scripps Institution of Oceanography).
\label{profileandsection}
\vspace{-3ex}
\end{figure}
\vspace{-2ex}
\section{Comments on Geostrophic Currents}
Now that we know how to calculate geostrophic currents\index{geostrophic currents!comments on}
from hydrographic data\index{hydrographic data!and geostrophic currents}, let's consider some
of the limitations of the theory and techniques.
\paragraph{Converting Relative Velocity to Velocity}
\index{geostrophic currents!relative to the earth}Hydrographic data give geo\-stro\-phic
currents relative to geostrophic currents\index{geostrophic currents!relative}
at some reference level. How can we convert the relative geostrophic velocities to velocities
relative to the earth?
\begin{enumerate}
\vitem \textit{Assume a Level of no Motion}: Traditionally, oceanographers assume there
is a \textit{\textit{level of no motion}}, \index{reference surface|textbf}sometimes
called a \textit{\textit{reference surface}}, roughly 2,000 m below the surface. This is
the assumption used to derive the currents in table 10.4. Currents are assumed to be
zero at this depth, and relative currents are integrated up to the surface and down to
the bottom to obtain current velocity as a function of depth. There is some experimental
evidence that such a level exists on average for mean currents (see for example, Defant,
1961: 492).
Defant recommends choosing a reference level where the current shear in the vertical is
smallest. This is usually near 2 km. This leads to useful maps of surface currents
because surface currents tend to be faster than deeper currents. Figure 10.9 shows the
geopotential anomaly and surface currents in the Pacific relative to the 1,000 dbar
pressure level.
\begin{figure}[t!]
\makebox[120mm][c]{\includegraphics{wyrtkiplot}}
\footnotesize
Figure 10.9. Mean \rule{0mm}{4ex}geopotential anomaly relative to the 1,000
dbar surface in the Pacific based on 36,356 observations. Height of the anomaly is in
geopotential centimeters. If the velocity at 1,000 dbar were zero, the map would be the
surface topography of the Pacific. After Wyrtki (1979).
\label{fig:wyrtkiplot}
\vspace{-4ex}
\end{figure}
\vitem \textit{Use known currents:} The known currents could be measured by
current meters or by satellite altimetry. Problems arise if the currents are not
measured at the same time as the hydrographic data\index{hydrographic data!and geostrophic
currents}. For example, the hydrographic data may have been collected over a period of months
to decades, while the currents may have been measured over a period of only a few months.
Hence, the hydrography may not be consistent with the current measurements.
Sometimes currents and hydrographic data are measured at nearly the same
time (figure 10.10). In this example, currents were measured continuously
by moored current meters (points) in a deep western boundary current and calculated from
\textsc{ctd}\index{CTD} data taken just after the current meters were deployed and just before
they were recovered (smooth curves). The solid line is the current assuming a level
of no motion at 2,000 m, the dotted line is the current adjusted using the current
meter observations smoothed for various intervals before or after the \textsc{ctd}
casts.
\begin{figure}[t!]
\makebox[120mm][c]{\includegraphics{whitplot}}
\footnotesize
Figure 10.10 Current \rule{0mm}{3ex }meter measurements
can be used with \textsc{ctd}\index{CTD} measurements to determine current as a function of
depth avoiding the need for assuming a depth of no motion. Solid line: profile
assuming a depth of no motion at 2000 decibars. Dashed line: profile
adjusted to agree with currents measured by current meters 1--7 days before the
\textsc{ctd} measurements. (Plots from Tom Whitworth, Texas A\&M University)
\label{fig:whitplot}
\vspace{-3ex}
\end{figure}
\vitem \textit{Use Conservation Equations}: Lines of hydrographic stations\index{hydrographic
stations} across a strait or an ocean basin may be used with conservation
of mass and salt to calculate currents. This is an example of an inverse problem (Wunsch,
1996 describes the application of inverse methods in oceanography). See Mercier et al. (2003)
for a description of how they determined the circulation in the upper layers of the eastern
basins of the south Atlantic using hydrographic data from the World Ocean Circulation
Experiment and direct measurements of current in a box model\index{box model} constrained by inverse
theory.
\end{enumerate}
\paragraph{Disadvantage of Calculating Currents from Hydrographic Data}
\index{hydrographic data!disadvantage of}Currents calculated from hydrographic
data\index{hydrographic data!and geostrophic currents} have been used to make maps of ocean currents
since the early 20th century. Nevertheless, it is important to review
the limitations of the technique.
\begin{enumerate}
\vitem Hydrographic data\index{hydrographic data!and geostrophic currents} can be used to
calculate only the current relative to a current at another level.
\vitem The assumption of a level of no motion may be suitable in the deep ocean,
but it is usually not a useful assumption when the water is shallow such
as over the continental shelf.
\vitem Geostrophic currents cannot be calculated from hydrographic stations\index{hydrographic
data!and geostrophic currents} that are close together. Stations must be tens of kilometers
apart.
\end{enumerate}
\paragraph{Limitations of the Geostrophic Equations}
\index{geostrophic equations!limitations of}I began this section\index{geostrophic
balance!limitations of} by showing that the geostrophic balance applies with good
accuracy\index{accuracy!equations!geostrophic} to flows that exceed a few tens of kilometers
in extent and with periods greater than a few days. The balance cannot, however, be perfect.
If it were, the flow in the ocean would never change because the balance ignores any
acceleration of the flow. The important limitations of the geostrophic
assumption\index{geostrophic approximation} are:
\begin{enumerate}
\vitem Geostrophic currents\index{geostrophic currents!cannot change} cannot evolve with time because the
balance ignores acceleration of the flow. Acceleration dominates if the horizontal dimensions are less than
roughly 50 km and times are less than a few days. Acceleration is negligible, but not zero, over longer
times and distances.
\vitem The geostrophic balance\index{geostrophic balance!not near equator} does not apply within about
2\degrees\ of the equator where the Coriolis force goes to zero because $\sin \varphi \rightarrow 0$.
\vitem The geostrophic balance\index{geostrophic balance!ignores friction} ignores the
influence of friction.
\end{enumerate}
\paragraph{Accuracy} Strub et al. (1997) showed that
currents\index{geostrophic currents!measured by altimetry} calculated from satellite altimeter
measurements of sea-surface slope have an accuracy\index{accuracy!altimeter} of
$\pm$3--5 cm/s. Uchida, Imawaki, and Hu (1998) compared currents measured by
drifters\index{drifters!in Kuroshio} in the Kuroshio\index{Kuroshio!geostrophic balance in}
with currents calculated from satellite altimeter data assuming
geostrophic balance. Using slopes over distances of 12.5 km, they found the difference between
the two measurements was
$\pm$16 cm/s for currents up to 150 cm/s, or about 10\%. Johns, Watts, and Rossby (1989)
measured the velocity of the Gulf Stream\index{Gulf Stream!northeast of Cape Hatteras}
northeast of Cape Hatteras and compared the measurements with velocity calculated from
hydrographic data\index{hydrographic data!and geostrophic currents} assuming geostrophic
balance. They found that the measured velocity in the core of the stream, at depths less than
500 m, was 10--25 cm/s faster than the velocity calculated from the geostrophic equations
using measured velocities at a depth of 2000 m\index{geostrophic currents!and level of no
motion}\index{geostrophic currents!in Gulf Stream}. The maximum velocity in the core was
greater than 150 cm/s, so the error was $\approx 10$\%. When they added the influence of the
curvature of the Gulf Stream, which adds an acceleration term to the geostrophic equations, the
difference in the calculated and observed velocity dropped to less than 5--10 cm/s ($\approx
5$\%).
\section{Currents From Hydrographic Sections}
\index{hydrographic sections}Lines of hydrographic data along ship tracks are often used to
produce contour plots of density in a vertical section along the track. Cross-sections of
currents sometimes show sharply dipping density surfaces with a large contrast in density on
either side of the current. The baroclinic currents in the section can be estimated using a
technique first proposed by Margules (1906) and described by Defant (1961: 453). The
technique allows oceanographers to estimate the speed and direction of currents perpendicular
to the section by a quick look at the section.
To derive Margules' equation, consider the slope $\partial z/\partial x$ of a stationary
interface between two water masses with densities $\rho_1$ and $\rho_2$ (see
figure 10.11). To calculate the change in velocity across the interface we assume homogeneous
layers of density $\rho_1 < \rho_2$ both of which are in geostrophic
equilibrium\index{geostrophic currents!from slope of density surfaces}. Although the ocean does
not have an idealized interface that we assumed, and the water masses do not have uniform
density, and the interface between the water masses is not sharp, the concept is still useful
in practice.
\begin{figure}[t!]
%\vspace{-3ex}
%\centering
\makebox[120mm][c]{\includegraphics{Fig10-10}}
\footnotesize
Figure 10.11 Slopes $\beta$ of the \rule{0mm}{4ex}sea surface and the slope
$\gamma$ of the interface between two homogeneous, moving layers, with density
$\rho_1$ and $\rho_2$ in the northern hemisphere. After Neumann and Pierson (1966: 166)
\label{fig:Fig10-10}
\vspace{-3ex}
\end{figure}
The change in pressure on the interface is:
\begin{equation}
\delta p = \frac{\partial p}{\partial x}\,\delta x + \frac{\partial p}{\partial
z}\, \delta z ,
\end{equation}
and the vertical and horizontal pressure gradients are obtained from (10.6):
\begin{equation}
\frac{\partial p}{\partial z}= - \rho_1 g + \rho_1 f v_1
\end{equation}
Therefore:
\begin{subequations}
\begin{align}
\delta p_1&=-\rho_1fv_1 \, \delta x + \rho_1 g \, \delta z \\
\delta p_2&=-\rho_2fv_2 \, \delta x + \rho_2 g \, \delta z \\ \notag
\end{align}
\end {subequations}
The boundary conditions require $\delta p_1 = \delta p_2$ on the interface if the interface is
not moving. Equating (10.20a) with (10.20b), dividing by $\delta x$, and solving for $\delta
z/\delta x$ gives:
\begin{displaymath}
\frac{\delta z}{\delta x}\equiv \tan \gamma =\frac{f}{g}\left(\frac{\rho_2\,v_2
- \rho_1\,v_1}{\rho_2 -\rho_1}\right)
\end{displaymath}
Because $\rho_1 \approx \rho_2$, and for small $\beta$ and $\gamma$,
\begin{subequations}
\begin{align}
\tan \gamma &\approx \frac{f}{g}\left(\frac{\rho_1}{\rho_2 - \rho_1}\right)(v_2-v_1) \\
\tan \beta_1&=-\frac{f}{g}\, v_1 \\
\tan \beta_2&=-\frac{f}{g}\, v_2
\end{align}
\end {subequations}
where $\beta$ is the slope of the sea surface, and $\gamma$ is the slope of the
interface between the two water masses. Because the internal differences in density are
small, the slope is approximately 1000 times larger than the slope of the constant
pressure surfaces.
Consider the application of the technique to the Gulf Stream\index{Gulf Stream} (figure
10.8). From the figure: $\varphi = 36$\degrees, $\rho_1 = 1026.7$ kg/m$^3$, and
$\rho_2 = 1027.5$ kg/m$^3$ at a depth of 500 decibars. If we use the $\sigma_t =
27.1$ surface to estimate the slope between the two water masses, we see that the
surface changes from a depth of 350 m to a depth of 650 m over a distance of 70
km. Therefore,
$\tan
\gamma = 4300 \times 10^{-6} = 0.0043$, and
$\Delta v = v_2 - v_1 = -0.38$ m/s. Assuming $v_2 = 0$, then $v_1 = 0.38$ m/s.
This rough estimate of the velocity of the Gulf Stream\index{Gulf Stream!velocity of} compares
well with velocity at a depth of 500m calculated from hydrographic data\index{hydrographic
data!across Gulf Stream} (table 10.4) assuming a level of no motion at 2,000 decibars.
The slope of the constant-density surfaces are clearly seen in figure 10.8. And
plots of constant-density surfaces can be used to quickly estimate current
directions and a rough value for the speed. In contrast, the slope of the sea
surface is $8.4 \times 10^{-6}$ or 0.84 m in 100 km if we use data from table
10.4.
Note that constant-density surfaces in the Gulf Stream\index{Gulf Stream!density surfaces}
slope downward to the east, and that sea-surface topography slopes upward to the east. Constant
pressure and constant density surfaces have opposite slope.
If the sharp interface between two water masses reaches the surface, it is an
oceanic front, which has properties that are very similar to atmospheric
fronts.
Eddies in the vicinity of the Gulf Stream\index{Gulf Stream!eddies} can have warm or cold cores
(figure 10.12). Application of Margules' method to these mesoscale eddies\index{mesoscale
eddies} gives the direction of the flow. Anticyclonic eddies (clockwise rotation in the
northern hemisphere) have warm cores ($\rho_1$ is deeper in the center of the eddy than
elsewhere) and the constant-pressure surfaces bow upward. In particular, the sea
surface is higher at the center of the ring. Cyclonic eddies are the reverse\index{geostrophic
currents!from hydrographic data|)}.
\begin{figure}[h!]
\vspace{-1ex}
\makebox[120mm][c]{\includegraphics{rings}}
\footnotesize
Figure 10.12 Shape of constant-pressure \rule{0mm}{4ex}surfaces $p_i$ and the
interface between two water masses of density $\rho_1, \rho_2$ if the upper is
rotating faster than the lower.
\textbf{Left:} Anticyclonic motion, warm-core eddy. \textbf{Right:} Cyclonic,
cold-core eddy. Note that the sea surface $p_0$ slopes up toward the center of the
warm-core ring, and the constant-density surfaces slope down toward the center. Circle with
dot is current toward the reader, circle with cross is current away from the reader. After
Defant (1961: 466).
\label{fig:rings}
\vspace{-4ex}
\end{figure}
\section{Lagrangian Measurements of Currents}
\index{Lagrangian measurements}Oceanography and fluid mechanics distinguish between
two techniques for measuring currents: Lagrangian and Eulerian. Lagrangian techniques
follow a water particle. Eulerian techniques measure the velocity of water at a fixed
position.
\paragraph{Basic Technique}
Lagrangian techniques track the position of a drifter designed to follow a water
parcel either on the surface or deeper within the water column. The mean
velocity over some period is calculated from the distance between positions at
the beginning and end of the period divided by the period. Errors are due to:
\begin{enumerate}
\vitem The failure of the drifter to follow a parcel of water. We assume the
drifter stays in a parcel of water, but wind blowing on the surface float of a surface drifter can cause the drifter to move relative to the water.
\vitem Errors in determining the position of the drifter.
\vitem Sampling errors. Drifters go only where drifters\index{drifters!accuracy of
current measurements} want to go. And drifters want to go to convergent zones. Hence drifters
tend to avoid areas of divergent flow.
\end{enumerate}
\begin{figure}[t!]
%\vspace{-2ex}
\makebox[120mm][c]{\includegraphics{argos}}
\footnotesize
Figure 10.13 System\rule{0mm}{4ex} Argos\index{Argos system} uses radio signals transmitted from surface buoys to determine the position of the buoy. A satellite receives the signal from the buoy B. The time rate of change of the signal, the Doppler shift $F$, is a function of buoy position and distance from the satellite's track. Note that a buoy at BB would produce the same Doppler shift as the buoy at B. The recorded Doppler signal is transmitted to ground stations E, which relays the information to processing centers A via control stations K. After Dietrich et al. (1980: 149).
\label{fig:argos}
\vspace{-4ex}
\end{figure}
\paragraph{Satellite Tracked Surface Drifters}
\index{Lagrangian measurements!satellite tracked surface drifters}\index{satellite tracked
surface drifters}\index{drifters}Surface drifters consist of a drogue plus a float. Its position is determined by the Argos\index{Argos system} system on meteorological satellites (Swenson and Shaw, 1990) or calculated from \textsc{gps} data recorded continuously by the buoy and relayed to shore.
Argos-tracked buoys carry a radio transmitter with a very stable frequency $F_0$. A receiver on the
satellite receives the signal and determines the Doppler shift $F$ as a function of time $t$
(figure 10.13). The Doppler frequency is
\begin{displaymath}
F=\frac{dR}{dt}\,\frac{F_0}{c} + F_0
\end{displaymath}
where $R$ is the distance to the buoy, $c$ is the velocity of light. The closer the buoy to the satellite the more rapidly the frequency changes. When $F = F_0$ the range is a minimum. This is the time of closest approach, and the satellite's velocity vector is perpendicular to the line from the satellite to the buoy. The time of closest approach and the time rate of change of Doppler frequency at that time gives the buoy's position relative to the orbit with a 180\degrees\ ambiguity (B and BB in the figure). Because the orbit is accurately known, and because the buoy can be observed many times, its position can be determined without ambiguity.
The accuracy\index{accuracy!Argos} of the calculated position depends on the stability of the
frequency transmitted by the buoy. The Argos system\index{Argos system} tracks buoys with an
accuracy\index{drifters!accuracy of current measurements} of $\pm$(1--2) km, collecting 1--8
positions per day depending on latitude. Because 1 cm/s $\approx$ 1 km/day, and because
typical values of currents in the ocean range from one to two hundred centimeters per second,
this is an very useful accuracy.
\paragraph{Holey-Sock Drifters}
\index{Lagrangian measurements!holey-sock drifters}\index{drifters!holey-sock}The most widely used, satellite-tracked drifter is the holey-sock drifter. It consists of a cylindrical drogue of cloth 1 m in diameter by 15 m long with 14 large holes cut in the sides. The weight of the drogue is supported by a float set 3 m below the surface. The submerged float is tethered to a partially submerged surface float carrying the Argos\index{Argos system} transmitter.
The buoy was designed for the Surface Velocity Program and extensively tested. Niiler et al. (1995) carefully measured the rate at which wind blowing on the
surface float pulls the drogue through the water, and they found that the
buoy moves $12\pm9$\degrees\ to the right of the wind at a speed
\begin{equation}
U_s = \left( 4.32\pm 0.67 \times\right) 10^{-2} \frac{U_{10}}{DAR} +
\left( 11.04\pm 1.63 \right) \frac{D}{DAR}
\end{equation}
where $DAR$ is the drag area ratio defined as the drogue's drag area divided by
the sum of the tether's drag area and the surface float's drag area, and $D$ is
the difference in velocity of the water between the top of the cylindrical drogue
and the bottom. Drifters typically have a $DAR$ of 40, and the drift $U_s < 1$ cm/s for $U_{10} < 10$
m/s.
\paragraph{Argo Floats}
The most widely used subsurface floats are the Argo floats.\index{floats!Argo} The floats (figure 10.14) are designed to cycle between the surface and some
predetermined depth. Most floats drift for 10 days at a depth of 1 km, sink to 2 km, then rise to the surface. While rising, they profile temperature and salinity as a function of pressure (depth). The floats remains on the surface for a few hours, relays data to shore via the Argos system\index{Argos system}, then sink again to 1 km. Each float carries enough power to repeat this cycle for several years. The float thus measures currents at 1 km depth and density distribution in the upper ocean. Three thousand Argo floats are being deployed in all parts of the ocean for the Global Ocean Data Assimilation Experiment \textsc{godae}\index{Global Ocean Data Assimilation Experiment!floats}.\index{floats}
\begin{figure}[h!]
\vspace{-2ex}
\makebox[120mm][c]{\includegraphics{alace}}
\footnotesize
Figure 10.14 The Autonomous \rule{0mm}{4ex}Lagrangian Circulation Explorer (ALACE) floats\index{floats!ALACE} is the prototype for the Argos floats. It measures currents at a depth of 1 km. \textbf{Left:} Schematic of the drifter. To ascend, the hydraulic pump moves oil from an
internal reservoir to an external bladder, reducing the drifter's density. To descend, the
latching valve is opened to allow oil to flow back into the internal reservoir.
The antenna is mounted to the end cap. \textbf{Right:} Expanded schematic of the
hydraulic system. The motor rotates the wobble plate actuating the piston which
pumps hydraulic oil. After Davis et al. (1992).
\label{fig:alace}
\vspace{-2ex}
\end{figure}
\paragraph{Lagrangian Measurements Using Tracers}
\index{Lagrangian measurements!tracers}\index{tracers}The most common method for measuring the flow in the deep ocean is to track parcels of water containing molecules not normally found in the ocean.
Thanks to atomic bomb tests in the 1950s and the recent exponential increase
of chlorofluorocarbons in the atmosphere, such tracers have been introduced into the
ocean in large quantities. See \S 13.4 for a list of tracers used in oceanography. The
distribution of trace molecules is used to infer the movement of the water. The
technique is especially useful for calculating velocity of deep water masses averaged
over decades and for measuring turbulent mixing discussed in \S 8.4.
\begin{figure}[t!]
\makebox[120mm][c]{\includegraphics{tritium}}
\footnotesize
Figure 10.15 Distribution of \rule{0pt}{3ex} tritium along a section through the western basins in the north Atlantic, measured in 1972 (\textbf{Top}) and remeasured in 1981 (\textbf{Bottom}). Units are tritium units, where one tritium unit is $10^{18}$ (tritium atoms)/(hydrogen atoms) corrected to the activity levels that would have been observed on 1 January 1981. Compare this figure to the density in the ocean shown in figure 13.10. After Toggweiler (1994).
\label{fig:tritium}
\vspace{-5ex}
\end{figure}
The distribution of trace molecules is calculated from the concentration of the
molecules in water samples collected on hydrographic sections\index{hydrographic sections} and
surveys. Because the collection of data is expensive and slow, there are few repeated
sections. Figure 10.15 shows two maps of the distribution of tritium in the north Atlantic
collected in 1972--1973 by the Geosecs Program and in 1981, a decade later. The sections show
that tritium, introduced into the atmosphere during the atomic bomb tests in the atmosphere in
the 1950s to 1972, penetrated to depths below 4 km only north of 40\degrees N by 1971 and to
35\degrees N by 1981. This shows that deep currents are very slow, about 1.6 mm/s in this
example.
Because the deep currents are so small, we can question what process are responsible
for the observed distribution of tracers. Both turbulent diffusion and advection by
currents can fit the observations. Hence, does figure 10.15 give mean currents in
the deep Atlantic, or the turbulent diffusion of tritium?
\begin{figure}[t!]
\makebox[120mm][c]{\includegraphics{Fig10-16-bw}}
\footnotesize
Figure 10.16 Ocean \rule{0mm}{4ex}temperature and current patterns
are combined in this \textsc{avhrr} \index{Advanced Very High Resolution Radiometer
(AVHRR)}analysis. Surface currents were computed by tracking the displacement of small thermal
or sediment features between a pair of images. A directional edge-enhancement filter was
applied here to define better the different water masses. Warm water is shaded darker. From
Ocean Imaging, Solana Beach, California, with permission.
\label{Fig10.16.bw}
\vspace{-4ex}
\end{figure}
Another useful tracer is the temperature and salinity of the water. I will
consider these observations in \S 13.4 where I describe the core method
for studying deep circulation. Here, I note that \textsc{avhrr} \index{Advanced Very High
Resolution Radiometer (AVHRR)}observations of surface temperature of the ocean are an
additional source of information about currents.
Sequential infrared images of surface temperature are used to calculate the
displacement of features in the images (figure 10.16). The technique is
especially useful for surveying the variability of currents near shore. Land
provides reference points from which displacement can be calculated accurately,
and large temperature contrasts can be found in many regions in some seasons.
There are two important limitations.
\begin{enumerate}
\vitem Many regions have extensive cloud cover, and the ocean cannot be seen.
\vitem Flow is primarily parallel to temperature fronts, and strong currents
can exist along fronts even though the front may not move. It is therefore
essential to track the motion of small eddies embedded in the flow near the front
and not the position of the front.
\end{enumerate}
\begin{figure}[b!]
\vspace{-3ex}
\makebox[120mm][c]{\includegraphics{duckies}}
\footnotesize
Figure 10.17 Trajectories that \rule{0mm}{5ex}spilled rubber duckies
would have followed had they been spilled on January 10 of different years. Five
trajectories were selected from a set of 48 simulations of the spill each year
between 1946 and 1993. The trajectories begin on January 10 and end two years
later (solid symbols). Grey symbols indicate positions on November 16 of the year
of the spill. The grey circle gives the location where rubber
ducks first came ashore near Sitka in 1992. The code at lower left gives the dates of the
trajectories. After Ebbesmeyer and Ingraham (1994).
\label{fig:duckies}
%\vspace{-2ex}
\end{figure}
\paragraph{The Rubber Duckie Spill}
\index{Rubber Duckie Spill}On January 10, 1992 a 12.2-m container with 29,000 bathtub
toys, including rubber ducks (called rubber duckies by children) washed overboard from a container ship at 44.7\degrees N, 178.1\degrees E (figure 10.17). Ten months later the toys began washing ashore near Sitka, Alaska. A similar accident on May 27, 1990 released 80,000 Nike-brand shoes at 48\degrees N, 161\degrees W when waves washed containers from the \textit{Hansa Carrier}.
The spills and eventual recovery of the toys and shoes proved to be good tests of a
numerical model for calculating the trajectories of oil spills developed by Ebbesmeyer
and Ingraham (1992, 1994). They calculated the possible trajectories of the spilled
toys using the Ocean Surface Current Simulations \textsc{oscurs} numerical model
driven by winds calculated from the Fleet Numerical Oceanography Center's daily
sea-level pressure data. After modifying their calculations by increasing the windage
coefficient by 50\% for the toys and by decreasing their angle of deflection function by 5\degrees , their calculations accurately predicted the arrival of the toys\index{drifters!rubber duckie} near Sitka, Alaska on November 16, 1992, ten months after the spill.
\begin{figure}[b!]
\vspace{-3ex}
\makebox[120mm][c]{\includegraphics{moorings}}
\footnotesize
Figure 10.18 \textbf{Left:} An example \rule{0mm}{3ex}of a surface
mooring of the type deployed by the Woods Hole Oceanographic Institution's Buoy
Group.
\textbf{Right:} An example of a subsurface mooring deployed by the same group.
After Baker (1981: 410--411).
\label{fig:moorings}
%\vspace{-3ex}
\end{figure}
\section{Eulerian Measurements}
\index{Eulerian measurements}Eulerian measurements are made by many different
types of instruments on ships and moorings.
Moorings (figure 10.18) are placed on the sea floor by ships. The moorings may last for months to longer than a year. Because the mooring must be deployed and recovered by deep-sea research ships, the technique is expensive and few moorings are now being deployed. The subsurface mooring shown on the right in the figure is preferred for several reasons: it does not have a surface float that is forced by high frequency, strong, surface currents; the mooring is out of sight and it does not attract the attention of fishermen; and the floatation is usually deep enough to avoid being caught by fishing nets. Measurements made from moorings have errors due to:
\begin{enumerate}
\vitem Mooring motion. Subsurface moorings move least. Surface moorings in strong
currents move most, and are seldom used.
\vitem Inadequate Sampling. Moorings tend not to last long enough to give
accurate estimates of mean velocity or interannual variability of the velocity.
\vitem Fouling of the sensors by marine organisms, especially instruments
deployed for more than a few weeks close to the surface.
\end{enumerate}
\paragraph{Acoustic-Doppler Current Meters and Profilers}
\index{Eulerian measurements!acoustic-doppler current
profiler}\index{acoustic-doppler current profiler}The most common Eulerian measurements of currents are made using sound. Typically, the current meter or profiler transmits sound in three or four narrow beams pointed in different directions. Plankton and tiny bubbles reflect the sound back to the instrument. The Doppler shift of the reflected sound is proportional to the radial component of the velocity of whatever reflects the sound. By combining data from three or four beams, the horizontal velocity of the current is calculated assuming the bubbles and plankton do not move very fast relative to the water.
Two types of acoustic current meters are widely used. The Acoustic-Doppler Current Profiler, called the \textsc{adcp}, measures the Doppler shift of sound reflected from water at various distances from the instrument using sound beams projected into the water just as a radar measures radio scatter as a function of range using radio beams projected into the air. Data from the beams are combined to give profiles of current velocity as a function of distance from the instrument. On ships, the beams are pointed diagonally downward at 3--4 horizontal angles relative to the ship's bow. Bottom-mounted meters use beams pointed diagonally upward.
Ship-board instruments are widely used to profile currents within 200 to 300 m of the sea surface while the ship steams between hydrographic stations\index{hydrographic stations!and acoustic Doppler current profiler}. Because a ship moves relative to the bottom, the ship's velocity and orientation must be accurately known. \textsc{gps} data have provided this information since the early 1990s.
Acoustic-Doppler current meters are much simpler than the \textsc{adcp}. They transmit continuous beams of sound to measure current velocity close to the meter, not as a function of distance from the meter. They are placed on moorings and sometimes on a \textsc{ctd}. Instruments on moorings record velocity as a function of time for many days or months. The Aanderaa current meter (figure 10.19) in the figure is an example of this type. Instruments on \textsc{ctd}s\index{CTD} profile currents from the surface to the bottom at hydrographic stations.
\begin{figure}[t!]
%\vspace{-3ex}
\makebox[120mm][c]{\includegraphics{RCM9}}
\footnotesize
Figure 10.19 An example of a \rule{0mm}{5ex}moored acoustic current meter, the \textsc{rcm 9}
produced by Aanderaa Instruments. Two components of horizontal velocity are
measured by an acoustic system, and the directions are referenced to north
using an internal Hall-effect compass. The electronics, data recorder, and battery
are in the pressure-resistant housing. Accuracy\index{accuracy!current meter} is $\pm$0.15
cm/s and
$\pm$5\degrees. (Courtesy Aanderaa Instruments)
\label{fig:RCM9}
\vspace{-2ex}
\end{figure}
\section{Important Concepts}
\begin{enumerate}
\item
Pressure distribution is almost precisely the hydrostatic pressure obtained
by assuming the ocean is at rest. Pressure is therefore calculated very
accurately from measurements of temperature and conductivity as a function of
pressure using the equation of state of seawater. Hydrographic data\index{hydrographic
data!and altimetry} give the relative, internal pressure field of the ocean.
\vitem
Flow in the ocean is in almost exact geostrophic balance\index{geostrophic balance}
except for flow in the upper and lower boundary layers. Coriolis force almost exactly balances the horizontal pressure gradient.
\vitem
Satellite altimetric observations of the oceanic topography give the
surface geostrophic current\index{geostrophic currents!measured by altimetry}. The calculation
of topography requires an accurate geoid\index{geoid}. If the geoid\index{geoid} is not known, altimeters can measure the change in topography as a function of time, which gives the change in surface geostrophic currents.
\vitem
Topex/Poseidon\index{Topex/Poseidon} and Jason\index{Jason} are the most accurate
altimeter systems, and they can measure the topography\index{topography!measured by altimetry}
or changes in topography with an accuracy\index{accuracy!topography} of
$\pm$4 cm.
\vitem
Hydrographic data\index{hydrographic data!and geostrophic currents} are used to
calculate the internal geostrophic currents\index{geostrophic currents!relative to level of no
motion} in the ocean relative to known currents at some level. The level can be surface
currents measured by altimetry or an assumed level of no motion at depths below 1--2 km.
\vitem
Flow in the ocean that is independent of depth is called barotropic flow,
flow that depends on depth is called baroclinic flow. Hydrographic data\index{hydrographic
data!and geostrophic currents} give only the baroclinic flow.
\vitem Geostrophic flow cannot change with time, so the flow in the ocean is not
exactly geostrophic. The geostrophic method does not apply to flows at the
equator where the Coriolis force vanishes.
\vitem Slopes of constant density or temperature surfaces seen in a
cross-section of the ocean can be used to estimate the speed of flow through the
section.
\vitem Lagrangian techniques measure the position of a parcel of water in the
ocean. The position can be determined using surface drifters or subsurface
floats\index{drifters}, or chemical tracers such as tritium.
\vitem Eulerian techniques measure the velocity of flow past a point in the ocean.
The velocity of the flow can be measured using moored current meters or acoustic
velocity profilers on ships, \textsc{ctd}s\index{CTD} or moorings.
\end{enumerate}
\chapter{Wind Driven Ocean Circulation}
What drives the ocean currents? At first, we might answer, the winds. But if we think more carefully about the question, we might not be so sure. We might notice, for example, that strong currents, such as the North Equatorial Countercurrents in the Atlantic and Pacific Ocean go upwind. Spanish navigators in the 16th century noticed strong northward currents along
the Florida coast that seemed to be unrelated to the wind. How can this happen? And, why are strong currents found offshore of east coasts but not offshore of west coasts?
Answers to the questions can be found in a series of three remarkable papers published from 1947 to 1951. In the first, Harald Sverdrup (1947) showed that the circulation in the upper kilometer or so of the ocean is directly related to the curl of the wind stress\index{wind stress!curl of} if the Coriolis force varies with latitude. Henry Stommel (1948) showed that the circulation in oceanic gyres is asymmetric also because the Coriolis force varies with latitude. Finally, Walter Munk (1950) added eddy viscosity and calculated the circulation of the upper layers of the Pacific. Together the three oceanographers laid the foundations for a modern theory of ocean circulation.
\section{Sverdrup's Theory of the Oceanic Circulation}
\index{Sverdrup's Theory}\index{oceanic
circulation!Sverdrup's Theory}\index{circulation!Sverdrup's Theory}While Sverdrup was analyzing observations of equatorial currents, he came upon (11.6) below relating the curl of the wind stress\index{wind stress!curl of} to mass transport\index{transport!Sverdrup} within the upper ocean. To derive the relationship, Sverdrup assumed that the flow is stationary, that lateral friction and molecular viscosity are small, that non-linear terms such as $u\, \partial{u} / \partial{x}$ are small, and that turbulence\index{turbulent!stress} near the sea surface can be described using a vertical eddy viscosity. He also assumed that the wind-driven circulation vanishes at some depth of no motion. With these assumptions, the horizontal components of the momentum equation from 8.9 and 8.12 become:
\begin{subequations}
\begin{equation}
\frac{\partial{p}}{\partial{x}} = f\,\rho\,v + \frac{\partial{T_{xz}}}{\partial{z}}
\end{equation}
\begin{equation}
\frac{\partial{p}}{\partial{y}} = -f\,\rho\,u + \frac{\partial{T_{yz}}}{\partial{z}}
\end{equation}
\end{subequations}
Sverdrup integrated these equations from the surface to a depth $-D$ equal
to or greater than the depth at which the horizontal pressure gradient becomes
zero. He defined:
\begin{subequations}
\begin{alignat}{2}
\frac{\partial{P}}{\partial{x}} &= \int\limits_{-D}^{0} \frac{\partial{p}}{\partial{x}}\,dz, &\qquad
\frac{\partial{P}}{\partial{y}} &= \int\limits_{-D}^{0} \frac{\partial{p}}{\partial{y}}\,dz, \\
M_x &\equiv \int\limits_{-D}^{0} \rho\,u(z)\,dz, &\qquad M_y &\equiv \int\limits_{-D}^{0} \rho\,v(z)\,dz,
\end{alignat}
\end{subequations}
where $M_x, M_y$ are the mass transports\index{transport!Sverdrup} in the wind-driven layer
extending down to an assumed depth of no motion.
The horizontal boundary condition at the sea surface is the wind stress\index{wind stress}. At depth $-D$ the stress is zero because the currents go to zero:
\begin{alignat}{2}
T_{xz}(0) &= T_x & \qquad T_{xz}(-D) &= 0
\notag \\
T_{yz}(0) &= T_y & \qquad T_{yz}(-D) &= 0
\end{alignat}
where $T_x$ and $T_y$ are the components of the wind stress\index{wind stress!components}.
Using these definitions and boundary conditions, (11.1) become:
\begin{subequations}
\begin{align}
\frac{\partial{P}}{\partial{x}} &= f\,M_y + T_x \\
\frac{\partial{P}}{\partial{y}} &= -f\,M_x + T_y
\end{align}
\end{subequations}
In a similar way, Sverdrup integrated the continuity equation (7.19) over the
same vertical depth, assuming the vertical velocity at the surface and at depth
$-D$ are zero, to obtain:
\begin{equation}
\frac{\partial{M_x}}{\partial{x}} + \frac{\partial{M_y}}{\partial{y}} = 0
\end{equation}
Differentiating (11.4a) with respect to $y$ and (11.4b) with respect to $x$,
subtracting, and using (11.5) gives:
\begin{align}
\beta\,M_y &= \frac{\partial{T_y}}{\partial{x}} - \frac{\partial{T_x}}{\partial{y}}
\notag \\
\beta\,M_y &= \text{curl}_z(T)
\end{align}
where $\beta \equiv \partial{f}/\partial{y}$ is the rate of change of Coriolis
parameter with latitude, and where curl$_z(T)$ is the vertical component
of the curl of the wind stress\index{wind stress!curl of}.
This is an important and fundamental result---the northward mass
transport\index{transport!northward} of wind driven currents is equal to the curl of the wind
stress. Note that Sverdrup allowed $f$ to vary with latitude. We will see later that this is
essential.
We calculate $\beta $ from
\begin{equation}
\beta \equiv \frac{\partial{f}}{\partial{y}} = \frac{2\,\Omega\cos\varphi}{R}
\end{equation}
where $R$ is earth's radius and $\varphi$ is latitude.
Over much of the open ocean, especially in the tropics, the wind is zonal and
$\partial{T_y}/\partial{x}$ is sufficiently small that
\begin{equation}
M_y \approx -\frac{1}{\beta}\,\frac{\partial T_x}{\partial y}
\end{equation}
Substituting (11.8) into (11.5), assuming $\beta$ varies with latitude, Sverdrup obtained:
\begin{equation}
\frac{\partial{M_x}}{\partial{x}}=-
\frac{1}{2\,\Omega\cos\varphi}\left(\frac{\partial{T_x}}{\partial{y}}\tan\varphi +
\frac{\partial^2{T_x}}{\partial{y}^2}R \right)
\end{equation}
\begin{figure}[t!]
%\vspace{-1ex}
\makebox[120mm] [c]{\includegraphics{sverdrupmap}}
\centering
\footnotesize
Figure 11.1 Streamlines of \rule{0mm}{3ex}mass transport\index{transport!in Pacific} in the
eastern Pacific calculated\\from Sverdrup's theory using mean annual wind stress\index{wind
stress!annual average}. After Reid (1948).
\label{fig:sverdrupmap}
\vspace{-3ex}
\end{figure}
Sverdrup integrated this equation from a north-south eastern boundary at $ x =
0$, assuming no flow into the boundary. This requires $M_x = 0$ at $x = 0$. Then
\begin{equation}
M_x =- \frac{\Delta{x}}{2\,\Omega\cos\varphi}\left[ \left<
\frac{\partial{T_x}}{\partial{y}}\right > \tan\varphi +
\left< \frac{\partial^2{T_x}}{\partial{y}^2} \right> R \right]
\end{equation}
where $\Delta x$ is the distance from the eastern boundary of the ocean basin,
and brackets indicate zonal averages of the wind stress\index{wind stress!zonal average}
(figure 11.1).
To test his theory, Sverdrup compared transports\index{transport!Sverdrup} calculated from
known winds in the eastern tropical Pacific with transports calculated from hydrographic
data\index{hydrographic data!from Carnegie} collected by the \textit{Carnegie} and
\textit{Bushnell} in October and November 1928, 1929, and 1939 between 34\degrees N and
10\degrees S and between 80\degrees W and 160\degrees W. The
hydrographic data\index{hydrographic data!and Sverdrup transport} were used to compute
$P$ by integrating from a depth of $D = -1000$ m. The comparison, figures 11.2,
showed not only that the transports can be accurately calculated from the wind,
but also that the theory predicts wind-driven currents going upwind.
\begin{figure}[t!]
\makebox[120mm] [c]{\includegraphics{sverdrup}}
\footnotesize
Figure 11.2 Mass \rule{0mm}{3ex}transport\index{transport!in Pacific} in the eastern Pacific
calculated from Sverdrup's theory using observed winds with 11.8 and 11.10 (solid
lines) and pressure calculated from hydrographic data\index{hydrographic data!and Sverdrup
transport} from ships with 11.4 (dots). Transport is in tons per second through a section one
meter wide extending from the sea surface to a depth of one kilometer. Note the difference in
scale between $M_y$ and $M_x$. After Reid (1948).
\label{fig:windpacific}
\vspace{-4ex}
\end{figure}
\paragraph{Comments on Sverdrup's Solutions}
\begin{enumerate}
\vitem Sverdrup assumed\index{Sverdrup's assumptions} i) The internal flow in the
ocean is geostrophic; ii) there is a uniform depth of no motion; and iii) Ekman's
transport\index{Ekman transport} is correct. I examined Ekman's theory in Chapter
9, and the geostrophic balance in Chapter 10. We know little about the depth of no
motion in the tropical Pacific.
\vitem The solutions are limited to the east side of the ocean because
$M_x$ grows with $x$. The result comes from neglecting friction which would
eventually balance the wind-driven flow. Nevertheless, Sverdrup solutions have
been used for describing the global system of surface currents. The solutions are
applied throughout each basin all the way to the western limit of the basin.
There, conservation of mass is forced by including north-south currents confined
to a thin, horizontal boundary layer (figure 11.3).
\vitem Only one boundary condition can be satisfied, no flow through the eastern
boundary. More complete descriptions of the flow require more complete equations.
\vitem The solutions do not give the vertical distribution of the
current.
\vitem Results were based on data from two cruises plus average wind data
assuming a steady state. Later calculations by Leetmaa, McCreary, and Moore
(1981) using more recent wind data produces solutions with seasonal variability
that agrees well with observations provided the level of no motion is at 500 m. If
another depth were chosen, the results are not as good.
\vitem Wunsch (1996: \S 2.2.3) after carefully examining the evidence for a
Sverdrup balance in the ocean concluded we do not have sufficient information to
test the theory. He writes
\begin{quote} \small
The purpose of this extended discussion has not been to disapprove the validity of
Sverdrup balance. Rather, it was to emphasize the gap commonly existing in
oceanography between a plausible and attractive theoretical idea and the ability to
demonstrate its quantitative applicability to actual oceanic flow fields.---Wunsch
(1996).
\end{quote}
\begin{figure}[t!]
\makebox[120mm] [c]{\includegraphics{sverdrupxport}}
\footnotesize
Figure 11.3 Depth-integrated \rule{0mm}{3ex}Sverdrup transport \index{transport!global
Sverdrup} applied globally using the wind stress\index{wind stress!and Sverdrup transport}
from Hellerman and Rosenstein (1983). Contour interval is 10 Sverdrups. After Tomczak and
Godfrey (1994: 46).
\label{fig:sverdrupxport}
\vspace{-4ex}
\end{figure}
Wunsch, however, notes
\begin{quote} \small
Sverdrup's relationship is so central to theories of the ocean circulation that
almost all discussions assume it to be valid without any comment at all and proceed
to calculate its consequences for higher-order dynamics\dots it is difficult to
overestimate the importance of Sverdrup balance.---Wunsch (1996).
\end{quote}
But the gap is shrinking. Measurements of mean stress in the equatorial
Pacific (Yu and McPhaden, 1999) show that the flow there is in Sverdrup balance.
\end{enumerate}
\vspace{-2ex}
\paragraph{Stream Lines, Path Lines, and the Stream Function}
Before discussing more about the ocean's wind-driven circulation, we need to introduce the concept of stream lines
and the stream function (see Kundu, 1990: 51 \& 66).
At each instant in time, we can represent a flow field by a vector velocity
at each point in space. The instantaneous curves that are everywhere tangent to the
direction of the vectors are called the \textit{stream lines}\index{stream
lines|textbf} of the flow. If the flow is unsteady, the pattern of stream lines change
with time.
The trajectory of a fluid particle, the path followed by a Lagrangian drifter, is called the \textit{path
line}\index{path line|textbf} in fluid mechanics. The path line is the same as the stream line for steady flow, and
they are different for an unsteady flow.
We can simplify the description of two-dimensional, incompressible flows by using the \textit{stream
function}\index{stream function|textbf} $\psi$ defined by:
\begin{equation}
u \equiv \frac{\partial{\psi}}{\partial{y}}, \qquad v \equiv -
\frac{\partial{\psi}}{\partial{x}},
\end{equation}
The stream function is often used because it is a scalar from which the vector
velocity field can be calculated. This leads to simpler equations for some flows.
Stream functions are also useful for visualizing the flow. At each instant, the
flow is parallel to lines of constant $\psi$. Thus if the flow is steady, the
lines of constant stream function are the paths followed by water parcels.
The volume rate of flow between any two stream lines of a steady flow is $d\psi$,
and the volume rate of flow between two stream lines $\psi _1$ and $\psi _2$ is
equal to $\psi _1 - \psi _2$. To see this, consider an arbitrary line $dx =
(dx, dy)$ between two stream lines (figure 11.4). The volume rate of flow
between the stream lines is:
\begin{figure}[t!]
\makebox[120mm] [c]{\includegraphics{volxportsketch}}
\centering
\footnotesize
Figure 11.4 Volume transport\index{transport!volume} \rule{0mm}{3ex}between stream lines in
a\\two-dimensional, steady flow. After Kundu (1990: 68).
\label{fig:volxportsketch}
\vspace{-4ex}
\end{figure}
\begin{equation}
v\,dx+(-u)\,dy=-\frac{\partial{\psi}}{\partial{x}}\,dx-\frac{\partial{\psi}}{\partial{y}}\,dy=-d\psi
\end{equation}
and the volume rate of flow between the two stream lines is numerically equal to
the difference in their values of $\psi$.
Now, lets apply the concepts to satellite-altimeter maps of the oceanic
topography. In \S 10.3 I wrote (10.10)
\vspace{-1.5ex}
\begin{align}
u_s &= -\frac{g}{f}\,\frac{\partial{\zeta}}{\partial{y}} \notag \\
v_s &= \frac{g}{f}\,\frac{\partial{\zeta}}{\partial{x}}
\end{align}
Comparing (11.13) with (11.11) it is clear that
\begin{equation}
\psi = -\frac{g}{f}\,\zeta
\end{equation}
and the sea surface is a stream function scaled by $g/f$. Turning to figure 10.5,
the lines of constant height are stream lines, and flow is along the lines. The
surface geostrophic transport\index{geostrophic transport}\index{transport!geostrophic mass} is
proportional to the difference in height, independent of distance between the stream lines.
The same statements apply to figure 10.9, except that the transport\index{transport!surface
mass} is relative to transport at the 1000 decibars surface, which is roughly one kilometer
deep.
In addition to the stream function, oceanographers use the mass-transport
stream\index{transport!stream function} function $\Psi$ defined by:
\begin{equation}
M_x \equiv \frac{\partial{\Psi}}{\partial{y}}, \qquad M_y \equiv -
\frac{\partial{\Psi}}{\partial{x}}
\end{equation}
This is the function shown in figures 11.2 and 11.3.
\section[Western Boundary Currents]{Stommel's Theory of Western Boundary Currents}
\index{western boundary currents!Stommels Theory}\index{Stommel's Theory}At the same time Sverdrup was beginning to
understand circulation in the eastern Pacific, Stommel was beginning to understand why western boundary currents
occur in ocean basins. To study the circulation in the north Atlantic, Stommel (1948) used essentially the same
equations used by Sverdrup (11.1, 11.2, and 11.3) but he added a bottom stress proportional to velocity to
(11.3):
\begin{subequations}
\begin{alignat}{2}
\left(A_z \frac{\partial{u}}{\partial{z}}\right)_0 &= -T_x =-F\cos(\pi \,y/b)& \qquad \left(A_z \frac{\partial{u}}{\partial{z}}\right)_D &= -R\,u \\
\left(A_z \frac{\partial{v}}{\partial{z}}\right)_0 &= -T_y =0 & \qquad \left(A_z \frac{\partial{v}}{\partial{z}}\right)_D &= -R\,v
\end{alignat}
\end{subequations}
where $F$ and $R$ are constants.
Stommel calculated steady-state solutions for flow in a rectangular basin $0\leq
y\leq b$, $0\leq x\leq \lambda$ of constant depth $D$ filled with water of
constant density. His first solution was for a non-rotating earth. This solution
had a symmetric flow pattern with no western boundary current (figure 11.5,
left). Next, Stommel assumed a constant rotation, which again led to a symmetric
solution with no western boundary current. Finally, he assumed that the Coriolis
force varies with latitude. This led to a solution with western intensification
(figure 11.5, right). Stommel suggested that the crowding of stream lines in the
west indicated that the variation of Coriolis force with latitude may explain
why the Gulf Stream\index{Gulf Stream!Stommel's theory for} is found in the ocean. We now know
that the variation of Coriolis force with latitude is required for the existence of the western
boundary current, and that other models for the flow which use different
formulations for friction, lead to western boundary currents with different
structure. Pedlosky (1987, Chapter 5) gives a very useful, succinct, and
mathematically clear description of the various theories for western boundary
currents.
In the next chapter, we will see that Stommel's results can also be
explained in terms of vorticity---wind produces clockwise torque (vorticity),
which must be balanced by a counterclockwise torque produced at the western
boundary.
\begin{figure}[h!]
%\vspace{-1ex}
\makebox[120mm] [c]{\includegraphics{stommelcurrents}}
\footnotesize
Figure 11.5 Stream function \rule{0pt}{3ex}for flow in a basin as calculated by
Stommel (1948).
\textbf{Left:} Flow for non-rotating basin or flow for a basin with constant
rotation. \textbf{Right:} Flow when rotation varies linearly with y.
\label{fig:stommelcurrents}
\vspace{-3ex}
\end{figure}
\section{Munk's Solution}
\index{Munk's solution}Sverdrup's and Stommel's work suggested the
dominant processes producing a basin-wide, wind-driven circulation. Munk (1950)
built upon this foundation, adding information from Rossby (1936) on lateral eddy
viscosity, to obtain a solution for the circulation within an ocean basin. Munk
used Sverdrup's idea of a vertically integrated mass transport flowing over a
motionless deeper layer. This simplified the mathematical problem, and it is more
realistic. The ocean currents are concentrated in the upper kilometer of the ocean,
they are not barotropic and independent of depth. To include friction, Munk used
lateral eddy friction with constant $A_H = A_x = A_y$. Equations (11.1) become:
\begin{subequations}
\begin{align}
\frac{1}{\rho}\, \frac{\partial{p}}{\partial{x}}
&=\quad f \,v+\frac{\partial}{\partial{z}}\left(A_z
\frac{\partial{u}}{\partial{z}}\right) + A_H\,
\frac{\partial^2{u}}{\partial{x}^2} + A_H\, \frac{\partial^2{u}}{\partial{y}^2} \\
\frac{1}{\rho}\, \frac{\partial{p}}{\partial{y}} &=\:-f \,u+\frac{\partial}{\partial{z}}\left(A_z
\frac{\partial{v}}{\partial{z}}\right) + A_H\, \frac{\partial^2{v}}{\partial{x}^2} + A_H\,
\frac{\partial^2{v}}{\partial{y}^2}
\end{align}
\end{subequations}
Munk integrated the equations from a depth $-D$ to the surface at $z = z_0$ which
is similar to Sverdrup's integration except that the surface is not at $z = 0$.
Munk assumed that currents at the depth $-D$ vanish, that (11.3) apply at the
horizontal boundaries at the top and bottom of the layer, and that $A_H$ is
constant.
To simplify the equations, Munk used the mass-transport stream function\index{transport!stream
function} (11.15), and he proceeded along the lines of Sverdrup. He eliminated the pressure
term by taking the $y$ derivative of (11.17a) and the $x$ derivative of (11.17b) to obtain
the equation for mass transport:
\begin{equation}
\underbrace{A_H \nabla^{4}
\Psi\vphantom{\frac{\partial\Psi}{\partial{x}}}}_{\text{Friction}}\:-\:\underbrace{\beta\,\frac{\partial\Psi}{\partial{x}} =-\,\text{curl}_z T
\,}_{\text{Sverdrup Balance}}
\end{equation}
where
\begin{equation}
\nabla^4 =\frac{\partial^4}{\partial{x}^4}+2\,\frac{\partial^4}{\partial{x}^2
\,\partial{y}^2} + \frac{\partial^4}{\partial{y}^4}
\end{equation}
is the biharmonic operator. Equation (11.18) is the same as (11.6) with the
addition of the lateral friction term $A_H$. The friction term is large
close to a lateral boundary where the horizontal derivatives of the velocity
field are large, and it is small in the interior of the ocean basin. Thus in the
interior, the balance of forces is the same as that in Sverdrup's solution.
\begin{figure}[t!]
\includegraphics{munkcurrents}
\footnotesize
Figure 11.6 \textbf{Left:} Mean \rule{0mm}{4ex}annual wind stress\index{wind stress!annual
average}
$T_x (y)$ over the Pacific and the curl of the wind stress. $\varphi _b$ are the
northern and southern boundaries of the gyres, where $M_y = 0$ and curl $\tau = 0$.
$\varphi _0$ is the center of the gyre.
\textbf{Upper Right:} The mass transport stream function\index{transport!stream function} for a
rectangular basin calculated by Munk (1950) using observed wind stress for the Pacific. Contour
interval is 10 Sverdrups. The total transport between the coast and any point
$x,y$ is $\psi (x,y)$.The transport in the relatively narrow northern section is
greatly exaggerated. \textbf{Lower Right:} North-South component of the mass
transport. After Munk (1950).
\label{fig:munkcurrents}
\vspace{-3ex}
\end{figure}
Equation (11.18) is a fourth-order partial differential equation, and four
boundary conditions are needed. Munk assumed the flow at a boundary is parallel
to a boundary and that there is no slip at the boundary:
\begin{equation}
\Psi_{bdry} = 0, \qquad \left(\frac{\partial{\Psi}}{\partial{n}}\right)_{bdry} = 0
\end{equation}
where $n$ is normal to the boundary. Munk then solved (11.18) with (11.20)
assuming the flow was in a rectangular basin extending from $x = 0$ to
$x = r$, and from $y = -s$ to $y = +s$. He further assumed that the wind
stress\index{wind stress} was zonal and in the form:
\begin{align}
T&=a\,\cos ny + b\,\sin ny \,+\, c \notag \\
n&=j\,\pi/s, \qquad j=1,\,2,\,\dots
\end{align}
Munk's solution (figure 11.6) shows the dominant features of the gyre-scale
circulation in an ocean basin. It has a circulation similar to Sverdrup's in the
eastern parts of the ocean basin and a strong western boundary current in the
west. Using $A_H = 5 \times 10^{3}$ m$^2$/s gives a boundary current roughly 225
km wide with a shape similar to the flow observed in the Gulf Stream and the
Kuroshio\index{Kuroshio!width of}.
The transport in western boundary currents\index{transport!in western boundary currents} is
independent of
$A_H$, and it depends only on (11.6) integrated across the width of the ocean basin. Hence, it
depends on the width of the ocean, the curl of the wind stress\index{wind stress!curl of}, and
$\beta$. Using the best available estimates of the wind stress, Munk calculated that the Gulf
Stream\index{Gulf Stream!transport}\index{transport!by Gulf Stream} should have a transport of
36 Sv and that the Kuroshio\index{Kuroshio!transport of} should have a transport of 39 Sv. The
values are about one half of the measured values of the flow available to Munk. This is very
good agreement considering the wind stress\index{wind stress} was not well known.
Recent recalculations show good agreement except for the
region offshore of Cape Hatteras where there is a strong recirculation. Munk's
solution was based on wind stress\index{wind stress} averaged aver 5\degrees squares. This
underestimated the curl of the stress. Leetmaa and Bunker (1978) used modern drag
coefficient\index{drag!coefficient} and 2\degrees $\times$ 5\degrees\ averages of stress to
obtain 32 Sv transport in the Gulf Stream\index{Gulf Stream!transport}, a value very close to
that calculated by Munk.
\section{Observed Surface Circulation in the Atlantic}
The theories by Sverdrup, Munk, and Stommel describe an idealized flow. But the ocean is much more complicated. To see just how complicated the flow is at the surface, let's look at a whole ocean basin, the north Atlantic. I have chosen this region because it is the best observed, and because mid-latitude processes in the Atlantic are similar to mid-latitude processes in the other ocean. Thus, for example, I use the Gulf Stream as an example of a western boundary current.
Let's begin with the Gulf Stream\index{Gulf Stream!observations of} to see how our understanding of ocean surface currents has evolved. Of course, we can't look at all aspects of theflow. You can find out much more by reading Tomczak and Godfrey (1994) book on\textit{Regional Oceanography: An Introduction}.
\paragraph{North Atlantic Circulation}
\index{circulation!North Atlantic}The north Atlantic is the most thoroughly studied ocean basin. There is an extensive body of theory to describe most aspects of the circulation, including flow at the surface, in the thermocline\index{thermocline!in North Atlantic}, and at depth, together with an extensive body of field observations. By looking at figures showing the circulation, we can learn more about the circulation, and by looking at the figures produced over the past few decades we can trace an ever more complete
understanding of the circulation.
Let's begin with the traditional view of the time-averaged surface flow in the north Atlantic based mostly on hydrographic observations\index{hydrographic data!and north Atlantic circulation} of the density field (figure 2.7). It is a contemporary view of the mean circulation of the entire ocean based on a century of more of observations. Because the figure includes all the ocean, perhaps it is overly simplified. So, let's look then at a similar view of the mean circulation of just the north Atlantic (figure 11.7).
The figure shows a broad, basin-wide, mid latitude gyre as we expect from Sverdrup's theory described in \S 11.1. In the west, a western boundary current, the Gulf Stream\index{Gulf Stream!as a western boundary current}, completes the gyre. In the north a subpolar gyre includes the Labrador current. An equatorial current system and countercurrent are found at low latitudes with flow similar to that in the
Pacific. Note, however, the strong cross equatorial flow in the west which flows along the northeast coast of Brazil toward the Caribbean.
\begin{figure}[t!]
\makebox[121 mm] [c] {\includegraphics{NAtlcur1}}
\centering
\footnotesize
Figure 11.7 Sketch \rule{0mm}{3ex}of the major surface currents in the North Atlantic. Values are transport\index{transport!in North Atlantic} in units of $10^6$ m$^3$/s. After Sverdrup, Johnson, and Fleming (1942: fig. 187).
\label{fig:NAtlcur1}
\vspace{-4ex}
\end{figure}
\begin{figure}[t!]
\makebox[121 mm][c] {\includegraphics{NATLcur}}
\footnotesize
Figure 11.8 Detailed schematic \rule{0mm}{3ex}of named currents in the north Atlantic.
The numbers give the transport\index{transport!in North Atlantic} in units on $10^6 m^3/s$ from
the surface to a depth of 1 km. \textbf{Eg}: East Greenland Current; \textbf{Ei}: East Iceland
Current; \textbf{Gu}: Gulf Stream\index{Gulf Stream};
\textbf{Ir}: Irminger Current; \textbf{La}: Labrador Current; \textbf{Na}: North
Atlantic Current; \textbf{Nc}: North Cape Current; \textbf{Ng}: Norwegian Current;
\textbf{Ni}: North Iceland Current; \textbf{Po}: Portugal Current; \textbf{Sb}:
Spitsbergen Current; \textbf{Wg}: West Greenland Current. Numbers within squares
give sinking water in units on $10^6 m^3 /s$. Solid Lines: Warmer
currents. Broken Lines: Colder currents. After Dietrich et al. (1980: 542).
\label{fig:NATLcur1}
\vspace{-3ex}
\end{figure}
If we look closer at the flow in the far north Atlantic (figure 11.8) we see that
the flow is still more complex. This figure includes much more detail of a region
important for fisheries and commerce. Because it is based on an extensive base of
hydrographic observations, is this reality? For example, if we were to drop a
Lagrangian float into the Atlantic would it follow the streamline shown in the
figure?
To answer the question, let's look at the tracks of a 110 buoys drifting on the
sea surface compiled by Phil Richardson (figure 11.9 top). The tracks give a very
different view of the currents in the north Atlantic. It is hard to distinguish
the flow from the jumble of lines, sometimes called spaghetti tracks. Clearly, the
flow is very turbulent, especially in the Gulf Stream\index{Gulf Stream!mapped by
floats}, a fast, western-boundary
current. Furthermore, the turbulent eddies seem to have a diameter of a few
degrees. This is much different than turbulence\index{turbulence} in the atmosphere. In the
air, the large eddies are called storms, and storms have diameters of
10\degrees --20\degrees. Thus oceanic ``storms'' are much smaller than atmospheric
storms.
Perhaps we can see the mean flow if we average the drifter tracks. What happens
when Richardson averages the tracks through $2^{\circ}
\times 2^{\circ}$ boxes? The averages (figure 11.9 bottom) begin to show some
trends, but note that in some regions, such as east of the Gulf Stream, adjacent
boxes have very different means, some having currents going in different
directions. This indicates the flow is so variable, that the average is not
stable. Forty or more observations do not yields a stable mean value. Overall,
Richardson finds that the kinetic energy of the eddies is 8 to 37 times larger
than the kinetic energy of the mean flow. Thus oceanic turbulence\index{turbulence!oceanic} is
very different than laboratory turbulence\index{turbulence!laboratory}. In the lab, the mean
flow is typically much faster than the eddies.
Further work by Richardson (1993) based on subsurface buoys freely drifting at
depths between 500 and 3,500 m, shows that the current extends deep below the
surface, and that typical eddy diameter is 80 km.
\begin{figure}[t!]
\makebox[121 mm] [c] {\includegraphics{drifters}}
\footnotesize
Figure 11.9 \textbf{Top} Tracks \rule{0mm}{2ex}of 110 drifting buoys deployed in
the western north Atlantic\index{floats!in North Atlantic}.
\textbf{Bottom} Mean velocity of currents in $2^{\circ} \times 2^{\circ}$ boxes
calculated from tracks above. Boxes with fewer than 40 observations were omitted.
Length of arrow is proportional to speed. Maximum values are near 0.6 m/s in the
Gulf Stream near 37\degrees N 71\degrees W. After Richardson (1981).
\label{fig:drifters}
\vspace{-5ex}
\end{figure}
\paragraph{Gulf Stream Recirculation Region}
\index{Gulf Stream!recirculation region}\index{circulation!Gulf Stream
\rule{0mm}{4ex}recirculation region}\index{oceanic circulation!Gulf Stream
recirculation region}If we look closely at figure 11.7 we see that the
transport\index{transport!by Gulf Stream} in the Gulf Stream increases from 26 Sv in the
Florida Strait (between Florida and Cuba) to 55 Sv offshore of Cape Hatteras. Later
measurements showed the transport increases from 30 Sv in the Florida Strait to 150 Sv near
40\degrees N.
The observed increase, and the large transport off Hatteras, disagree with
transports calculated from Sverdrup's theory. Theory predicts a much smaller
maximum transport of 30 Sv, and that the maximum ought to be near 28\degrees N.
Now we have a problem: What causes the high transports near 40\degrees N?
\begin{figure}[t!]
%\vspace{-3ex}
\makebox[120mm][c]{\includegraphics{ringformation}}
\centering
\footnotesize Figure 11.10 Gulf Stream \rule{0mm}{5ex}meanders lead to the
formation of a spinning eddy, a ring.\\Notice that rings have a diameter of about
1\degrees. After Ring Group (1981).
\label{fig:ringformation}
\vspace{-3ex}
\end{figure}
Niiler (1987) summarizes the theory and observations. First, there is no
hydrographic evidence for a large influx of water from the Antilles Current that
flows north of the Bahamas and into the Gulf Stream. This rules out the
possibility that the Sverdrup flow is larger than the calculated value, and
that the flow bypasses the Gulf of Mexico. The flow seems to come primarily from
the Gulf Stream itself. The flow between 60\degrees W and 55\degrees W is to the
south. The water then flows south and west, and rejoins the Stream between
65\degrees W and 75\degrees W. Thus, there are two subtropical gyres: a small
gyre directly south of the Stream centered on 65\degrees W, called the Gulf Stream recirculation
region, and the broad, wind-driven gyre near the surface seen in figure 11.7
that extends all the way to Europe.
\begin{figure}[t!]
%\vspace{-3ex}
\makebox[120mm][c]{\includegraphics{ringsmap}}
%\centering
\footnotesize Figure 11.11 Sketch \rule{0mm}{3ex}of the position of the
Gulf Stream\index{Gulf Stream!sketch of}, warm core, and cold core eddies observed in infrared
images of the sea surface collected by the infrared radiometer on
\textsc{noaa}-5 in October and December 1978. After Tolmazin (1985: 91).
\label{fig:ringsmap}
\vspace{-4ex}
\end{figure}
The Gulf Stream recirculation carries two to three times the mass of the broader
gyre. Current meters deployed in the recirculation region show that the flow
extends to the bottom. This explains why the recirculation is weak in the maps
calculated from hydrographic data\index{hydrographic data!across Gulf Stream}. Currents
calculated from the density distribution give only the baroclinic component of the flow, and
they miss the component that is independent of depth, the barotropic component.
The Gulf Stream recirculation is driven by the potential energy of the steeply
sloping thermocline\index{thermocline!below Gulf Stream} at the Gulf Stream. The depth of the
27.00 sigma-theta ($\sigma_{\theta}$) surface drops from 250 meters near 41\degrees N
in figure 10.8 to 800 m near 38\degrees N south of the Stream. Eddies in the
Stream convert the potential energy to kinetic energy through baroclinic
instability. The instability leads to an interesting phenomena: negative
viscosity. The Gulf Stream accelerates not decelerates. It acts as
though it were under the influence of a negative viscosity. The same process
drives the jet stream in the atmosphere. The steeply sloping density surface
separating the polar air mass from mid-latitude air masses at the atmosphere's
polar front also leads to baroclinic instability. For more on this topic
see Starr's (1968) book on
\textit{Physics of Negative Viscosity Phenomena}.
Let's look at this process in the Gulf Stream (figure 11.10). The strong current
shear in the Stream causes the flow to begin to meander. The meander intensifies,
and eventually the Stream throws off a ring. Those on the south side drift
southwest, and eventually merge with the stream several months later (figure
11.11). The process occurs all along the recirculation region, and satellite
images show nearly a dozen or so rings occur north and south of the stream
(figure 11.11).
\section{Important Concepts}
\begin{enumerate}
\item The theory for wind-driven, geostrophic currents\index{geostrophic currents!Sverdrup's
theory for} was first outlined in a series of papers by Sverdrup, Stommel, and Munk between
1947 and 1951.
\vitem They showed that realistic currents can be calculated only if the
Coriolis parameter\index{Coriolis parameter} varies with latitude.
\vitem Sverdrup showed that the curl of the wind stress\index{wind stress!curl of} drives a
northward mass transport\index{transport!northward}, and that this can be used to calculate
currents in the ocean away from western boundary currents.
\vitem Stommel showed that western boundary currents are required for flow to
circulate around an ocean basin when the Coriolis parameter\index{Coriolis parameter} varies
with latitude.
\vitem Munk showed how to combine the two solutions to calculate the
wind-driven geostrophic circulation\index{geostrophic currents!Munk's theory for} in an ocean
basin. In all cases, the current is driven by the curl of the wind stress\index{wind
stress!curl of}.
\vitem The observed circulation in the ocean is very turbulent. Many years of
observations may need to be averaged together to obtain a stable map of the mean
flow.
\vitem The Gulf Stream\index{Gulf Stream} is a region of baroclinic instability in which
turbulence\index{turbulence!in Gulf Stream} accelerates the stream. This creates a Gulf Stream
recirculation. Transports in the recirculation region are much larger than
transports\index{transport!by Gulf Stream} calculated from the Sverdrup-Munk theory.
\end{enumerate}
\chapter{Vorticity in the Ocean}
Most of the fluid flows with which we are familiar, from bathtubs to swimming
pools, are not rotating, or they are rotating so slowly that rotation is not
important except maybe at the drain of a bathtub as water is let out. As a
result, we do not have a good intuitive understanding of rotating flows. In the
ocean, rotation and the conservation of vorticity strongly influence flow over
distances exceeding a few tens of kilometers. The consequences of the rotation
leads to results we have not seen before in our day-to-day dealings with fluids.
For example, did you ask yourself why the curl of the wind stress\index{wind stress!curl of}
leads to a mass transport\index{transport!mass} in the north-south direction and not in the
east-west direction? What is special about north-south motion? In this chapter, I will
explore some of the consequences of rotation for flow in the ocean.
\section{Definitions of Vorticity}
In simple words, vorticity\index{vorticity} is the rotation of the fluid. The rate of
rotation can be defined various ways. Consider a bowl of water sitting on a table in
a laboratory. The water may be spinning in the bowl. In addition to the spinning of
the water, the bowl and the laboratory are rotating because they are on a rotating
earth. The two processes are separate and lead to two types of vorticity.
\paragraph{Planetary Vorticity}
Everything on earth, including the ocean, the atmosphere, and bowls of water, rotates
with the earth. This rotation is the \textit{planetary
vorticity}\index{planetary vorticity|textbf} $f$. It is twice the local rate of
rotation of earth:
\begin{equation}
\boxed{f \equiv 2\,\Omega \sin \varphi \,\, \text{(radians/s)} = 2 \sin \varphi
\,\, \text{(cycles/day)}}
\end{equation}
Planetary vorticity is the Coriolis parameter\index{Coriolis parameter} I used earlier to
discuss flow in the ocean. It is greatest at the poles where it is twice the rotation rate of
earth. Note that the vorticity vanishes at the equator and that the vorticity in
the southern hemisphere is negative because $\varphi$ is negative.
\paragraph{Relative Vorticity}
The ocean and atmosphere do not rotate at exactly the same rate as earth. They have some
rotation relative to earth due to currents and winds. \textit{Relative vorticity}\index{relative
vorticity|textbf} $\zeta$ is the vorticity due to currents in the ocean. Mathematically it is:
\begin{equation}
\boxed{ \zeta \equiv \text{curl}_z\, \textbf{V} =
\frac{\partial{v}}{\partial{x}}-\frac{\partial{u}}{\partial{y}} }
\end{equation}
where $\textbf{V} = (u, v)$ is the horizontal velocity vector, and where we have
assumed that the flow is two-dimensional. This is true if the flow extends over
distances greater than a few tens of kilometers. $\zeta $ is the vertical
component of the three-dimensional vorticity vector $\omega$, and it is sometimes
written
$\omega{_z}$. $\zeta$ is positive for counter-clockwise rotation viewed from
above. This is the same sense as earth's rotation in the northern hemisphere.
\textit{Note on Symbols} Symbols commonly used in one part of oceanography
often have very different meaning in another part. Here I use $\zeta$ for
vorticity, but in Chapter 10, I used $\zeta$ to mean the height of the sea
surface. I could use $\omega_z$ for relative vorticity, but $\omega$ is also
commonly used to mean frequency in radians per second. I have tried to
eliminate most confusing uses, but the dual use of $\zeta$ is one we will
have to live with. Fortunately, it shouldn't cause much confusion.
For a rigid body rotating at rate $\Omega$, curl\textbf{V} $= 2\,\Omega$. Of
course, the flow does not need to rotate as a rigid body to have relative
vorticity. Vorticity can also result from shear. For example, at a north/south
western boundary in the ocean,
$u=0$, $v=v(x)$ and $\zeta = \partial{v(x)}/\partial{x}$.
$\zeta$ is usually much smaller than $f$, and it is greatest at the edge of fast
currents such as the Gulf Stream\index{Gulf Stream!vorticity}. To obtain some understanding of
the size of $\zeta$, consider the edge of the Gulf Stream off Cape Hatteras where
the velocity decreases by 1 m/s in 100 km at the boundary. The curl of the current
is approximately (1 m/s)/(100 km) = 0.14 cycles/day = 1 cycle/week. Hence even
this large relative vorticity is still almost seven times smaller than $f$. A more
typical values of relative vorticity, such as the vorticity of eddies, is a
cycle per month.
\paragraph{Absolute Vorticity}\index{absolute vorticity}\index{vorticity!absolute}
The sum of the planetary and relative vorticity is called
\textit{absolute vorticity}\index{absolute vorticity|textbf}:
\begin{equation}
\boxed{ \text{Absolute Vorticity} \equiv (\zeta + f) }
\end{equation}
We can obtain an equation for absolute vorticity in the ocean by manipulating the equations of motion for frictionless flow. We begin with:
\begin{subequations}
\begin{align}
\frac{Du}{Dt} -f\,v &= -\frac{1}{\rho}\,\frac{\partial{p}}{\partial{x}} \\
\frac{Dv}{Dt} +f\,u &= -\frac{1}{\rho}\,\frac{\partial{p}}{\partial{y}}
\end{align}
\end{subequations}
If we expand the substantial derivative, and if we subtract
$\partial\:/\partial{y}$ of (12.4a) from $\partial\:/\partial{x}$ of (12.4b) to
eliminate the pressure terms, we obtain after some algebraic manipulations:
\begin{equation}
\boxed{ \frac{D}{Dt}\left(\zeta +f \right) + \left(\zeta +f \right)
\left(\frac{\partial{u}}{\partial{x}} +
\frac{\partial{v}}{\partial{y}} \right) = 0 }
\end{equation}
In deriving (12.5) we used:
\begin{equation}
\frac{Df}{Dt} = \frac{\partial{f}}{\partial{t}}
+u\,\frac{\partial{f}}{\partial{x}} +v\,\frac{\partial{f}}{\partial{y}} = \beta
\,v \notag
\end{equation}
because $f$ is independent of time $t$ and eastward distance $x$.
\paragraph{Potential Vorticity}
The rotation rate of a column of fluid changes as the column is expanded or contracted. This changes the vorticity through changes in $\zeta$. To see how this happens, consider barotropic, geostrophic\index{geostrophic currents!vorticity} flow in an ocean with depth $H(x, y, t)$, where $H$ is the distance from the sea surface to the bottom. That is, we allow the surface to have topography (figure 12.1).
\begin{figure}[t!]
\centering
\makebox[120mm] [c]{\includegraphics{vorticitysketch}}
\footnotesize
Figure 12.1 Sketch of fluid flow used \rule{0mm}{3ex}for deriving conservation of\\potential vorticity. After Cushman-Roisin (1994: 55).
\label{fig:vorticitysketch}
\vspace{-3ex}
\end{figure}
Integrating the continuity equation (7.19) from the bottom to the top of the ocean gives (Cushman-Roisin, 1994):
\begin{equation}
\left( \frac{\partial{u}}{\partial{x}} + \frac{\partial{v}}{\partial{y}}\right) \int_{b}^{b+H} dz + w \bigr|_{b}^{b+H} = 0
\end{equation}
where $b$ is the topography of the bottom, and $H$ is the depth of the water. Notice that $\partial{u}/\partial{x}$ and $\partial{v}/\partial{y}$ are independent of $z$ because they are barotropic, and the terms can be taken outside the integral.
The boundary conditions require that flow at the surface and the bottom be along the surface and the bottom. Thus the vertical velocities at the top and the bottom are:
\begin{align}
w(b+H) &= \frac{\partial{(b+H)}}{\partial{t}} + u\,\frac{\partial{(b+H)}}{\partial{x}}+v\, \frac{\partial{(b+H)}}{\partial{y}} \\
w(b) &= u\,\frac{\partial{(b)}}{\partial{x}}+v\,\frac{\partial{(b)}}{\partial{y}}
\end{align}
where we used $\partial{b}/\partial{t} = 0$ because the bottom does not move, and $\partial{H}/\partial{z} = 0$. Substituting (12.7) and (12.8) into (12.6) we obtain
\begin{displaymath}
\left( \frac{\partial{u}}{\partial{x}} + \frac{\partial{v}}{\partial{y}}\right) + \frac{1}{H}\,\frac{DH}{Dt} = 0
\end{displaymath}
Substituting this into (12.5) gives:
\begin{displaymath}
\frac{D}{Dt}\left(\zeta +f \right) -\frac{\left(\zeta +f
\right)}{H}\,\frac{DH}{Dt} = 0
\end{displaymath}
which can be written:
\begin{displaymath}
\frac{D }{Dt}\,\left( \frac{\zeta + f}{H} \right) = 0
\end{displaymath}
The quantity within the parentheses must be constant. It is called
\textit{potential vorticity}\index{potential vorticity|textbf} $\Pi$. Potential
vorticity is conserved along a fluid trajectory:
\begin{equation}
\boxed{\text{Potential Vorticity} = \Pi \equiv \frac{\zeta + f}{H} }
\end{equation}
For baroclinic flow in a continuously stratified fluid, the potential vorticity
can be written (Pedlosky, 1987: \S 2.5):
\begin{equation}
\Pi = \frac{\zeta + f}{\rho} \cdot \nabla \lambda
\end{equation}
where $\lambda$ is any conserved quantity for each fluid element. In, particular,
if $\lambda = \rho$ then:
\begin{equation}
\Pi = \frac{\zeta + f}{\rho}\,\frac{\partial{\rho}}{\partial{z}}
\end{equation}
assuming the horizontal gradients of density are small compared
with the vertical gradients, a good assumption in the thermocline\index{thermocline}. In most
of the interior of the ocean, $f \gg \zeta$ and (12.11) is written (Pedlosky, 1996, eq
3.11.2):
\begin{equation}
\Pi = \frac{f}{\rho}\,\frac{\partial{\rho}}{\partial{z}}
\end{equation}
This allows the potential vorticity of various layers of the ocean to be
determined directly from hydrographic data\index{hydrographic data!and potential vorticity}
without knowledge of the velocity field.
\section{Conservation of Vorticity}
\index{vorticity!conservation of}The angular momentum of any isolated spinning body is
conserved. The spinning body can be an eddy in the ocean or the earth in space. If the
the spinning body is not isolated, that is, if it is linked to another body, then
angular momentum can be transferred between the bodies. The two bodies need not be in
physical contact. Gravitational forces can transfer momentum between bodies in space.
I will return to this topic in Chapter 17 when I discuss tides in the ocean. Here,
let's look at conservation of vorticity in a spinning ocean.
Friction is essential for the transfer of momentum in a fluid. Friction transfers
momentum from the atmosphere to the ocean through the thin, frictional, Ekman layer at
the sea surface\index{Ekman layer}. Friction transfers momentum from the ocean to the
solid earth through the Ekman layer at the sea floor. Friction along the sides of
sub-sea mountains leads to pressure differences on either side of the mountain which
causes another kind of drag called \textit{form drag}\index{form
drag|textbf}\index{drag!form|textbf}. This is the same drag that causes wind force on
cars moving at high speed. In the vast interior of the ocean, however, the flow is
frictionless, and vorticity is conserved. Such a flow is said to be
\textit{conservative}\index{flow!conservative|textbf}\index{conservative
flow|textbf}\index{conservative|textbf}.
\paragraph{Conservation of Potential Vorticity}
\index{potential vorticity!conservation of}The conservation of potential vorticity couples
changes in depth, relative vorticity, and changes in latitude. All three interact.
\begin{enumerate}
\vitem Changes in the depth $H$ of the flow changes in the relative
vorticity. The concept is analogous with the way figure skaters decreases their
spin by extending their arms and legs. The action increases their moment of
inertia and decreases their rate of spin (figure 12.2).
\begin{figure}[h!]
%\vspace{-2ex}
\makebox[120mm] [c]{\includegraphics{vortexsketch}}
\footnotesize
Figure 12.2 Sketch of the \rule{0pt}{4ex}production of relative
vorticity by the changes in the height of a fluid column. As the vertical
fluid column moves from left to right, vertical stretching reduces the moment of
inertia of the column, causing it to spin faster.
\label{fig:spinsketch}
\vspace{-2ex}
\end{figure}
\vitem Changes in latitude require a corresponding change in $\zeta$. As a column
of water moves equatorward, $f$ decreases, and $\zeta$ must increase (figure
12.3). If this seems somewhat mysterious, von Arx (1962) suggests we consider a
barrel of water at rest at the north pole. If the barrel is moved southward, the
water in it retains the rotation it had at the pole, and it will appear to
rotate counterclockwise at the new latitude where $f$ is smaller.
\end{enumerate}
\begin{figure}[b!]
\centering
%\vspace{-2ex}
\makebox[120mm] [c]{\includegraphics{planetaryvorticity}}
\footnotesize
Figure 12.3 Angular \rule{0pt}{6ex}momentum tends to be conserved
as columns of water change latitude. This changes the relative vorticity
of the columns. After von Arx (1962: 110).
\label{fig:planetaryvorticity}
%\vspace{-3ex}
\end{figure}
\section{Influence of Vorticity}
\index{potential vorticity!conservation!consequences of}The concept
of conservation of potential vorticity has far reaching consequences, and its
application to fluid flow in the ocean gives a deeper understanding of ocean
currents.
\paragraph{Flow Tends to be Zonal} In the ocean $f$ tends to be much larger than
$\zeta$ and thus $f/H = $ constant. This requires that the flow in an ocean of constant depth be
zonal. Of course, depth is not constant, but in general, currents tend to be east-west rather
than north south. Wind makes small changes in $\zeta$, leading to a small meridional
component to the flow (see figure 11.3).
\paragraph{Topographic Steering} Barotropic flows are diverted by sea
floor features. Consider what happens when a flow that extends from the surface to
the bottom encounters a sub-sea ridge (figure 12.4). As the depth decreases,
$\zeta + f$ must also decrease, which requires that $f$ decrease, and the flow is
turned toward the equator. This is called \textit{topographic
steering}\index{topographic steering|textbf}. If the change in depth is sufficiently
large, no change in latitude will be sufficient to conserve potential vorticity, and
the flow will be unable to cross the ridge. This is called \textit{topographic
blocking}\index{topographic blocking|textbf}.
\begin{figure}[h!]
\centering
\vspace{-1ex}
\makebox[120mm] [c]{\includegraphics{ridgevorticity}}
\footnotesize
Figure 12.4 Barotropic \rule{0pt}{3ex} flow over a sub-sea ridge is turned equatorward\\to conserve potential vorticity. After Dietrich et al. (1980: 333).
\label{fig:ridgevorticity}
\vspace{-3ex}
\end{figure}
\paragraph{Western Boundary Currents} The balance of vorticity provides an alternate explanation for the existence of western boundary currents. Consider the gyre-scale flow in an ocean basin (figure 12.5), say in the north Atlantic from 10\degrees N to 50\degrees N. The wind blowing over the Atlantic adds negative vorticity $\zeta_{\tau}$. As the water flows around the gyre, the vorticity of the gyre must remain nearly constant, else the flow would spin faster or slower. Overall, the negative vorticity input by the wind must be balanced by a source of positive vorticity.
Throughout most of the basin the negative vorticity input by the wind is balanced by an increase in relative vorticity. As the flow moves southward throughout the basin, $f$ decreases and $\zeta$ must increase according to (12.9) because $H$, the depth of the wind-driven circulation, does not change much.
The balance breaks down, however, in the west where the flow returns northward. In the west, $f$ increases, $\zeta$ decreases, and a source of positive vorticity is needed. The positive vorticity $\zeta_{b}$ is produced by the western boundary boundary current.
\begin{figure}[t!]
\makebox[122mm] [c]{\includegraphics{westbdycurrent}}
\footnotesize
Figure 12.5 The balance \rule{0pt}{4ex}of potential vorticity
can clarify why western boundary currents are necessary.
\textbf{Left:} Vorticity input by the wind $\zeta_{\tau}$ balances the
change in relative vorticity $\zeta$ in the east as the
flow moves southward and $f$ decreases. The two do not balance in the west
where $\zeta$ must decrease as the flow moves northward and $f$ increases.
\textbf{Right:} Vorticity in the west is balanced by relative vorticity
$\zeta_b$ generated by shear in the western boundary current.
\label{fig:westbdycurrent}
\vfill
\vspace{-4ex}
\end{figure}
\section{Vorticity and Ekman Pumping}
\index{Ekman pumping}Rotation places another very interesting constraint on the
geostrophic flow\index{geostrophic currents!vorticity constraints} field. To help understand
the constraints, let's first consider flow in a fluid with constant rotation. Then we will look
into how vorticity constrains the flow of a fluid with rotation that varies with latitude. An
understanding of the constraints leads to a deeper understanding of Sverdrup's and Stommel's
results discussed in the last chapter.
\paragraph{Fluid dynamics on the \textbf{\textit{f}} Plane: the Taylor-Proudman
Theorem} \index{f-plane@\textit{f}-plane!fluid dynamics
on}\index{f-plane@\textit{f}-plane!Taylor-Proudman Theorem}The influence of vorticity due to
earth's rotation is most striking for geostrophic flow of a fluid with constant density
$\rho{_0}$ on a plane with constant rotation $f = f_0$. From Chapter 10, the three components
of the geostrophic equations (10.4) are:
\begin{subequations}
\begin{align}
f\,v &= \;\;\, \frac{1}{\rho_{0}}\,\frac{\partial{p}}{\partial{x}} \\
f\,u &= -\frac{1}{\rho_{0}}\,\frac{\partial{p}}{\partial{y}} \\
g &= -\frac{1}{\rho_{0}}\,\frac{\partial{p}}{\partial{z}}
%0 &= \frac{\partial{u}}{\partial{x}} + \frac{\partial{v}}{\partial{y}} +
%\frac{\partial{w}}{\partial{z}}
\end{align}
%\end{subequations}
and the continuity equations (7.19) is:
\begin{equation}
0 = \frac{\partial{u}}{\partial{x}} + \frac{\partial{v}}{\partial{y}} +
\frac{\partial{w}}{\partial{z}}
\end{equation}
\end{subequations}
Taking the $z$ derivative of (12.13a) and using (12.13c) gives:
\begin{align}
-f_0\,\frac{\partial{v}}{\partial{z}} &= -\frac{1}{\rho_{0}}\,\frac{\partial
}{\partial{z}}\,
\left(\frac{\partial{p}}{\partial{x}}\right) = \frac{\partial
}{\partial{x}}\left(-\frac{1}{\rho_{0}}\,\frac{\partial{p}}{\partial{z}}\right) =
\frac{\partial g}{\partial x}= 0
\notag \\
f_{0} \frac{\partial{v}}{\partial{z}} &= 0 \notag \\
\therefore \quad \frac{\partial{v}}{\partial{z}} &= 0 \notag
\end{align}
Similarly, for the u-component of velocity (12.13b). Thus, the vertical derivative
of the horizontal velocity field must be zero.
\begin{equation}
\boxed{ \frac{\partial{u}}{\partial{z}} = \frac{\partial{v}}{\partial{z}} =0 }
\end{equation}
This is the \textit{Taylor-Proudman Theorem}\index{Taylor-Proudman Theorem|textbf}, which applies to slowly varying
flows in a homogeneous, rotating, inviscid fluid. The theorem places strong constraints on the flow:
\begin{quotation} \small
If therefore any small motion be communicated to a rotating fluid the resulting
motion of the fluid must be one in which any two particles originally in a line
parallel to the axis of rotation must remain so, except for possible small
oscillations about that position---Taylor (1921).
\end{quotation}
Hence, rotation greatly stiffens the flow! Geostrophic flow\index{geostrophic
currents!vorticity constraints} cannot go over a seamount, it must go around it. Taylor (1921)
explicitly derived (12.14) and (12.16) below. Proudman (1916) independently derived the same
theorem but not as explicitly.
Further consequences of the theorem can be obtained by eliminating the pressure
terms from (12.13a \& 12.13b) to obtain:
\begin{subequations}
\begin{align}
\frac{\partial{u}}{\partial{x}} + \frac{\partial{v}}{\partial{y}} &= -\frac{\partial }{\partial{x}} \left(\frac{1}{f_{0}\,\rho_{0}}\,
\frac{\partial{p}}{\partial{y}} \right) + \frac{\partial }{\partial{y}} \left(\frac{1}{f_{0}\,\rho_{0}}\,
\frac{\partial{p}}{\partial{x}} \right) \\
\frac{\partial{u}}{\partial{x}} + \frac{\partial{v}}{\partial{y}} &= \frac{1}{f_{0}\,\rho_{0}} \left( -\frac{\partial ^2 p}{\partial{x}\,\partial{y}} + \frac{\partial ^2
p}{\partial{x}\,\partial{y}} \right) \\
\frac{\partial{u}}{\partial{x}} + \frac{\partial{v}}{\partial{y}} &= 0
\end{align}
\end{subequations}
Because the fluid is incompressible, the continuity equation (12.13d) requires
\begin{equation}
\frac{\partial{w}}{\partial{z}} = 0
\end{equation}
Furthermore, because $w = 0$ at the sea surface and at the sea floor, if the bottom
is level, there can be no vertical velocity on an $f$--plane. Note that the
derivation of (12.16) did not require that density be constant. It requires only
slow motion in a frictionless, rotating fluid.
\paragraph{Fluid Dynamics on the Beta Plane: Ekman Pumping}
\index{B-plane@$\beta$-plane!fluid dynamics on}\index{B-plane@$\beta$-plane!Ekman
Pumping}If (12.16) is true, the flow cannot expand or contract in the vertical
direction, and it is indeed as rigid as a steel bar. There can be no gradient of
vertical velocity in an ocean with constant planetary vorticity. How then can the
divergence of the Ekman transport\index{transport!Ekman} at the sea surface lead to vertical
velocities at the surface or at the base of the Ekman layer? The answer can only be that one
of the constraints used in deriving (12.16) must be violated. One constraint that can be
relaxed is the requirement that $f = f_0$.
Consider then flow on a beta plane. If $f = f_0 + \beta\,y$, then (12.15a)
becomes:
\begin{align}
\frac{\partial{u}}{\partial{x}} + \frac{\partial{v}}{\partial{y}} &= - \frac{1}{f\,\rho_{0}}\, \frac{\partial ^2 p}{\partial{x}\,\partial{y}} +
\frac{1}{f\,\rho_{0}} \, \frac{\partial ^2 p}{\partial{x}\,\partial{y}} - \frac{\beta}{f} \,\frac{1}{f\,\rho_{0}}\,\frac{\partial{p}}{\partial{x}} \\
f \left( \frac{\partial{u}}{\partial{x}} + \frac{\partial{v}}{\partial{y}} \right) &= - \beta \, v
\end{align}
where we have used (12.13a) to obtain $v$ in the right-hand side of (12.18).
Using the continuity equation, and recalling that $\beta\, y \ll f_0$
\begin{equation}
f_0 \frac{\partial{w_G}}{\partial{z}} = \beta \, v
\end{equation}
where we have used the subscript $G$ to emphasize that (12.19) applies to the
ocean's interior, geostrophic flow\index{geostrophic currents!in ocean's interior}. Thus the
variation of Coriolis force with latitude allows vertical velocity gradients in the
geostrophic interior of the ocean, and the vertical velocity leads to north-south currents.
This explains why Sverdrup and Stommel both needed to do their calculations on a
$\beta$-plane\index{B-plane@$\beta$-plane}.
\paragraph{Ekman Pumping in the Ocean}
\index{Ekman pumping|(}In Chapter 9, we saw that the curl of the wind
stress\index{wind stress!curl of} $\mathbf{T}$ produced a divergence of the Ekman
transports\index{transport!Ekman} leading to a vertical velocity $w_E (0)$ at the top of the
Ekman layer. In Chapter 9 we derived
\begin{equation}
w_E (0) = -\text{curl}\left(\frac{\mathbf{T}}{\rho f} \right)
\end{equation}
which is (9.30b) where $\rho$ is density and $f$ is the Coriolis parameter\index{Coriolis
parameter}. Because the vertical velocity at the sea surface must be zero, the Ekman vertical
velocity must be balanced by a vertical geostrophic velocity\index{geostrophic
currents!vertical and Ekman pumping} $w_G(0)$.
\begin{equation}
w_E (0) = - w_G (0) = -\text{curl}\left(\frac{\mathbf{T}}{\rho f} \right)
\end{equation}
Ekman pumping ($w_E (0)$) drives a vertical geostrophic current ($-w_G (0)$) in
the ocean's interior. But why does this produce the northward current calculated by
Sverdrup (11.6)? Peter Niiler (1987: 16) gives an explanation.
\begin{quotation} \small
Let us postulate there exists a deep level where horizontal and vertical motion
of the water is much reduced from what it is just below the mixed
layer\index{mixed layer!and Ekman pumping} [figure 12.6]$\ldots$ Also let us assume that
vorticity is conserved there (or mixing is small) and the flow is so slow that accelerations
over the earth's surface are much smaller than Coriolis accelerations. In such a
situation a column of water of depth $H$ will conserve its spin per unit
volume, $f/H$ (relative to the sun, parallel to the earth's axis of rotation).
A vortex column which is compressed from the top by wind-forced sinking ($H$
decreases) and whose bottom is in relatively quiescent water would tend to
shorten and slow its spin. Thus because of the curved ocean surface it has to
move southward (or extend its column) to regain its spin. Therefore, there
should be a massive flow of water at some depth below the surface to the south
in areas where the surface layers produce a sinking motion and to the north
where rising motion is produced. This phenomenon was first modeled correctly by
Sverdrup (1947) (after he wrote ``ocean'') and gives a dynamically plausible
explanation of how wind produces deeper circulation in the ocean.
\end{quotation}
Peter Rhines (1984) points out that the rigid column of water trying to escape the
squeezing imposed by the atmosphere escapes by moving southward. The southward
velocity is about 5,000 times greater than the vertical Ekman velocity\index{Ekman
pumping|)}.
\begin{figure}[t]
%\centering
\makebox[120mm] [c]{\includegraphics{NiilerPlot}}
\footnotesize
Figure 12.6 Ekman pumping\index{Ekman pumping} \rule{0pt}{2ex} that produces a
downward velocity at the base of the Ekman layer forces the fluid in the
interior of the ocean to move southward. See text for why this happens. After Niiler (1987).
\label{fig:vorticity}
\vfill
\vspace{-3ex}
\end{figure}
\paragraph{Ekman Pumping: An Example}
\index{Ekman pumping!example}Now let's see how Ekman pumping drives geostrophic
flow\index{geostrophic currents!and Ekman pumping} in say the central north Pacific (figure
12.7) where the curl of the wind stress\index{wind stress!curl of} is negative. Westerlies in
the north drive a southward transport\index{transport!southward in westerlies}, the trades in
the south drive a northward transport\index{transport!northward in trades}. The converging
Ekman transports must be balanced by downward geostrophic velocity (12.21).
\begin{figure}[t]
\makebox[120mm] [c]{\includegraphics{EkmanPumping}}
\footnotesize
Figure 12.7 Winds \rule{0pt}{5ex} at the sea surface drive Ekman transports\index{transport!Ekman} to the right of the wind in this northern hemisphere example (bold arrows in shaded Ekman layer). The converging Ekman transports driven by the trades and westerlies drives a downward geostrophic flow just below the Ekman layer (bold vertical arrows), leading to downward bowing constant density
surfaces $\rho_i$. Geostrophic currents associated with the warm water are shown by bold arrows. After Tolmazin (1985: 64).
\label{fig:EkmanPumping}
\vspace{-4ex}
\end{figure}
\begin{figure}[b!]
\vspace{-3ex}
\makebox[120mm] [c]{\includegraphics{zonalmeanwind}}
\footnotesize
Figure 12.8 An example of how \rule{0pt}{5ex}winds produce
geostrophic currents running upwind. Ekman transports\index{transport!Ekman} due to winds in
the north Pacific (\textbf{Left}) lead to Ekman pumping\index{Ekman pumping}
(\textbf{Center}), which sets up north-south pressure gradients in the upper ocean. The
pressure gradients are balanced by the Coriolis force due to east-west geostrophic
currents\index{geostrophic currents!and Ekman transports} (\textbf{Right}). Horizontal lines
indicate regions where the curl of the zonal wind stress\index{wind stress!curl of} changes
sign. \textbf{AK}: Alaskan Current,
\textbf{NEC}: North Equatorial Current, \textbf{NECC}: North Equatorial Counter Current.
\label{fig:zonalmeanwinds}
\vfill
%\vspace{-4ex}
\end{figure}
Because the water near the surface is warmer than the deeper water, the vertical
velocity produces a pool of warm water. Much deeper in the ocean, the wind-driven
geostrophic current must go to zero (Sverdrup's hypothesis) and the deep pressure
gradients must be zero. As a result, the surface must dome upward because a column
of warm water is longer than a column of cold water having the same weight (they
must have the same weight, otherwise, the deep pressure would not be constant, and
there would be a deep horizontal pressure gradient). Such a density distribution
produces north-south pressure gradients at mid depths that must be balanced by
east-west geostrophic currents. In short, the divergence of the Ekman
transports\index{transport!Ekman} redistributes mass within the frictionless interior of the
ocean leading to the wind-driven geostrophic currents.
Now let's continue the idea to include the entire north Pacific to see how winds
produce currents flowing upwind. The example will give a deeper understanding of
Sverdrup's results we discussed in \S11.1.
Figure 12.8 shows shows the mean zonal winds in the Pacific, together with the
north-south Ekman transports\index{transport!in Pacific} driven by the zonal winds. Notice that
convergence of transport\index{transport!convergence of} leads to downwelling, which produces a
thick layer of warm water in the upper kilometer of the water column, and high sea level.
Figure 12.6 is a sketch of the cross section of the region between 10\degrees N and 60\degrees
N, and it shows the pool of warm water in the upper kilometer centered on 30\degrees N.
Conversely, divergent transports leads to low sea level. The mean north-south
pressure gradients associated with the highs and lows are balanced by the Coriolis
force of east-west geostrophic\index{geostrophic currents!in Pacific} currents in the upper
ocean (shown at the right in the figure).
\section{Important Concepts}
\begin{enumerate}
\item Vorticity strongly constrains ocean dynamics.
\vitem Vorticity due to earth's rotation is much greater than other sources of
vorticity.
\vitem Taylor and Proudman showed that vertical velocity is impossible in a
uniformly rotating flow. The ocean is rigid in the direction parallel to the
rotation axis. Hence Ekman pumping\index{Ekman pumping} requires that planetary vorticity vary
with latitude. This explains why Sverdrup and Stommel found that realistic oceanic
circulation, which is driven by Ekman pumping, requires that $f$ vary with latitude.
\vitem The curl of the wind stress\index{wind stress!curl of} adds relative vorticity to
central gyres of each ocean basin. For steady state circulation in the gyre, the ocean must
lose vorticity in western boundary currents.
\vitem Positive wind stress curl leads to divergent flow in the Ekman layer.
The ocean's interior geostrophic circulation adjusts through a northward mass
transport.
\vitem Conservation of absolute vorticity\index{absolute vorticity}\index{vorticity!absolute}
in an ocean with constant density leads to the conservation of potential vorticity. Thus
changes in depth in an ocean of constant density requires changes of latitude of the current.
\end{enumerate}
\chapter{Deep Circulation in the Ocean}
The direct forcing of the oceanic circulation by wind discussed in the last few
chapters is strongest in the upper kilometer of the water column. Below a
kilometer lies the vast water masses of the ocean extending to depths of 4--5 km.
The water is everywhere cold, with a potential temperature less than 4\degrees C.
The water mass is formed when cold, dense water sinks from the surface to great
depths at high latitudes. It spreads out from these regions to fill the ocean
basins. Deep mixing\index{mixing!of deep waters} eventually pulls the water up through the
thermocline\index{thermocline!mixing in}\index{mixing!in thermocline} over large areas of the
ocean. It is this upward mixing\index{mixing!of deep waters} that drives the deep circulation.
The vast deep ocean is usually referred to as the
\textit{abyss}\index{abyss|textbf}, and the circulation as the \textit{abyssal
circulation}\index{abyssal circulation|textbf}\index{circulation!abyssal|textbf}\index{oceanic
circulation!abyssal|textbf}.
The densest water at the sea surface, water that is dense enough to sink to the bottom, is formed when frigid air blows across the ocean at high latitudes in winter in the Atlantic between Norway and Greenland and
near Antarctica. The wind cools and evaporates water. If the wind is cold enough, sea ice forms, further increasing the salinity of the water because ice is fresher than sea water. Bottom water\index{bottom water!North Atlantic} is produced only in these two regions. Cold, dense water is formed in the North Pacific, but it is not salty enough to sink to the bottom.
At mid and low latitudes, the density, even in winter, is sufficiently low that the water cannot sink more than a few hundred meters into the ocean. The only exception are some seas, such as the Mediterranean Sea, where evaporation is so great that the salinity of the water is sufficiently great for the water to sink to the bottom of these seas. If these seas can exchange water with the open ocean, the waters formed in winter in the seas mixes with the water in the open ocean and it spreads out along intermediate depths in the open ocean.
\section{Defining the Deep Circulation}
Many terms have been used to describe the deep circulation. They include:
1) \textit{abyssal circulation}\index{abyssal circulation}\index{circulation!abyssal}\index{oceanic
circulation!abyssal},
2) \textit{thermohaline circulation}
3) \textit{meridional overturning circulation}\index{circulation!meridional overturning|textbf}\index{oceaniccirculation!Meridional Overturning|textbf}, and
4) \textit{global conveyor}.
The term thermohaline circulation was once widely used, but it has almost entirely disappeared from the oceanographic literature (Toggweiler and Russell, 2008). It is no longer used because it is now clear that the flow is not density driven, and because the concept has not been clearly defined (Wunsch, 2002b).
The meridional overturning circulation is better defined. It is the zonal average of the flow plotted as a function of depth and latitude. Plots of the circulation show where vertical flow is important, but they show no information about how circulation in the gyres influences the flow.
Following Wunsch (2002b), I define the deep circulation as the circulation of mass. Of course, the mass circulation also carries heat, salt, oxygen, and other properties. But the circulation of the other properties is not the same as the mass transport\index{transport!mass}. For example, Wunsch points out that the north Atlantic imports heat but exports oxygen.
\section{Importance of the Deep Circulation}
\index{deep circulation!importance of}\index{circulation!deep!importance of}\index{oceanic
circulation!deep!importance of}The deep circulation carries heat, salinity,
oxygen, CO$_2$, and other properties from high latitudes in winter to lower
latitudes throughout the world. This has very important consequences.
\begin{enumerate}
\vitem The contrast between the cold deep water and the warm surface waters determines the stratification of the ocean, which strongly influences ocean dynamics.
\vitem The volume of deep water is far larger than the volume of surface
water. Although currents in the deep ocean are relatively weak, they have
transports\index{transport!volume, in deep ocean} comparable to the surface transports.
\vitem The fluxes of heat and other variables carried by the deep circulation influences
earth's heat budget and climate. The fluxes vary from decades to centuries
to millennia, and this variability modulates climate over such time intervals. The
ocean may be the primary cause of variability over times ranging from years to
decades, and it may have helped modulate ice-age climate.
\end{enumerate}
Two aspects of the deep circulation are especially important for understanding
earth's climate and its possible response to increased carbon dioxide
CO$_2$ in the atmosphere: i) the ability of cold water to store CO$_2$ and heat absorbed from the
atmosphere, and ii) the ability of deep currents to modulate the heat
transported\index{transport!heat} from the tropics to high latitudes.
\paragraph{The ocean as a Reservoir of Carbon Dioxide}
\index{carbon dioxide}The ocean are the primary reservoir of readily available CO$_2$, an
important greenhouse gas. The ocean contain 40,000 GtC of dissolved, particulate,
and living forms of carbon. The land contains 2,200 GtC, and the atmosphere contains
only 750 GtC. Thus the ocean hold 50 times more carbon than the air. Furthermore,
the amount of new carbon put into the atmosphere since the industrial revolution,
150 GtC, is less than the amount of carbon cycled through the marine ecosystem in
five years. (1 GtC = 1 gigaton of carbon = $10^{12}$ kilograms of carbon.) Carbonate
rocks such as limestone, the shells of marine animals, and coral are other, much
larger, reservoirs. But this carbon is locked up. It cannot be easily exchanged with
carbon in other reservoirs.
More CO$_2$ dissolves in cold water than in warm water. Just imagine shaking and
opening a hot can of Coke$^{\mbox{\textsf{\scriptsize TM}}}$. The CO$_2$ from a
hot can will spew out far faster than from a cold can. Thus the cold deep water
in the ocean is the major reservoir of dissolved CO$_2$ in the ocean.
New CO$_ 2$ is released into the atmosphere when fossil fuels and trees are burned. Very quickly, 48\% of the CO$_2$ released into the atmosphere dissolves into the ocean (Sabine et al, 2004), much of which ends up deep in the ocean.
Forecasts of future climate change depend strongly on how much CO$_2$ is stored
in the ocean and for how long. If little is stored, or if it is stored and later
released into the atmosphere, the concentration in the atmosphere will change,
modulating earth's long-wave radiation balance. How much and how long CO$_2$ is
stored in the ocean depends on the transport\index{transport!carbon dioxide} of CO$_2$ by the
deep circulation and on the net flux of carbon deposited on the sea floor. The amount that
dissolves depends on the temperature of the deep water, the storage time in the deep ocean
depends on the rate at which deep water is replenished, and the deposition depends on
whether the dead plants and animals that drop to the sea floor are oxidized.
Increased ventilation of deep layers, and warming of the deep layers could
release large quantities of the gas to the atmosphere.
The storage of carbon in the ocean also depends on the dynamics of marine ecosystems, upwelling\index{upwelling!and carbon storage}, and the amount of dead plants and animals stored in sediments. But I won't consider these processes.
\paragraph{Oceanic Transport of Heat}
\index{heat transport!oceanic}\index{transport!heat}The ocean carry about half the heat
out of the tropics needed to maintain earth's temperature. Heat carried by
the Gulf Stream\index{Gulf Stream!transport of heat by} and the north Atlantic drift keeps the
far north Atlantic ice free, and it helps warm Europe. Norway, at 60\degrees N is far warmer
than southern Greenland or northern Labrador at the same latitude. Palm trees grow on the west
coast of Ireland, but not in Newfoundland which is further south.
Wally Broecker (1987), working at Lamont-Doherty Geophysical Observatory of Columbia University, calls the oceanic component of the heat-transport\index{transport!heat} system the \textit{Global Conveyor Belt}\index{Global Conveyer Belt}\index{heat transport!Global Conveyer Belt}. The basic idea is that surface currents carry heat to the far north Atlantic (figure 13.1). There the surface water releases heat and water to the atmosphere, and it becomes sufficiently cold, salty, and dense that it sinks to the bottom\index{bottom water!North Atlantic} in the Norwegian and Greenland Seas. It then flows southward in cold, bottom currents. Some of the water remains on the surface and returns to the south in cool surface currents such as the Labrador Current and Portugal Current (see figure 11.8). Richardson (2008) has written a very useful paper surveying our understanding of the global conveyor belt.
The deep bottom water from the north Atlantic\index{bottom water!North Atlantic} is mixed upward in
other regions and ocean, and eventually it makes its way back to the Gulf Stream and the North
Atlantic. Thus most of the water that sinks in the north Atlantic must be replaced by water from the far south Atlantic. As this surface water moves northward across the equator and eventually into the Gulf Stream, it carries heat out of the south Atlantic.
So much heat is pulled\index{transport!heat} northward by the formation of north Atlantic bottom water in winter that heat transport in the Atlantic is entirely northward, even in the southern hemisphere (figure 5.11). Much of the solar heat absorbed by the tropical Atlantic is shipped north to warm Europe and the northern hemisphere. Imagine then what might happen if the supply of heat is shut off. I will get back to that topic in the next section.
\begin{figure}[t!]
\makebox[120mm] [c]{\includegraphics{fig13-1}}
\footnotesize
Figure 13.1 The surface (narrow dashes) \rule{0mm}{3ex}and deep (wide dashes) currents
in the north Atlantic. The North Atlantic Current brings warm water northward where it cools.
Some sinks and returns southward as a cold, deep, western-boundary current. Some returns
southward at the surface. From Woods Hole Oceanographic Institution.
\label{fig:fig13-1}
\vspace{-4ex}
\end{figure}
We can make a crude estimate of the importance of the north Atlantic surface and deep
circulation from a calculation based on what we know about waters in the Atlantic
compiled by Bill Schmitz (1996) in his wonderful summary of his life's work. The
Gulf Stream\index{Gulf Stream!transport} carries\index{transport!by Gulf Stream} 40 Sv of
18\degrees C water northward. Of this, 14 Sv return southward in the deep western boundary
current at a temperature of 2\degrees C. The water must therefore lose 0.9 petawatts (1
petawatt $= 10^{15}
$ watt) in the north Atlantic north of 24\degrees N. Although the calculation is very crude,
it is remarkably close to the value of
$1.2\,\pm\,0.2$ petawatts estimated much more carefully by Rintoul and Wunsch
(1991).
Note that if the water remained on the surface and returned as an eastern
boundary current, it would be far warmer than the deep current when it returned
southward. Hence, the heat transport\index{transport!heat} would be much reduced and it would probably not keep the far north Atlantic ice free.
The production of bottom water\index{bottom water} is influenced by the surface salinity and winds in the north Atlantic (Toggweiler and Russell, 2008). It is also influenced by the rate of upwelling\index{upwelling} due to mixing\index{mixing!of deep waters} in other oceanic areas. First, let's look at the influence of salinity.
Saltier surface waters form denser water in winter than less salty water. At first you may think that temperature is also important, but at high latitudes water in all ocean basins gets cold enough to freeze, so all ocean produce $-2$\degrees C water at the surface. Of this, only the most salty will sink, and the saltiest water is in the Atlantic and under the ice on the continental shelves around Antarctica.
The production of bottom water is remarkably sensitive to small changes in salinity. Rahmstorf (1995), using a numerical model of the meridional over\-turning circulation\index{circulation!meridional overturning}, showed that a $\pm$0.1Sv variation of the flow of fresh water into the north Atlantic can switch on or off the deep circulation of 14 Sv. If the deep-water production is shut off during times of low salinity, the 1 petawatt of heat may also be shut off. Weaver and Hillaire-Marcel (2004) point out that the shutdown of the production of bottom water is unlikely, and if it did happen, it would lead to a colder Europe, not a new ice age, because of the higher concentrations of CO$_2$ now in the atmosphere.
I write \textit{may be shut off} because the ocean is a very complex system. We
don't know if other processes will increase heat transport\index{transport!heat} if the deep
circulation is disturbed. For example, the circulation at intermediate depths
may increase when deep circulation is reduced.
The production of bottom water is also remarkably sensitive to small changes in mixing\index{mixing!of deep waters} in the deep ocean. Munk and Wunsch (1998) calculate that 2.1 TW (terawatts $= 10^{12}$ watts) are required to drive the deep circulation, and that this small source of mechanical mixing\index{mixing!and poleward heat flux} drives a poleward heat flux\index{heat flux!poleward} of 2000 TW. Some of the energy for mixing\index{mixing!energy for} comes from winds which can produce turbulent mixing\index{mixing!by winds} throughout the ocean. Some energy comes from the dissipation of tidal currents\index{mixing!tidal}, which depend on the distribution of the continents. Some of the energy comes from the flow of deep water past the mid-ocean ridge system. Thus during the last ice age, when sea level was much lower, tides, tidal currents, tidal dissipation, winds, and deep circulation all differed from present values.
\paragraph{Role of the Ocean in Ice-Age Climate Fluctuations}
\index{ice-age|(}What might happen if the production of deep water in the Atlantic is shut off? Information contained in Greenland and Antarctic ice sheets, in north Atlantic sediments, and in lake sediments provide important clues.
Several ice cores through the Greenland and Antarctic ice sheets provide a continuous record of atmospheric conditions\index{atmospheric conditions!finding historical} over Greenland and Antarctica extending back more than 700,000 years before the present in some cores. Annual layers in the core are counted to get age. Deeper in the core, where annual layers are hard to see, age is calculated from depth and from dust layers from well-dated volcanic eruptions. Oxygen-isotope ratios of the ice give air temperature at the glacier surface when the ice was formed. Deuterium concentrations give ocean-surface temperature at the moisture source region. Bubbles in the ice give atmospheric CO$_2$ and methane concentration. Pollen, chemical composition, and particles give information about volcanic eruptions, wind speed, and direction. Thickness of annual layers gives snow accumulation rates. And isotopes of some elements give solar and cosmic ray activity (Alley, 2000).
Cores through deep-sea sediments in the north Atlantic made by the Ocean Drilling Program give information about i) surface and deep temperatures and salinity at the location of above the core, ii) the production of north Atlantic deep water, iii) ice volume in glaciers, and iv) production of icebergs. Ice-sheet and deep-sea cores have allowed reconstructions of climate for the past few hundred thousand years.
\begin{enumerate}
\vitem The oxygen-isotope and deuterium records in the ice cores show abrupt climate variability many times over the past 700,000 years. Many times during the last ice age temperatures near Greenland warmed rapidly over periods of 1--100 years, followed by gradual cooling over longer periods. (Dansgaard et al, 1993). For example, $\approx 11,500$ years ago, temperatures over Greenland warmed by $\approx 8$\degrees C in 40 years in three steps, each spanning 5 years (Alley, 2000). Such abrupt warming is called a Dansgaard/Oeschger event\index{Dansgaard/Oeschger event}. Other studies have shown that much of the northern hemisphere warmed and cooled in phase with temperatures calculated from the ice core.
\vitem The climate of the past 8,000 years was constant with very little variability. Our perception of climate change is thus based on highly unusual circumstances. All of recorded history has been during a period of warm and stable climate.
\vitem Hartmut Heinrich and colleagues (Bond et al. 1992), studying the sediments in the north Atlantic found periods when coarse material was deposited on the bottom in mid ocean. Only icebergs can carry such material out to sea, and the find indicated times when large numbers of icebergs were released into the north Atlantic. These are now called Heinrich events\index{Heinrich events}.
\begin{figure}[t!]
%\vspace{-3ex}
\makebox[121mm] [c]{\includegraphics{NAiceage}}
\footnotesize
Figure 13.2 Periodic \rule{0mm}{3ex}surges of icebergs during the last ice age appear to have modulated temperatures of the northern hemisphere by lowering the salinity of the far north Atlantic and reducing the meridional overturning circulation\index{circulation!meridional overturning}. Data from cores through the Greenland ice sheet (1), deep-sea sediments (2,3), and alpine-lake sediments (4) indicate that:
\textbf{Left:} During recent times the circulation has been stable, and the polar front which separates warm and cold water masses has allowed warm water to penetrate beyond Norway.
\textbf{Center:} During the last ice age, periodic surges of icebergs reduced salinity and reduced the meridional overturning circulation, causing the polar front to move southward and keeping warm water south of Spain.
\textbf{Right:} Similar fluctuations during the last interglacial appear to have caused rapid, large changes in climate. The \textbf{Bottom} plot is a rough indication of temperature in the region, but the scales are not the same. After Zahn (1994).
\label{fig:NAiceage}
\vspace{-3ex}
\end{figure}
\begin{figure}[b!]
\vspace{-2ex}
\makebox[121mm] [c]{\includegraphics{hysteresis}}
\footnotesize
Figure 13.3 The \rule{0mm}{3ex}meridional-overturning circulation\index{circulation!meridional overturning} in the north Atlantic may be stable near \textit{2} and \textit{4}. But, the switching from a warm, salty regime to a cold, fresh regime and back has hysteresis. This means that as the warm salty ocean in an initial state \textit{1} freshens, and becomes more fresh than \textit{2} it quickly switches
to a cold, fresh state
\textit{3}. When the area again becomes salty, it must move past state
\textit{4} before it can switch back to \textit{1}.
\label{fig:hysteresis}
%\vspace{-5ex}
\end{figure}
\vitem The correlation of Greenland temperature with iceberg production is related to the deep circulation. When icebergs melted, the surge of fresh water increased the stability of the water column shutting off the production of north Atlantic Deep Water\index{North Atlantic Deep Water}. The shut-off of deep-water formation greatly reduced the northward transport\index{transport!northward heat} of warm water into the north Atlantic, producing very cold northern hemisphere climate (figure 13.2). The melting of the ice pushed the polar front, the boundary between cold and warm water in the north Atlantic further south than its present position. The location of the front, and the time it was at different positions can be determined from analysis of bottom sediments.
\vitem When the meridional overturning circulation\index{circulation!meridional overturning} shuts down, heat normally carried from the south Atlantic to the north Atlantic becomes available to warm the southern hemisphere. This results in a climate 'sea-saw' between northern and southern hemispheres.
\vitem The switching on and off of the deep circulation has large hysteresis (figure 13.3). The circulation has two stable states. The first is the present circulation. In the second, deep water is produced mostly near Antarctica, and upwelling\index{upwelling!in North Pacific} occurs in the far north Pacific (as it does today) and in the far north Atlantic. Once the circulation is shut off, the system switches to the second stable state. The return to normal salinity does not cause the circulation to turn on. Surface waters must become saltier than average for the first state to return (Rahmstorf, 1995).
\vitem Heinrich events seem to precede the largest Dansgaard/Oeschger events (Stocker and Marchal, 2000). Here's what seems to happen. The Heinrich event shuts off the Atlantic deep circulation which leads to a very cold north Atlantic (Martrat et al, 2007). This is followed about 1000 years later by a Dansgaard/Oeschger event with rapid warming.
\vitem Dansgaard/Oeschger--Heinrich tandem events have global influence, and they are related to warming events seen in Antarctic ice cores. Temperatures changes in the two hemispheres are out of phase. When Greenland warms, Antarctica cools. Recent data from the European Project for Ice Coring in Antarctica (\textsc{epica}) shows that in the period between 20,000 and 90,000 years ago, 40\% of the variance in the Greenland temperature data can be explained by Antarctic temperature data (Steig, 2006).
\vitem A weakened version of this process with a period of about 1000 years may be modulating present-day climate in the north Atlantic, and it may have been responsible for the Little Ice Age from 1100 to 1800.
\end{enumerate}
The relationship between variations in salinity, air temperature, deep-water formation, and the atmospheric circulation is not yet understood. For example, we don't know if changes in the atmospheric circulation trigger changes in the meridional overturning circulation, or if changes in the meridional overturning circulation trigger changes in the atmospheric circulation (Brauer et al, 2008). Furthermore, surges may result from warmer temperatures caused by increased water vapor from the tropics (a greenhouse gas) or from an internal instability of a large ice sheet. We do know, however, that climate can change very abruptly, and that circulation in the northern hemisphere has a very sensitive threshold, that when crossed, causes large changes in the circulation pattern.
For example, Steffensen (2008) found that 11,704, 12,896, and 14,694 years before 2000 \textsc{ad} the temperature of the source water for Greenland precipitation warmed 2--4\degrees C in 1--3 years. This indicates a very rapid reorganization of the atmospheric circulation at high latitudes in the northern hemisphere and a shift in the location of the source region. During the earliest event air temperature over Greenland warmed by $\approx$ 10\degrees C in 3 years. At the later events, air temperature over Greenland changed more slowly, over 60 to 200 years. Brauer et al (2008) found an abrupt change in storminess over Germany at almost exactly the same time, 12,679 years ago.
\section{Theory for the Deep Circulation}
\index{deep circulation!theory
for|(}\index{circulation!deep!theory for|(}\index{oceanic
circulation!deep!theory for|(}Stommel, Arons, and Faller in a series of papers from 1958 to 1960 described a simple theory of the abyssal circulation\index{abyssal circulation}\index{circulation!abyssal}\index{oceanic circulation!abyssal} (Stommel 1958; Stommel, Arons, and Faller, 1958; Stommel and Arons, 1960). The theory differed so greatly from what was expected that Stommel and Arons devised laboratory experiments with rotating fluids to confirmed their theory. The theory for the deep circulation has been further discussed by Marotzke (2000) and Munk and Wunsch (1998).
The Stommel, Arons, Faller theory \index{Stommel, Arons, Faller theory|(}is based on three
fundamental ideas\index{deep circulation!fundamental ideas}\index{circulation!deep!fundamental
ideas}\index{oceanic circulation!deep!fundamental ideas}:
\begin{enumerate}
\vitem Cold, deep water is supplied by deep convection at a few high-latitude
locations in the Atlantic, notably in the Irminger and Greenland Seas in the
north and the Weddell Sea in the south.
\vitem Uniform mixing\index{mixing!of deep waters} in the ocean brings the cold, deep water
back to the surface.
\vitem The deep circulation is strictly geostrophic in the interior\index{geostrophic
currents!deep interior} of the ocean, and therefore potential vorticity is conserved.
\end{enumerate}
Notice that the deep circulation\index{deep circulation} is driven by mixing\index{mixing!of
deep waters}, not by the sinking of cold water at high latitudes. Munk and Wunsch (1998)
point out that deep convection by itself leads to a deep, stagnant, pool of cold
water. In this case, the deep circulation is confined to the upper layers of
the ocean. Mixing or upwelling\index{upwelling!and deep circulation} is required to pump cold
water upward through the thermocline\index{thermocline!mixing in}\index{mixing!in thermocline}
and drive the deep circulation. Winds and tides are the primary source of energy driving the
mixing\index{mixing!tidal}.
Notice also that convection and sinking are not the same, and they do not occur in the same place (Marotzke and Scott, 1999). Convection occurs in small regions a few kilometers on a side. Sinking, driven by Ekman pumping\index{Ekman pumping} and geostrophic currents, can occur over far larger areas. In this chapter, we are discussing mostly sinking of water.
To describe the simplest aspects of the flow, we begin with the Sverdrup equation
applied to a bottom current of thickness $H$ in an ocean of constant depth:
\begin{equation}
\beta\,v =f\,\frac{\partial{w}}{\partial{z}}
\end{equation}
where $f =2\,\Omega\,\sin \varphi$, $\beta = \left(2\Omega\,\cos \varphi
\right)/{R}$, $\Omega$ is earth's rotation rate, $R$ earth's radius, and
$\varphi$ is latitude. Integrating (13.1) from the bottom of the ocean to the top
of the abyssal circulation\index{abyssal
circulation}\index{circulation!abyssal}\index{oceanic circulation!abyssal} gives:
\begin{align}
V &= \int_{0}^{H} v\,dz = \int_{0}^{H}
\frac{f}{\beta}\,\frac{\partial{w}}{\partial{z}}\,dz \notag \\
V &= R \tan \varphi \,W_0
\end{align}
where $V$ is the vertical integral of the northward velocity, and $W_0$ is the
velocity at the base of the thermocline\index{thermocline!vertical velocity in}. $W_0$ must be
positive (upward) almost everywhere to balance the downward mixing\index{mixing!of heat
downward} of heat. Then $V$ must be everywhere toward the poles. This is the abyssal flow in
the interior of the ocean sketched by Stommel in figure 13.4. The $U$ component of the flow
is calculated from
$V$ and $w$ using the continuity equation.
\begin{figure}[t!]
%\centering
\makebox[120mm] [c]{\includegraphics{stommeldeep}}
\footnotesize
Figure 13.4 Idealized sketch of \rule{0mm}{4ex}the deep circulation
due to deep convection in the Atlantic (dark circles) and
upwelling\index{upwelling!and deep circulation} through the
thermocline\index{thermocline!vertical velocity in} elsewhere. The real circulation
is much different than the circulation shown in this sketch. After Stommel (1958).
\label{fig:stommeldeep}
\vspace{-3ex}
\end{figure}
To connect the streamlines of the flow in the west, Stommel added a deep western
boundary current. The strength of the western boundary current depends on the
volume of water $S$ produced at the source regions.
Stommel and Arons calculated
the flow for a simplified ocean bounded by the Equator and two meridians (a pie
shaped ocean). First they placed the source $S_0$ near the pole to approximate
the flow in the north Atlantic. If the volume of water sinking at the source
equals the volume of water upwelled in the basin, and if the upwelled velocity is
constant everywhere, then the transport\index{transport!in western boundary currents} $T_w$ in
the western boundary current is:
\begin{equation}
T_w = -2\,S_0 \sin \varphi
\end{equation}
The transport in the western boundary current at the poles is twice the volume of
the source, and the transport diminishes to zero at the Equator (Stommel and
Arons, 1960a: eq, 7.3.15; see also Pedlosky, 1996: \S 7.3). The flow driven by
the upwelling\index{upwelling!and deep circulation} water adds a recirculation equal to the
source. If
$S_0$ exceeds the volume of water upwelled in the basin, then the western boundary current
carries water across the Equator. This gives the western boundary current
sketched in the north Atlantic in figure 13.4.
Next, Stommel and Arons calculated the transport\index{transport!calculated by Stommel and
Arons} in a western boundary current in a basin with no source. The transport is:
\begin{equation}
T_w = S \left[ 1 - 2 \, \sin \varphi \right]
\end{equation}
where $S$ is the transport\index{transport!across equator} across the Equator from the other
hemisphere. In this basin Stommel notes:
\begin{quote} \small
A current of recirculated water equal to the source strength starts at the pole
and flows toward the source $\ldots$ [and] gradually diminishes to zero at
$\varphi = 30$\degrees north latitude. A northward current of equal strength
starts at the equatorial source and also diminishes to zero at 30\degrees north
latitude.
\end{quote}
This gives the western boundary current as sketched in the north Pacific in
figure 13.4.
\begin{figure}[t!]
\centering
\makebox[120mm] [c]{\includegraphics{deepindian}}
\footnotesize
Figure 13.5 Deep flow\rule{0mm}{3ex} in the Indian Ocean inferred from the temperature,
given in \degrees C. Note that the flow is constrained by the deep mid-ocean ridge system.
After Tchernia (1980).
\label{fig:deepindian}
\vspace{-3ex}
\end{figure}
Note that the Stommel-Arons theory assumes a flat bottom. The mid-ocean ridge system divides the deep ocean into a series of basins connected by sills through which the water flows from one basin to the next. As a result, the flow in the deep ocean is not as simple as that sketched by Stommel. Boundary
current flow along the edges of the basins, and flow in the eastern basins in the Atlantic comes through the mid-Atlantic ridge from the western basics. Figure 13.5 shows how ridges control the flow in the Indian Ocean.
Finally, Stommel-Arons theory gives some values for time required for water to move from the source regions to the base of the thermocline\index{thermocline} in various basins. The time varies from a few hundred years for basins near the sources to nearly a thousand years for the north Pacific, which is farther from the sources\index{Stommel, Arons, Faller theory|)}.
\paragraph{Some Comments on the Theory for the Deep Circulation} Our understanding of the deep circulation is still evolving.
\begin{enumerate}
\vitem Marotzke and Scott (1999) points out that deep convection and mixing\index{mixing!of
deep waters} are very different processes. Convection reduces the potential energy of the water column, and it is self powered. Mixing in a stratified fluid increases the potential energy, and it must be driven by an external process.
\vitem Numerical models show that the deep circulation is very sensitive to the assumed value of vertical eddy diffusivity in the thermocline\index{thermocline!eddy diffusivity in} (Gargett and Holloway, 1992).
\vitem Numerical calculations by Marotzke and Scott (1999) indicate that the mass transport\index{transport!mass} is not limited by the rate of deep convection, but it is sensitive to the assumed value of vertical eddy diffusivity, especially near side boundaries.
\vitem Cold water is mixed upward at the ocean's boundaries, above seamounts\index{mixing!above seamounts} and mid-ocean ridges, along strong currents such as the Gulf Stream\index{Gulf Stream!and deep mixing}, and in the Antarctic Circumpolar Current (Toggweiler and Russell, 2008; Garabato et al, 2004, 2007). Because mixing is strong over mid-ocean ridges and small in nearby areas, flow is zonal in the ocean basins and poleward along the ridges (Hogg et al. 2001). A map of the circulation will not look like figure 13.4. Numerical models\index{numerical models!deep circulation} and measurements of deep flow by floats show the flow is indeed zonal.
\vitem Because the transport of mass, heat, and salt are not closely related the transport of heat into the north Atlantic may not be as sensitive to surface salinity as described above\index{deep circulation!theory for|)}\index{circulation!deep!theory for|)}\index{oceanic circulation!deep!theory for|)}.
\end{enumerate}
\section{Observations of the Deep Circulation}
\index{deep circulation!observations of}\index{circulation!deep!observations of}\index{oceanic
circulation!deep!observations of}The abyssal circulation\index{abyssal
circulation}\index{circulation!abyssal}\index{oceanic circulation!abyssal} is less well known than the
upper-ocean circulation. Direct observations from moored current meters or deep-drifting
floats were difficult to make until recently, and there are few long-term direct measurements
of current. In addition, the measurements do not produce a stable mean value for the deep
currents. For example, if the deep circulation takes roughly 1,000 years to transport\index{transport!by Antarctic Circumpolar Current} water from the north Atlantic to the Antarctic Circumpolar Current\index{Antarctic Circumpolar Current} and then to the north Pacific, the mean flow is about 1 mm/s. Observing this small mean flow in the presence of typical deep currents having variable velocities of up to 10 cm/s or greater, is very difficult.
Most of our knowledge of the deep circulation is inferred from measured distribution of water masses with their distinctive temperature and salinity and their concentrations of oxygen, silicate, tritium, fluorocarbons and other tracers. These measurements are much more stable than direct current measurements, and observations made decades apart can be used to trace the circulation. Tomczak (1999) carefully describes how the techniques can be made quantitative and how they can be applied in practice.
\paragraph{Water Masses}
The concept of water masses originates in meteorology. Vilhelm Bjerknes, a
Norwegian meteorologist, first described the cold air masses that form in the
polar regions. He showed how they move southward, where they collide with
warm air masses at places he called fronts, just as masses of troops collide at
fronts in war (Friedman, 1989). In a similar way, water masses are formed in
different regions of the ocean, and the water masses are often separated by
fronts. Note, however, that strong winds are associated with fronts in the
atmosphere because of the large difference in density and temperature on either
side of the front. Fronts in the ocean sometimes have little contrast in
density, and these fronts have only weak currents.
Tomczak (1999) defines a \textit{water mass}\index{water mass|textbf} as a
\begin{quote} \small
body of water with a common formation history, having its origin in a physical
region of the ocean. Just as air masses in the atmosphere, water masses are
physical entities with a measurable volume and therefore occupy a finite
volume in the ocean. In their formation region they have exclusive occupation
of a particular part of the ocean. Elsewhere they share the ocean with other
water masses with which they mix. The total volume of a water mass is given by
the sum of all its elements regardless of their location.
\end{quote}
\begin{figure}[b!]
\vspace{-2ex}
\makebox[120mm] [c]{\includegraphics{GulfStreamTSDPlot}}
\footnotesize
Figure 13.6 Temperature \rule{0mm}{3ex}and salinity measured at
hydrographic stations\index{hydrographic data!across Gulf Stream} on either side of the
Gulf Stream\index{Gulf Stream!cross section of}. Data are from tables 10.2 and 10.4.
\textbf{Left:} Temperature and salinity plotted as a function of depth.
\textbf{Right:} The same data, but salinity is plotted as a function of
temperature in a \textit{T-S} plot. Notice that temperature and salinity are
uniquely related below the mixed layer\index{mixed layer!T-S plot}. A few depths are noted next
to data points.
\label{fig:GulfStreamTSDPlot}
%\vspace{-3ex}
\end{figure}
Plots of salinity as a function of temperature, called \textit{T-S} plots, are
used to delineate water masses and their geographical distribution, to describe
mixing\index{mixing!between water masses} among water masses, and to infer motion of water in
the deep ocean. Here's why the plots are so useful: water properties, such as temperature
and salinity, are formed only when the water is at the surface or in the mixed
layer\index{mixed layer!water mass formation within}. Heating, cooling, rain, and evaporation
all contribute. Once the water sinks below the mixed layer\index{mixed layer} temperature and
salinity can change only by mixing\index{mixing!between water masses} with adjacent water
masses. Thus water from a particular region has a particular temperature associated with a
particular salinity, and the relationship changes little as the water moves through the deep
ocean.
Thus temperature and salinity are not independent variables. For
example, the temperature and salinity of the water at different depths below
the Gulf Stream\index{Gulf Stream!T-S plots} are uniquely related (figure 13.6, right),
indicating they came from the same source region, even though they do not appear related if
temperature and salinity are plotted independently as a function of depth
(figure 13.6, left).
Temperature\index{temperature!conservation of} and salinity\index{salinity!conservation of} are
\textit{conservative properties}\index{conservative properties|textbf} because there are no
sources or sinks of heat and salt in the interior of the ocean. Other properties, such as
oxygen are non-conservative. For example, oxygen content may change slowly due to oxidation of
organic material and respiration by animals.
Each point in the \textit{T-S} plot is a \textit{water
type}\index{water!type|textbf}\index{water!type|textbf}. This is a mathematical ideal. Some
water masses may be very homogeneous and they are almost points on the plot. Other water
masses are less homogeneous, and they occupy regions on the plot.
\begin{figure}[t!]
%\vspace{-3ex}
\makebox[120mm] [c]{\includegraphics{TSsketch}}
\footnotesize
Figure 13.7 \textbf{Upper:} Mixing \rule{0mm}{3ex}of two water
masses produces a line on a \textit{T-S} plot. \textbf{Lower:} Mixing among three
water masses produces intersecting lines on a \textit{T-S} plot, and the apex at
the intersection is rounded by further mixing\index{mixing!among water masses}. After Defant
(1961: 205).
\label{fig:TSsketch}
\vspace{-3ex}
\end{figure}
Mixing two water types\index{water!type mixing} leads to a straight line on a \textit{T-S}
diagram (figure 13.7). Because the lines of constant density on a \textit{T-S} plot are
curved, mixing\index{mixing!increases density} increases the density of the water. This is
called
\textit{densification}\index{densification|textbf} (figure 13.8).
\paragraph{Water Masses and the Deep Circulation}
\index{water mass!deep circulation}Let's use these ideas of water masses and
mixing\index{mixing!and deep circulation} to study the deep circulation. We start in the south
Atlantic because it has very clearly defined water masses. A \textit{T-S} plot calculated from
hydrographic data\index{hydrographic data!and water masses} collected in the south
Atlantic (figure 13.9) shows three important water masses listed in order of decreasing depth
(table 13.1): Antarctic Bottom Water\index{bottom water!Antarctic} \textsc{aab}, North
Atlantic Deep Water\index{North Atlantic Deep Water} \textsc{nadw}, and Antarctic Intermediate Water\index{Antarctic Intermediate Water} \textsc{aiw}. All are deeper than
one kilometer. The mixing\index{mixing!among water masses} among three water masses shows the
characteristic rounded apexes shown in the idealized case shown in figure 13.7.
The plot indicates that the same water masses can be found throughout the
western basins in the south Atlantic. Now let's use a cross section of salinity
to trace the movement of the water masses using the core method.
\begin{figure}[t!]
\centering
%\vspace{-3ex}
\makebox[120mm] [c]{\includegraphics{densification}}
\footnotesize
Figure 13.8 Mixing \rule{0pt}{3ex} of two water types\index{water!type mixing} of the same
density(L and G) produces water that is denser (M) than either water type. After
Tolmazin (1985: 137).
\label{fig:densification}
%\vspace{-3ex}
\end{figure}
\paragraph{Core Method}
The slow variation from place to place in the ocean of a tracer such as salinity can be used to
determine the source of the waters masses such as those in figure 13.9. This is called the
\textit{core method}\index{core method|textbf}. The method may also be used to track the slow
movement of the water mass. Note, however, that a slow drift of the water and horizontal
mixing\index{mixing!and core method} both produce the same observed properties in the plot, and
they cannot be separated by the core method.
\begin{table}[b!]\small %\centering
\vspace{-3ex}
\begin{tabular*}{121mm}{@{}llc|r|r@{}}
\multicolumn{5}{@{}l@{}}{\bfseries Table 13.1 Water Masses \rule[-1ex]{0mm}{1ex}of
the south Atlantic between 33\degrees S and 11\degrees N} \\
\hline
& \rule{0ex}{2.5ex} & & Temp. & Salinity \\
& & & (\degrees C) & \\
\hline
Antarctic water & Antarctic Intermediate Water\rule{0ex}{3ex} & \textsc{aiw} & 3.3 & 34.15 \\
& Antarctic Bottom Water & \textsc{abw} & 0.4 & 34.67 \\
North Atlantic water & North Atlantic Deep Water \rule{0ex}{3ex} & \textsc{nadw} & 4.0 & 35.00 \\
& North Atlantic Bottom Water & \textsc{nabw} & 2.5 & 34.90 \\
Thermocline water & Subtropical Lower Water \rule{0ex}{3ex} & \textsc{u} & 18.0 & 35.94 \\ [0.5ex]
\hline
\end{tabular*} \\ [0.5ex]
\footnotesize{\ From Defant (1961: table 82)} \rule{0ex}{1.5ex} \hfill
\
%\vspace{-4ex}
\end{table}
\begin{figure}[t!]
%\vspace{-1ex}
\makebox[120mm] [c]{\includegraphics{westernbasinsts}}
\footnotesize
Figure 13.9 \textit{T-S} plot \rule{0mm}{3ex}of data collected at
various latitudes in the western basins of the south Atlantic. Lines drawn
through data from 5\degrees N, showing possible mixing\index{mixing!between water masses}
between water masses:
\textsc{nadw}--North Atlantic Deep Water, \textsc{aiw}--Antarctic
Intermediate Water, \textsc{aab} Antarctic Bottom Water\index{bottom water!Antarctic}, \textsc{u}
Subtropical Lower Water.
\label{fig:WesternBasinsTS}
\vspace{-4ex}
\end{figure}
A \textit{core}\index{core|textbf} is a layer of water with extreme value (in the mathematical
sense) of salinity or other property as a function of depth. An extreme value is a local
maximum or minimum of the quantity as a function of depth. The method assumes that the flow is
along the core. Water in the core mixes with the water masses above and below the core and it
gradually loses its identity. Furthermore, the flow tends to be along surfaces of constant
potential density.
Let's apply the method to the data from the south Atlantic to find the source of the water
masses\index{water mass}. As you might expect, this will explain their names.
We start with a north-south cross section of salinity in the western basins of the
Atlantic (figure 13.10). It we locate the maxima and minima of salinity as a function
of depth at different latitudes, we can see two clearly defined cores. The upper
low-salinity core starts near 55\degrees S and it extends northward at depths near
1000 m. This water originates at the Antarctic Polar Front zone. This is the Antarctic
Intermediate Water\index{water mass!Antarctic Intermediate Water}\index{Antarctic Intermediate Water}. Below this water mass is a core of salty water originating in the far north Atlantic. This is the North
Atlantic Deep Water\index{water mass!North Atlantic Deep Water}\index{North Atlantic Deep Water}. Below this is the most dense water, the Antarctic Bottom Water\index{water mass!Antarctic Bottom Water}. It originates in winter when cold, dense, saline water forms in the Weddell Sea and
other shallow seas around Antarctica. The water sinks along the continental slope and
mixes with Circumpolar Deep Water. It then fills the deep basins of the south Pacific,
Atlantic, and Indian ocean.
\begin{figure}[t!]
%\vspace{-3ex}
\makebox[120mm] [c]{\includegraphics{Cores}}
\footnotesize
Figure 13.10 Contour \rule{0pt}{3ex}plot of salinity as a function of depth in the western
basins of the Atlantic from the Arctic Ocean to Antarctica. The plot clearly shows extensive
cores, one at depths near 1000 m extending from 50\degrees S to 20\degrees N, the other at is
at depths near 2000 m extending from 20\degrees N to 50\degrees S. The upper is the Antarctic
Intermediate Water, the lower is the North Atlantic Deep Water\index{North Atlantic Deep Water}. The arrows mark the assumed
direction of the flow in the cores. The Antarctic Bottom Water\index{bottom water!Antarctic}
fills the deepest levels from 50\degrees S to 30\degrees N. \textsc{pf} is the polar front,
\textsc{saf} is the subantarctic front. See also figures 10.15 and 6.10. After Lynn and Reid (1968).
\label{fig:Cores}
\vspace{-3ex}
\end{figure}
The Circumpolar Deep Water\index{water mass!Circumpolar Deep Water} is mostly North
Atlantic Deep Water that has been carried around Antarctica. As it is carried along,
it mixes with deep waters of the Indian and Pacific Ocean to form the circumpolar
water.
The flow is probably not along the arrows shown in figure 13.10. The
distribution of properties in the abyss\index{abyss} can be explained by a combination
of slow flow in the direction of the arrows plus horizontal mixing\index{mixing!along
constant-density surfaces} along surfaces of constant potential density with some weak vertical
mixing\index{mixing!vertical}. The vertical mixing\index{mixing!vertical} probably
occurs at the places where the density surface reaches the sea bottom at a lateral boundary
such as seamounts, mid-ocean ridges, and along the western boundary. Flow in a plane
perpendicular to that of the figure may be at least as strong as the flow in the plane of the
figure shown by the arrows.
The core method\index{core method} can be applied only to a tracer that does not influence density. Hence temperature is usually a poor choice. If the tracer controls den\-sity, then flow will be around the core according to ideas of geostrophy, not along core as assumed by the core method.
The core method works especially well in the south Atlantic with its clearly
defined water masses. In other ocean basins, the \textit{T-S} relationship is
more complicated. The abyssal waters in the other basins are a complex mixture
of waters coming from different areas in the ocean (figure 13.11). For example,
warm, salty water from the Mediterranean Sea enters the north Atlantic and
spreads out at intermediate depths displacing intermediate water from
Antarctica in the north Atlantic, adding additional complexity to the flow as
seen in the lower right part of the figure.
\begin{figure}[t!]
%\vspace{-2ex}
\centering
\makebox[120mm] [c]{\includegraphics{TSplots}}
\footnotesize
Figure 13.11 \textit{T-S} plots \rule{0pt}{3ex}of water in the
various ocean basins. After Tolmazin (1985: 138).
\label{fig:TSplots}
\vspace{-3ex}
\end{figure}
\paragraph{Other Tracers}
\index{core method!tracers}\index{tracers}I have illustrated the core method using
salinity as a tracer, but many other tracers are used. An ideal tracer is easy to
measure even when its concentration is very small; it is conserved, which means that
only mixing\index{mixing!of tracers} changes its concentration; it does not influence the
density of the water; it exists in the water mass we wish to trace, but not in other adjacent
water masses; and it does not influence marine organisms (we don't want to release toxic
tracers).
Various tracers meet these criteria to a greater or lesser extent, and
they are used to follow the deep and intermediate water in the ocean. Here are
some of the most widely used tracers.
\begin{enumerate}
\vitem
Salinity is conserved, and it influences density much less than
temperature.
\vitem Oxygen is only partly conserved. Its concentration is reduced by the
respiration by marine plants and animals and by oxidation of organic carbon.
\vitem
Silicates are used by some marine organisms. They are conserved at
depths below the sunlit zone.
\vitem
Phosphates are used by all organisms, but they can provide additional
information.
\vitem
$^3$He is conserved, but there are few sources, mostly at deep-sea volcanic areas
and hot springs.
\vitem
$^3$H (tritium) was produced by atomic bomb tests in the atmosphere in the
1950s. It enters the ocean through the mixed layer\index{mixed layer}, and it is useful for
tracing the formation of deep water. It decays with a half life of 12.3 y and it
is slowly disappearing from the ocean. Figure 10.16 shows the slow advection or
perhaps mixing\index{mixing!of tritium} of the tracer into the deep north Atlantic. Note
that after 25 years little tritium is found south of 30\degrees\ N. This implies a mean
velocity of less than a mm/s.
\vitem
Fluorocarbons (Freon used in air conditioning) have been recently
injected into atmosphere. They can be measured with very great sensitivity, and
they are being used for tracing the sources of deep water.
\vitem
Sulphur hexafluoride SF$_6$ can be injected into sea water, and the
concentration can be measured with great sensitivity for many months.
\end{enumerate}
Each tracer has its usefulness, and each provides additional information about
the flow.
\paragraph{North Atlantic Meridional Overturning Circulation}
The great importance of the meridional overturning circulation for European climate has led to programs to monitor the circulation. The Rapid Climate Change/Meridional Overturning Circulation and Heat Flux Array \textsc{rapid/mocha} deployed an array of instruments that measured bottom pressure plus temperature and salinity throughout the water column at 15 locations along 24\degrees N near the western and eastern boundaries and on either side of the mid-Atlantic ridge beginning in 2004 (Church, 2007). At the same time, flow of the Gulf Stream was measured through the Strait of Florida, and wind stress, which gives the Ekman transports, was measured along 24\degrees N by satellite instruments. The measurements show that transport across 24\degrees N was zero, within the accuracy of the measurements, as expected. The one-year average of the Meridional Overturning Circulation was $18.7 \pm 5.6$ Sv, with variability ranging from 4.4 to 35.3 Sv. Accuracy of the measurement was $\pm $ 1.5 Sv.
\section{Antarctic Circumpolar Current}
\index{Antarctic Circumpolar Current}\index{deep circulation!Antarctic Circumpolar Current}\index{circulation!deep!Antarctic Circumpolar Current}\index{oceanic circulation!deep!Antarctic Circumpolar Current}The Antarctic Circumpolar Current is an important feature of the ocean's deep circulation because it transports\index{transport!by Antarctic Circumpolar Current} deep and intermediate water
between the Atlantic, Indian, and Pacific Ocean, and because Ekman pumping driven by westerly winds is a major driver of the deep circulation. Because it is so important for understanding the deep circulation in all ocean, let's look at what is known about this current.
\begin{figure}[t!] %\centering
\makebox[121 mm] [c] {\includegraphics{woce21density}}
\footnotesize
Figure 13.12 Cross \rule{0mm}{3ex}section of neutral density across the Antarctic Circumpolar Current in the Drake Passage from the World Ocean Circulation Experiment\index{World Ocean Circulation Experiment} section A21 in 1990. The current has three streams associated with the three fronts (dark shading): \textsc{sf} = Southern \textsc{acc} Front, \textsc{pf} = Polar Front, and \textsc{saf} = Subantarctic Front. Hydrographic station\index{hydrographic stations!across Antarctic Circumpolar Current} numbers are given at the top, and transports\index{transport!by Antarctic Circumpolar Current} are relative to 3,000 dbar. Circumpolar deep water is indicated by light shading. Data from Alex Orsi, Texas A\&M
University.
\label{fig:P16}
\vspace{-5ex}
\end{figure}
\begin{figure}[t!] %\centering
\makebox[121 mm] [c] {\includegraphics{aacmap}}
\footnotesize
Figure 13.13 Distribution \rule{0mm}{3ex}of fronts around Antarctica:
\textbf{STF}: Subtrobical Front; \textbf{SAF}: Subantarctic Front;
\textbf{PF}: Polar Front; \textbf{SACC}: Southern Antarctic Circumpolar Front. Shaded areas
are shallower than 3 km. From Orsi (1995).
\label{fig:AACx-section}
\vspace{-3ex}
\end{figure}
\begin{figure}[b!] %\centering
\vspace{-3ex}
\makebox[121 mm] [c] {\includegraphics{aacxport}}
\footnotesize
Figure 13.14 Variability of \rule{0mm}{5ex}the transport in the Antarctic\index{transport!by Antarctic Circumpolar Current} Circumpolar Current\index{Antarctic Circumpolar Current} as measured by an array of current meters deployed across the Drake Passage. The heavier line is smoothed, time-averaged transport. From Whitworth (1988).
\label{fig:aacxport}
%\vspace{-3ex}
\end{figure}
A plot of density across a line of constant longitude in the Drake Passage (figure 13.12) shows three fronts. They are, from north to south: 1) the Subantarctic Front, 2) the Polar Front, and 3) the Southern \textsc{acc} Front. Each front is continuous around Antarctica (figure 13.13). The plot also shows
that the constant-density surfaces slope at all depths, which indicates that the currents extend to the bottom.
Typical current speeds are around 10 cm/s with speeds of up to 50 cm/s near some fronts. Although the currents are slow, they transport\index{transport!in Southern Ocean} much more water than western boundary currents because the flow is deep and wide. Whitworth and Peterson (1985) calculated transport through the Drake Passage\index{transport!through Drake Passage} using several years of data from an array of 91 current meters on 24 moorings spaced approximately 50 km apart along a line spanning the passage. They also used measurements of bottom pressure measured by gauges on either side of the passage. They found that the average transport through the Drake Passage was
$125 \pm 11$ Sv, and that the transport varied from 95 Sv to 158 Sv. The maximum transport tended to occur in late winter and early spring (figure 13.14).
Because the antarctic currents reach the bottom, they are influenced by topographic steering. As the current crosses ridges such as the Kerguelen Plateau, the Pacific-Antarctic Ridge, and the Drake Passage, it is deflected by the ridges.
The core of the current is composed of Circumpolar Deep Water\index{Circumpolar Deep
Water!composition}, a mixture of deep water from all ocean. The upper branch of the
current contains oxygen-poor water from all ocean. The lower (deeper) branch
contains a core of high-salinity water from the Atlantic, including contributions
from the north Atlantic deep water mixed with salty Mediterranean Sea water. As the
different water masses circulate around Antarctica they mix with other water masses
with similar density. In a sense, the current is a giant `mix-master' taking deep
water from each ocean, mixing\index{mixing!in Circumpolar Current} it with deep water from
other ocean, and then redistributing it back to each ocean (Garabato et al, 2007).
The coldest, saltiest water in the ocean is produced on the continental shelf around Antarctica in winter, mostly from the shallow Weddell and Ross seas. The cold salty water drains from the shelves, entrains some deep water, and spreads out along the sea floor. Eventually, 8--10 Sv of bottom water are formed (Orsi, Johnson, and Bullister, 1999). This dense water then seeps into all the ocean basins. By definition, this water is too dense to cross through the Drake Passage, so it is not circumpolar water.
The Antarctic currents are wind driven. Strong west winds with maximum speed near 50\degrees S drive the currents (see figure 4.2), and the north-south gradient of wind speed produces convergence and divergence of Ekman transports\index{Ekman transport}. Divergence south of the zone of maximum wind speed, south of 50\degrees S leads to upwelling\index{upwelling!of Circumpolar deep Water} of the Circumpolar Deep Water. Convergence north of the zone of maximum winds leads to downwelling of
the Antarctic intermediate water. The surface water is relatively fresh but cold, and when they sink they define characteristics of the Antarctic intermediate water.
The position of the circumpolar current relative to the maximum of the westerly winds influences the meridional overturning circulation and climate. North of the maximum, Ekman transports converge\index{Ekman transport}, pushing water downward into the Antarctic Intermediate Water north of the Polar Front\index{Antarctic Polar Front}. South of the maximum winds, Ekman transports diverge, pulling Circumpolar Atlantic Deep Water to the surface south of the Polar Front, which helps drive the deep circulation (figure 13.10). When the maximum winds are further from the pole, less deep water is pulled upward, and the deep circulation is weak, as it was during the last ice age. As the earth warmed after the ice age, the maximum winds shifted south. The winds were more aligned with the Circumpolar Current, and they pulled more deep water to the surface. Since 1960, the winds have strengthened and shifted southward, further strengthening Circumpolar Current and the deep circulation Toggweiler and Russell, 2008).
Because wind constantly transfers momentum to the Antarctic Circumpolar Current\index{Antarctic
Circumpolar Current},
causing it to accelerate, the acceleration must be balanced by drag,
and we are led to ask: What keeps the flow from accelerating to very high speeds?
Munk and Palmen (1951), suggest form drag dominates. Form drag\index{form
drag}\index{drag!form} is due to the current crossing sub-sea ridges, especially at
the Drake Passage. Form drag is also the drag of the wind on a fast moving car. In
both cases, the flow is diverted, by the ridge or by your car, creating a low
pressure zone downstream of the ridge or down wind of the car. The low pressure zone
transfers momentum into the solid earth, slowing down the current.
\section{Important Concepts}
\begin{enumerate}
\item
The deep circulation of the ocean is very important because it determines
the vertical stratification of the ocean and because it modulates climate.
\vitem
The ocean absorbs CO$_2$ from the atmosphere reducing atmospheric CO$_2$ concentrations.
The deep circulation carries the CO$_2$ deep into the ocean temporarily keeping it from
returning to the atmosphere. Eventually, however, most of the CO$_2$ must be released back to
the atmosphere. But, some remains in the ocean. Phytoplankton convert CO$_2$ into organic
carbon, some of which sinks to the sea floor and is buried in sediments. Some CO$_2$ is used to
make sea shells, and it too remains in the ocean.
\vitem
The production of deep bottom waters in the north Atlantic\index{bottom water!North Atlantic} draws
a\index{transport!heat in North Atlantic} petawatt of heat into the northern hemisphere which helps
warm Europe.
\vitem
Variability of deep water formation in the north Atlantic has been tied
to large fluctuations of northern hemisphere temperature during the last ice
ages.
\vitem
Deep convection which produces bottom water occurs only in the far north Atlantic and at
a few locations around Antarctica.
\vitem
The deep circulation is driven by vertical mixing\index{mixing!of deep waters}, which is
largest above mid-ocean ridges, near seamounts, and in strong boundary currents.
\vitem
The deep circulation is too weak to measure directly. It is inferred from
observations of water masses defined by their temperature and salinity
and from observation of tracers.
\vitem
The Antarctic Circumpolar Current\index{Antarctic Circumpolar Current} mixes deep water from the Atlantic, Pacific, and Indian Ocean and redistributes it back to each ocean. The current is deep and slow with a
transport\index{transport!by Antarctic Circumpolar Current} of 125 Sv.
\end{enumerate}
\chapter{Equatorial Processes}
Equatorial processes are at the center of our understanding the influence of the ocean on the
atmosphere, and they dominate the interannual fluctuations in global weather patterns. The sun\index{sun!warms equatorial watewrs} warms the vast expanses of the tropical Pacific and Indian ocean, evaporating water. When the water condenses as rain it releases so much heat that these areas are the primary engine driving the atmospheric circulation\index{atmospheric circulation!causes} (figure 14.1). Rainfall\index{rainfall!equatorial} over extensive areas exceeds three meters per year (figure 5.5), and some oceanic regions receive more than five meters of rain per year. To put the numbers in perspective,
five meters of rain per year releases on average 400 W/m$^2$ of heat to the atmosphere. Equatorial currents modulate the air-sea interactions, especially through the phenomenon known as El Ni\~{n}o, with global consequences. I describe here first the basic equatorial processes, then the year-to-year variability of the processes and the influence of the variability on weather patterns.
\begin{figure}[b!]
\vspace{-3ex}
\makebox[120mm] [c]{\includegraphics{rainheat}}
\footnotesize
Figure 14.1 Average diabatic \rule{0pt}{3ex}heating between 700 and 50 mb in the
atmosphere during December, January and February calculated from \textsc{ecmwf} data for
1983--1989. Most of the heating is due to the release of latent heat by rain.
After Webster et al. (1992).
\label{fig:rainheat}
%\vspace{-4ex}
\end{figure}
\section{Equatorial Processes}
\index{equatorial processes}The tropical ocean is characterized by a thin, permanent, shallow layer of warm water
over deeper, colder water. In this respect, the vertical stratification is similar to the summer stratification at
higher latitudes. Surface waters are hottest in the west (figure 6.3) in the great Pacific warm pool. The mixed
layer\index{mixed layer!equatorial} is deep in the west and very shallow in the east (Figure
14.2).
\begin{figure}[t!]
\centering
\makebox[120mm] [c]{\includegraphics{equator}}
\footnotesize
Figure 14.2 The mean, \rule{0pt}{3ex}upper-ocean, thermal
structure along the equator in the Pacific from north of New Guinea to Ecuador
calculated from data in Levitus (1982).
\label{fig:equator}
\vspace{-2ex}
\end{figure}
The shallow thermocline\index{thermocline!shallow} has important consequences. The
southeast trade winds blow along the equator (figure 4.2) although they tend to be
strongest in the east. North of the equator, Ekman transport\index{Ekman
transport}\index{transport!equatorial} is northward. South of the equator it is southward. The
divergence of the Ekman flow causes upwelling\index{upwelling!equatorial} on the equator. In
the west, the upwelled water is warm. But in the east the upwelled water is cold because the
thermocline is so shallow. This leads to a cold tongue of water at the sea surface extending
from South America to near the dateline (figure 6.3).
Surface temperature\index{surface temperature}\index{temperature!surface} in the east is a balance among four
processes:
\begin{enumerate}
\vitem The strength of the upwelling\index{upwelling!equatorial}, which is determined by the
westward component of the wind.
\vitem The speed of westward currents which carry cold water from the coast of
Peru and Ecuador.
\vitem North-south mixing\index{mixing!equatorial} with warmer waters on either side of
the equator.
\vitem Heat fluxes through the sea surface along the equator.
\end{enumerate}
The east-west temperature gradient on the equator drives a zonal circulation
in the atmosphere, the Walker circulation. Thunderstorms over the warm pool
carry air upward, and sinking air in the east feeds the return flow at the
surface. Variations in the temperature gradient influences the Walker
circulation, which, in turn, influences the gradient. The feedback
can lead to an instability, the El Ni\~{n}o-Southern Oscillation
(\textsc{enso})\index{Southern Oscillation!Index}\index{Southern
Oscillation!El Ni\~{n}o Southern Oscillation (ENSO)} discussed in the next section.
\paragraph{Surface Currents}
\index{currents!surface}\index{surface currents}The strong stratification confines the
wind-driven circulation to the mixed layer\index{mixed layer!currents} and upper
thermocline\index{thermocline!upper}. Sverdrup's theory and Munk's extension, described in
\S11.1 and \S11.3, explain the surface currents in the tropical Atlantic, Pacific, and Indian
ocean. The currents include (figure 14.3):
\begin{figure}[t!]
%\vspace{-1ex}
\makebox[120mm] [c]{\includegraphics{EqCurr}}
\footnotesize
Figure 14.3 Average currents at 10 m \rule{0pt}{3ex}calculated from
the Modular Ocean Model driven by observed winds and mean heat fluxes\index{heat flux} from
1981 to 1994. The model, operated by the \textsc{noaa} National Centers for
Environmental Prediction, assimilates observed surface and subsurface
temperatures. After Behringer, Ji, and Leetmaa (1998).
\label{fig:EqCurr}
\vspace{-4ex}
\end{figure}
\begin{enumerate}
\vitem The North Equatorial Countercurrent between 3\degrees N and 10\degrees N, which flows eastward with a typical
surface speed of 50 cm/s. The current is centered on the band of weak winds, the
\textit{doldrums}\index{doldrums|textbf}, around 5--10\degrees N where the north and south trade winds converge, the
\textit{tropical convergence zone}\index{tropical convergence zone|textbf}. \vitem The North and South Equatorial
Currents which flow westward in the zonal band on either side of the countercurrent. The currents are shallow, less
than 200 m deep. The northern current is weak, with a speed less than roughly 20 cm/s.
The southern current has a maximum speed of around 100 cm/s, in the band between 3\degrees N
and the equator.
\end{enumerate}
The currents in the Atlantic are similar to those in the Pacific because the
trade winds in that ocean also converge near 5\degrees --10\degrees N.
The South Equatorial Current in the Atlantic continues northwest along the
coast of Brazil, where it is known as the North Brazil Current. In the Indian
Ocean, the doldrums occur in the southern hemisphere and only during the
northern-hemisphere winter. In the northern hemisphere, the currents reverse
with the monsoon winds.
There is, however, much more to the story of equatorial currents.
\paragraph{Equatorial Undercurrent: Observations}
\index{equatorial processes!undercurrent}Just a few meters below the surface on the equator is
a strong eastward flowing current, the Equatorial Undercurrent, the last major oceanic current
to be discovered. Here's the story:
\begin{quotation} \small
In September 1951, aboard the U.S. Fish and Wildlife Service research vessel
long-line fishing on the equator south of Hawaii, it was noticed that the
subsurface gear drifted steadily to the east. The next year Cromwell, in
company with Montgomery and Stroup, led an expedition to investigate the
vertical distribution of horizontal velocity at the equator. Using floating
drogues at the surface and at various depths, they were able to establish the
presence, near the equator in the central Pacific, of a strong, narrow eastward
current in the lower part of the surface layer and the upper part of the
thermocline\index{thermocline!upper} (Cromwell, \textit{et. al.}, 1954). A few years later the
Scripps
\textit{Eastropac} Expedition, under Cromwell's leadership, found the current
extended toward the east nearly to the Galapagos Islands but was not present
between those islands and the South American continent.
The current is remarkable in that, even though comparable in transport\index{transport!by
equatorial undercurrent} to the Florida Current, its presence was unsuspected ten years ago.
Even now, neither the source nor the ultimate fate of its waters has been established. No
theory of oceanic circulation predicted its existence, and only now are such theories
being modified to account for the important features of its flow.---Warren S.
Wooster (1960).
\end{quotation}
The Equatorial Undercurrent in the Atlantic was first discovered by Buchanan in 1886, and in
the Pacific by the Japanese Navy in the 1920s and 1930s (McPhaden, 1986).
\begin{quote} \small
However, no attention was paid to these observations. Other earlier hints regarding this
undercurrent were mentioned by Matth\"{a}us (1969). Thus the old experience
becomes even more obvious which says that discoveries not attracting the
attention of contemporaries simply do not exist.---Dietrich et al. (1980).
\end{quote}
Bob Arthur (1960) summarized the major aspects of the flow:
\begin{enumerate}
\vitem
Surface flow may be directed westward at speeds of 25--75 cm/s;
\vitem
Current reverses at a depth of from 20 to 40 m;
\vitem
Eastward undercurrent extends to a depth of 400 meters with a
transport\index{transport!by equatorial undercurrent} of as much as 30 Sv $=30 \times 10^6$
m$^3$/s;
\vitem
Core of maximum eastward velocity (0.50--1.50 m/s) rises
from a depth of 100 m at 140\degrees W to 40 m at 98\degrees W, then dips down;
\vitem
Undercurrent appears to be symmetrical about the equator and
becomes much thinner and weaker at 2\degrees N and 2\degrees S.
\end{enumerate}
In essence, the Pacific Equatorial Undercurrent is a ribbon with dimensions of
$0.2 \text{ km} \times 300 \text{ km} \times 13,000 \text{ km}$ (figure 14.4).
\begin{figure}[t!]
\makebox[120mm] [c]{\includegraphics{equatorialxsec}}
\footnotesize
Figure 14.4 Cross \rule{0pt}{4ex}section of the Equatorial Undercurrent in the
Pacific calculated from Modular Ocean Model with assimilated surface data (See
\S 14.5). The section is an average from 160\degrees E to 170\degrees E from January
1965 to December 1999. Stippled areas are westward flowing. From Nevin S. Fu\v{c}kar.
\label{fig:equatorialxsec}
\vspace{-3ex}
\end{figure}
\paragraph{Equatorial Undercurrent: Theory}
\index{equatorial processes!undercurrent!theory}Although we do not yet have a complete theory
for the undercurrent, we do have a clear understanding of some of the more important processes
at work in the equatorial regions. Pedlosky(1996), in his excellent chapter on Equatorial
Dynamics of the Thermocline: The Equatorial Undercurrent, points out that the basic dynamical
balances we have used in mid latitudes break down near or on the equator.
Near the equator:
\begin{enumerate}
\vitem
The Coriolis parameter\index{Coriolis parameter!near equator} becomes very small, going to zero
at the equator:
\begin{equation}
f=2\Omega \sin\varphi = \beta y \approx 2\Omega \,\varphi
\end{equation}
where $\varphi$ is latitude, $\beta = \partial f/\partial y \approx 2\Omega/R$
near the equator, and $y=R\,\varphi$.
\vitem
Planetary vorticity $f$ is also small, and the advection of
relative vorticity cannot be neglected. Thus the Sverdrup balance (11.7) must
be modified.
\vitem
The geostrophic\index{geostrophic currents!not near equator} and vorticity balances fail when
the meridional distance $L$ to the equator is $O\left(\sqrt{U/\beta}\right)$, where $\beta =
\partial f / \partial y$. If
$U=1$ m/s, then $L=200$ km or 2\degrees\ of latitude. Lagerloeff et al (1999),
using measured currents, show that currents near the equator can be
described by the geostrophic balance\index{geostrophic balance!not near equator} for $|\varphi
| > 2.2^{\circ}$. They also show that flow closer to the equator can be described using a
$\beta
$-plane approximation\index{B-plane@$\beta$-plane} $f = \beta y$.
\vitem
The geostrophic balance for \textit{zonal} currents works so well near the
equator because $f$ and $\partial \zeta/\partial y \rightarrow 0$ as
$\varphi \rightarrow 0$, where $\zeta$ is sea surface topography.
\end{enumerate}
\vspace{-1.5ex}
Upwelled water along the equator produced by Ekman pumping\index{Ekman pumping} is
not part of a two-dimensional flow in a north-south, meridional plane. The flow is three-dimensional. Water tends to flow along the contours of constant density (isopycnal surfaces), close to the lines of constant temperature in figure 14.2. Cold water enters the undercurrent in the far west Pacific, and it moves eastward and upward along the equator. For example, the 25\degrees isotherm enters the undercurrent at a depth near 125 m in the western Pacific at 170\degrees E and eventually reaches the surface at 125\degrees W in the eastern Pacific.
The meridional geostrophic balance near the equator gives the speed of the zonal
currents, but it does not explain what drives the undercurrent. A very
simplified theory for the undercurrent is based on a balance of zonal pressure
gradients along the equator. Wind stress pushes water westward, producing the
deep thermocline\index{thermocline!deep} and warm pool in the west. The deepening of the
thermocline\index{thermocline!equatorial} causes the sea-surface topography $\zeta$ to be
higher in the west, assuming that flow below the thermocline is weak. Thus there is an
eastward pressure gradient along the equator in the surface layers to a
depth of a few hundred meters. The eastward pressure gradient at the surface (layer A in figure 14.5)
is balanced by the wind stress\index{wind stress!equatorial} $T_x $, and $T_x / H = -\partial p/\partial x$, where H is the mixed-layer depth
Below a few tens of meters in layer B, the influence of the wind stress is
small, and the pressure gradient is unbalanced, leading to an accelerated flow
toward the east, the equatorial undercurrent. Within this layer, the flow
accelerates until the pressure gradient is balanced by frictional forces which
tend to slow the current. At depths below a few hundred meters in layer C, the
eastward pressure gradient is too weak to produce a current, $\partial p / \partial x \approx 0$.
\begin{figure}[t!]
%\vspace{-1ex}
\makebox[120mm] [c]{\includegraphics{equatorsketch}}
\footnotesize
Figure 14.5 \textbf{Left:} \rule{0pt}{3ex}Cross-sectional sketch
of the thermocline\index{thermocline!equatorial} and sea-surface topography along the equator.
\textbf{Right:} Eastward pressure gradient in the central Pacific caused by the density
structure at left.
\label{fig:equatorsketch}
\vspace{-4ex}
\end{figure}
Coriolis forces keep the equatorial undercurrent centered on the equator. If
the flow strays northward, the Coriolis force deflects the current southward.
The opposite occurs if the flow strays southward.
\section[El Ni\~{n}o]{Variable Equatorial Circulation: El Ni\~{n}o/La
Ni\~{n}a} \index{equatorial processes!El Ni\~{n}o}\index{El Ni\~{n}o|(}\index{La
Ni\~{n}a|(}\index{equatorial processes!Ni\~{n}a}The trades are remarkably steady, but they do vary from month to month and year to year, especially in the western Pacific. One important source of variability are Madden-Julian waves in the atmosphere (McPhaden, 1999). If the trades in the west weaken or reverse, the air-sea system in the equatorial regions can be thrown into another state called El Ni\~{n}o. This disruption of the equatorial system in the Pacific is the most important cause of changing weather patterns around the globe.
Although the modern meaning of the term El Ni\~{n}o denotes a disruption of the
entire equatorial system in the Pacific, the term has been used in the past to
describe several very different processes. This causes a lot of confusion. To
reduce the confusion, let's learn a little history.
\paragraph{A Little History}
In the 19th century, the term was applied to conditions off the coast of Peru. The following quote comes from the introduction to Philander's (1990) excellent book \textit{El Ni\~{n}o, La Ni\~{n}a, and the Southern Oscillation}\index{Southern Oscillation}:
\begin{quotation} \small
In the year 1891, Se\~{n}or Dr. Luis Carranza of the Lima Geographical Society, contributed a small article to the Bulletin of that Society, calling attention to the fact that a counter-current flowing from north to south had been observed between the ports of Paita and Pacasmayo.
The Paita sailors, who frequently navigate along the coast in small craft, either to the north or the south of that port, name this counter-current the current of ``El Ni\~{n}o'' (the Child Jesus) because it has been observed to appear immediately after Christmas.
As this counter-current has been noticed on different occasions, and its appearance along the Peruvian coast has been concurrent with rains in latitudes where it seldom if ever rains to any great extent, I wish, on the present occasion, to call the attention of the distinguished geographers here assembled to this phenomenon, which exercises, undoubtedly, a very great influence over the climatic conditions of that part of the world.---Se\~{n}or Frederico Alfonso Pezet's address to the Sixth International Geographical Congress in Lima, Peru 1895.
\end{quotation}
The Peruvians noticed that in some years the El Ni\~{n}o current was stronger
than normal, it penetrated further south, and it is associated with heavy rains
in Peru. This occurred in 1891 when (again quoting from Philander's book)
\begin{quotation} \small
\ldots it was then seen that, whereas nearly every summer here and there there is
a trace of the current along the coast, in that year it was so visible, and its
effects were so palpable by the fact that large dead alligators and trunks of
trees were borne down to Pacasmayo from the north, and that the whole
temperature of that portion of Peru suffered such a change owing to the hot
current that bathed the coast. \ldots ---Se\~{n}or Frederico
Alfonso Pezet.
\ldots the sea is full of wonders, the land even more so. First of all the
desert becomes a garden \ldots . The soil is soaked by the heavy downpour, and
within a few weeks the whole country is covered by abundant pasture. The natural
increase of flocks is practically doubled and cotton can be grown in places
where in other years vegetation seems impossible.---From Mr. S.M. Scott
\& Mr. H. Twiddle quoted from a paper by Murphy, 1926.
\end{quotation}
The El Ni\~{n}o of 1957 was even more exceptional. So much so that it attracted
the attention of meteorologists and oceanographers throughout the Pacific basin.
\begin{quotation} \small
By the fall of 1957, the coral ring of Canton Island, in the memory of man ever
bleak and dry, was lush with the seedlings of countless tropical trees and
vines.
\begin{figure}[b!]
\vspace{-1ex}
\centering
\makebox[120mm] [c]{\includegraphics{ensocorrelations}}
\footnotesize
Figure 14.6 Correlation \rule{0pt}{3ex}coefficient of annual-mean sea-level
pressure with pressure at Darwin. --\ --\ --\ -- Coefficient $< -0.4$. After
Trenberth and Shea (1987).
\label{fig:ensocorrelations}
%\vspace{-3ex}
\end{figure}
One is inclined to select the events of this isolated atoll as epitomizing the
year, for even here, on the remote edges of the Pacific, vast concerted shifts
in the ocean and atmosphere had wrought dramatic change.
Elsewhere about the Pacific it also was common knowledge that the year had been
one of extraordinary climatic events. Hawaii had its first recorded typhoon;
the seabird-killing \textit{El Ni\~{n}o} visited the Peruvian coast; the ice
went out of Point Barrow at the earliest time in history; and on the Pacific's
western rim, the tropical rainy season lingered six weeks beyond its appointed
term---Sette and Isaacs (1960).
\end{quotation}
Just months after the event, in 1958, a distinguished group of oceanographers
and meteorologists assembled in Rancho Santa Fe, California to try to understand
the \textit{Changing Pacific Ocean in 1957 and 1958} (Sette and Isaacs (1960). There, for
perhaps the first time, they began the synthesis of atmospheric and oceanic events leading to
our present understanding of El Ni\~{n}o.
While oceanographers had been mostly concerned with the eastern equatorial
Pacific and El Ni\~{n}o, meteorologists had been mostly concerned with the
western tropical Pacific, the tropical Indian Ocean, and the
Southern Oscillation\index{Southern Oscillation}. Hildebrandsson, the Lockyers, and Sir
Gilbert Walker noticed in the early decades of the 20th century that pressure fluctuations
throughout that region are highly correlated with pressure fluctuations in many
other regions of the world (figure 14.6). Because variations in pressure are
associated with winds and rainfall, they wanted to find out if pressure in
one region could be used to forecast weather in other regions using the
correlations.
The early studies found that the two strongest centers of the variability are near
Darwin, Australia and Tahiti. The fluctuations at Darwin are opposite those at
Tahiti, and resemble an oscillation. Furthermore, the two centers had strong
correlations with pressure in areas far from the Pacific. Walker named the
fluctuations the \textit{Southern Oscillation}\index{Southern Oscillation|textbf}.
The \textit{Southern Oscillation Index}\index{Southern Oscillation!Index|textbf} is
sea-level pressure at Tahiti minus sea-level pressure at Darwin (figure 14.7)
normalized by the standard deviation of the difference. The index is related to the
trade winds. When the index is high, the pressure gradient between east and west
in the tropical Pacific is large, and the trade winds are strong. When the index is
negative trades, are weak.
\begin{figure}[t!]
%\vspace{-3ex}
\makebox[121mm] [c]{\includegraphics{NOAA-SOI}}
\footnotesize
Figure 14.7 Normalized Southern Oscillation \rule{0pt}{3ex}Index
from 1951 to 1999. The normalized index is sea-level pressure anomaly at Tahiti
divided by its standard deviation minus sea-level pressure anomaly at Darwin
divided by its standard deviation then the difference is divided by the standard
deviation of the difference. The means are calculated from 1951 to 1980. Monthly
values of the index have been smoothed with a 5-month running mean. Strong El
Ni\~{n}o events occurred in 1957--58, 1965--66, 1972--73, 1982--83, 1997--98. Data
from
\textsc{noaa}.
\label{fig:soi}
\vspace{-3ex}
\end{figure}
The connection between the Southern Oscillation\index{Southern Oscillation} and El Ni\~{n}o was made soon after the Rancho Santa Fe meeting. Ichiye and Petersen (1963) and Bjerknes (1966) noticed the relationship between equatorial temperatures in the Pacific during the 1957 El Ni\~{n}o and fluctuations in the trade winds associated with the Southern Oscillation. The theory was further developed by Wyrtki (1975).
Because El Ni\~{n}o and the Southern Oscillation\index{Southern Oscillation} are so closely related, the phenomenon is often referred to as the \textit{El Ni\~{n}o--Southern Oscillation}\index{Southern
Oscillation!El Ni\~{n}o Southern Oscillation (ENSO)|textbf} or \textsc{enso}. More recently, the oscillation is referred to as El Ni\~{n}o/La Ni\~{n}a, where La Ni\~{n}a refers to the positive phase of the oscillation when trade winds are strong, and water temperature in the eastern equatorial region is very cold.
\paragraph{Definition of El Ni\~{n}o}
\index{El Ni\~{n}o!defined}Philander (1990) points out that each El Ni\~{n}o is unique, with different temperature, pressure, and rainfall\index{rainfall!patterns} patterns. Some are strong, some are weak. So, exactly what events deserve to be called El Ni\~{n}o? The \textsc{icoads} \index{ICOADS (international comprehensive ocean-atmosphere data set)}data show that the best indicator of El Ni\~{n}o is sea-level pressure anomaly in the eastern equatorial Pacific from 4\degrees S to 4\degrees N and from 108\degrees W to 98\degrees W (Harrison and Larkin, 1996). It correlates better with sea-surface temperature in the central Pacific than with the Southern-Oscillation Index. Thus the importance of the El Ni\~{n}o is not exactly proportional to the Southern Oscillation Index\index{Southern Oscillation!Index}---the strong El Ni\~{n}o of 1957--58, has a weaker signal in figure 14.7 than the weaker El Ni\~{n}o of 1965--66.
Trenberth (1997) recommends that those disruptions of the equatorial system in the Pacific shall be called an El Ni\~{n}o only when the 5-month running mean of sea-surface temperature anomalies\index{anomalies!sea-surface temperature} in the region 5\degrees N--5\degrees S, 120\degrees W--170\degrees W exceeds 0.4\degrees C for six months or longer.
So El Ni\~{n}o, which started life as a change in currents off Peru each Christmas, has grown into a giant. It now means a disruption of the ocean-atmosphere system over the whole equatorial Pacific.
\paragraph{Theory of El Ni\~{n}o}
\index{El Ni\~{n}o!theory of}\index{La Ni\~{n}a!theory of}Wyrtki (1975) gives a clear description of El Ni\~{n}o.
\begin{quote} \small
During the two years preceding El Ni\~{n}o, excessively strong southeast
trades are present in the central Pacific. These strong southeast trades
intensify the subtropical gyre of the South Pacific, strengthen the South
Equatorial Current, and increase the east-west slope of sea level by building
up water in the western equatorial Pacific. As soon as the wind
stress\index{wind stress!equatorial} in the central Pacific relaxes, the accumulated water
flows eastward, probably in the form of an equatorial Kelvin
wave\index{waves!Kelvin}. This wave leads to the accumulation of warm water
off Ecuador and Peru and to a depression of the usually shallow
thermocline\index{thermocline!equatorial}. In total, El Ni\~{n}o is the result of the response
of the equatorial Pacific to atmospheric forcing by the trade winds.
\end{quote}
Sometimes the trades in the western equatorial Pacific not only weaken, they actually reverse direction for a few weeks to a month, producing \textit{westerly wind bursts}\index{westerly wind bursts|textbf} that quickly deepen the thermocline\index{thermocline!equatorial} there. The deepening of the thermocline\index{thermocline!equatorial} launches an eastward propagating Kelvin\index{waves!Kelvin} wave and a westward propagating Rossby wave\index{waves!Rossby}. (If you are asking, What are Kelvin and Rossby waves? I will answer that in a minute. So please be patient.)
\begin{figure}[p!]
\makebox[121mm] [c]{\includegraphics{elninoanomaliesR}}
\footnotesize
Figure 14.8 Anomalies\index{anomalies!sea-surface temperature} \rule{0pt}{3ex}of sea-surface
temperature (in \degrees C) during a typical El Ni\~{n}o obtained by averaging data from El Ni\~{n}os between 1950 and 1973. Months are after the onset of the event. After Rasmusson and Carpenter (1982).
\label{fig:elninoanomalies}
\vspace{-3ex}
\end{figure}
The Kelvin\index{waves!Kelvin} wave deepens the thermocline\index{thermocline!and Kelvin waves} as it moves eastward, and it carries warm water eastward. Both processes cause a deepening of the mixed layer\index{mixed layer!deepened by Kelvin waves} in the eastern equatorial Pacific a few months after the wave is launched in the western Pacific. The deeper thermocline\index{thermocline!equatorial} in the east leads to upwelling\index{upwelling!equatorial} of warm water, and the surface temperatures offshore of Ecuador and Peru warms by 2--4\degrees. The warm water reduces the temperature contrast between east and west, further reducing the trades. The strong positive feedback between sea-surface temperature and the trade winds causes rapid development of El Ni\~{n}o.
With time, the warm pool spreads east, eventually extending as far as 140\degrees W (figure 14.8). Plus, water warms in the east along the equator due to upwelling of warm water, and to reduced advection of cold water from the east due to weaker trade winds.
The warm waters along the equator in the east cause the areas of heavy rain to move eastward from Melanesia and Fiji to the central Pacific. Essentially, a major source of heat for the atmospheric circulation moves from the west to the central Pacific, and the whole atmosphere responds to the change. Bjerknes (1972), describing the interaction between the ocean and the atmosphere over the eastern equatorial Pacific concluded:
\begin{quote} \small
In the cold ocean case (1964) the atmosphere has a pronounced stable layer between 900 and 800 mb, preventing convection and rainfall\index{rainfall!over cold ocean}, and in the warm case (1965) the heat supply from the ocean eliminates the atmospheric stability and activates rainfall. \ldots A side effect of the widespread warming of the tropical belt of the atmosphere shows up in the increase of exchange of angular momentum with the neighboring subtropical belt, whereby the subtropical westerly jet strengthens \ldots The variability of the heat and moisture supply to the global atmospheric thermal engine from the equatorial Pacific can be shown to have far-reaching large-scale effects.
\end{quote}
Klaus Wyrtki (1985), drawing on extensive observations of El Ni\~{n}o, writes:
\begin{quote}\small
A complete El Ni\~{n}o cycle results in a net heat discharge from the tropical Pacific toward higher latitudes. At the end of the cycle the tropical Pacific is depleted of heat, which can only be restored by the slow accumulation of warm water in the western Pacific by the normal trade winds. Consequently, the time scale of the Southern Oscillation is given by the time required for the accumulation of warm water in the western Pacific.
\end{quote}
It is these far reaching events that make El Ni\~{n}o so important. Few people care about warm water off Peru around Christmas, many care about global changes the weather. El Ni\~{n}o is important because of its atmospheric influence.
When the Kelvin\index{waves!Kelvin} wave reaches the coast of Ecuador, part is reflected as an westward propagating Rossby wave\index{waves!Rossby}, and part propagates north and south as a coastal trapped Kelvin wave carrying warm water to higher latitudes. For example, during the 1957 El Ni\~{n}o, the northward propagating Kelvin wave produced unusually warm water off shore of California, and it eventually reached Alaska. This warming of the west coast of North America further influences climate in North America, especially in California.
As the Kelvin\index{waves!Kelvin} wave moves along the coast, it forces Rossby waves which move west across the Pacific at a velocity that depends on the latitude (14.4). The velocity is very slow at high latitudes and fastest on the equator, where the reflected wave moves back as a deepening of the thermocline\index{thermocline!equatorial}, reaching the central equatorial Pacific a year later. Similarly, the westward propagating Rossby wave\index{waves!Rossby} launched at the start of the El Ni\~{n}o in the west, reflects off Asia and returns to the central equatorial Pacific as a Kelvin wave, again about a year later.
El Ni\~{n}o ends when the Rossby waves reflected from Asia and Ecuador meet in the central Pacific about a year after the onset of El Ni\~{n}o (Picaut, Masia, and du Penhoat, 1997). The waves push the warm pool at the surface toward the west. At the same time, the Rossby\index{waves!Rossby} wave reflected from the western boundary causes the thermocline\index{thermocline!equatorial} in the central
Pacific to become shallower when the waves reaches the central Pacific. Then any strengthening of the trades causes upwelling\index{upwelling!equatorial} of cold water in the east, which increases the east-west temperature gradient, which increases the trades, which increases the upwelling (Takayabu et al 1999). The system is then thrown into the La Ni\~{n}a state with strong trades, and a very cold tongue along the equator in the east.
La Ni\~{n}a tends to last longer than El Ni\~{n}o, and the cycle from La Ni\~{n}a to El Ni\~{n}o and back takes about three years. The cycle isn't exact. El Ni\~{n}o comes back at intervals from 2-7 years, with an average near four years (figure 14.7)\index{El Ni\~{n}o|)}\index{La Ni\~{n}a|)}.
\paragraph{Equatorial Kelvin and Rossby Waves}
Kelvin and Rossby waves\index{waves!Rossby} are the ocean's way of adjusting to changes in forcing such as westerly wind bursts. The adjustment occurs as waves of current and sea level that are influenced by gravity, Coriolis force $f$, and the north-south variation of Coriolis force $\partial f/\partial y = \beta$. There are many kinds of these waves with different frequencies, wavelengths, and velocities. If gravity and $f$ are the restoring forces, the waves are called Kelvin and Poincar\'{e} waves. If $\beta $ is the restoring force, the waves are called planetary waves. One important type of planetary wave is the Rossby wave.
Two types of waves are especially important for El Ni\~{n}o: internal Kelvin\index{waves!Kelvin} waves and Rossby\index{waves!Rossby} waves. Both waves can have modes that are confined to a narrow, north-south region centered on the equator. These are \textit{equatorially trapped waves}\index{equatorially trapped waves|textbf}. Both exist in slightly different forms at higher latitudes.
Kelvin\index{waves!Kelvin} and Rossby wave theory is beyond the scope of this book, so I will just tell you what they are without deriving the properties of the waves. If you are curious, you can find the details in Philander (1990): Chapter 3; Pedlosky (1987): Chapter 3; and Apel (1987): \S6.10--6.12. If you know little about waves, their wavelength, frequency, group and phase velocities, skip to Chapter 16 and read \S16.1.
The theory for equatorial waves is based on a two-layer model of the
ocean (figure 14.9). Because the tropical ocean have a thin, warm, surface
layer above a sharp thermocline\index{thermocline!equatorial}, such a model is a good
approximation for those regions.
\begin{figure}[t!]
\centering
\makebox[121mm] [c]{\includegraphics{modelsketch}}
\footnotesize
Figure 14.9 Sketch \rule{0pt}{4ex}of the two-layer model of the equatorial ocean used to calculate planetary waves in those regions. After Philander (1990: 107).
\label{fig:modelsketch}
%\vspace{-3ex}
\end{figure}
Equatorial-trapped Kelvin\index{waves!Kelvin} waves are non-dispersive, with group velocity:
\begin{equation}
c_{Kg} = c \equiv \sqrt{g'H}; \qquad \text{where} \qquad
g' = \frac{\rho_2 - \rho_1}{\rho_1}\,g
\end{equation}
$g'$ is \textit{reduced gravity}\index{reduced gravity|textbf}, $\rho_1, \, \rho_2$ are the densities above and below the thermocline\index{thermocline!equatorial}, and $g$ is gravity. Trapped Kelvin waves propagate only to the east. Note, that $c$ is the phase and group velocity of a shallow-water, internal, gravity wave. It is the maximum velocity at which disturbances can travel along the thermocline. Typical values of the quantities in (14.2) are:
\begin{equation}
\frac{\rho_2 - \rho_1}{\rho_1} = 0.003; \qquad H=150 \text{ m;} \qquad c =2.1
\text{ m/s} \notag
\end{equation}
At the equator, Kelvin\index{waves!Kelvin} waves propagate eastward at speeds of up to 3 m/s, and they cross the Pacific in a few months. Currents associated with the wave are everywhere eastward with north-south component (figure 14.10).
\begin{figure}[t!]
%\vspace{-2ex}
\makebox[121mm] [c]{\includegraphics{rossbycurrents}}
\footnotesize
Figure 14.10 \textbf{Left:} Horizontal \rule{0pt}{4ex}currents associated with equatorially trapped waves generated by a bell-shaped displacement of the thermocline\index{thermocline!equatorial}.
\textbf{Right:} Displacement of the thermocline\index{thermocline!equatorial} due to the waves. The figures shows that after 20 days, the initial disturbance has separated into an westward propagating Rossby\index{waves!Rossby} wave (left) and an eastward propagating Kelvin\index{waves!Kelvin} wave (right). After Philander et al. (1984: 120).
\label{fig:rossbycurrents}
\vspace{-4ex}
\end{figure}
Kelvin waves can also propagate poleward as a trapped wave along an east coast of an ocean basin. Their group velocity is also given by (14.3), and they are confined to a coastal zone with width $x=c/\left(\beta\,y\right)$
The important Rossby\index{waves!Rossby} waves on the equator have frequencies much less than the Coriolis frequency. They can travel only to the west. The group velocity is:
\begin{equation}
c_{Rg} = - \frac{c}{\left(2\,n+1\right)}; \qquad n=1,\,2,\,3,\,\ldots
\end{equation}
The fastest wave travels westward at a velocity near 0.8 m/s. The currents associated with the wave are almost in geostrophic balance\index{geostrophic balance!and Rossby waves} in two counter-rotating eddies centered on the equator (figure
14.10).
Away from the equator, low-frequency, long-wavelength Rossby\index{waves!Rossby} waves also
travel only to the west, and the currents associated with the waves are again almost in geostrophic balance. Group velocity depends strongly on latitude:
\begin{equation}
c_{Rg} = -\frac{\beta\,g'\,H}{f^2}
\end{equation}
Wave dynamics in the equatorial regions differ markedly from wave dynamics at mid-latitudes. Baroclinic waves are much faster, and the response of the ocean to changes in wind forcing is much faster than at mid-latitudes. For planetary waves near the equator, we can speak of an \textit{equatorial wave guide}.
Now, let's return to El Ni\~{n}o and its ``far-reaching large-scale effects.''
\section{El Ni\~{n}o Teleconnections}
\textit{Teleconnections}\index{teleconnections|textbf}\index{El Ni\~{n}o!teleconnections|textbf}\index{La Ni\~{n}a!teleconnections|textbf} are statistically significant correlations between weather e\-vents that occur at different places on the earth. Figure 14.11 shows the dominant global teleconnections associated with the El Ni\~{n}o/Southern Oscillation\index{Southern Oscillation!El Ni\~{n}o Southern Oscillation (ENSO)}.
\begin{figure}[t!]
%\vspace{-1ex}
\makebox[121mm] [c]{\includegraphics{teleconnections}}
\footnotesize
Figure 14.11 Sketch \rule{0pt}{4ex}of regions receiving enhanced rain (dashed lines) or drought (solid lines) during an El Ni\~{n}o event. (0) indicates that rain changed during the year in which El Ni\~{n}o began, (+)indicates that rain changed during the year after El Ni\~{n}o began. After Ropelewski and
Halpert (1987).
\label{fig:teleconnections}
\vspace{-4ex}
\end{figure}
The influence of \textsc{enso}\index{Southern Oscillation!El Ni\~{n}o Southern Oscillation (ENSO)} is
through its influence on convection and associated latent heat release in the equatorial Pacific. As the area of heavy rain moves east, the source of atmospheric heating moves with the rain, leading to widespread changes in atmospheric circulation and weather patterns outside the tropical Pacific (McPhaden, Zebiak and Glantz, 2006), including perturbations in atmospheric pressure (figure 14.12). This sequence of events leads to some predictability of weather patterns a season in advance over North America, Brazil, Australia, South Africa and other regions.
The \textsc{enso} perturbations to mid-latitude and tropical weather systems leads to dramatic changes in rainfall\index{rainfall !and ENSO} in some regions (figure 14.11). As the convective regions migrate east along the equator, they bring rain to the normally arid, central-Pacific islands. The lack of rain the western Pacific leads to drought in Indonesia and Australia.
\begin{figure}[h!]
\vspace{-2ex}
\makebox[121mm] [c]{\includegraphics{pressureanomaly}}
\footnotesize
Figure 14.12 Changing \rule{0pt}{4ex}patterns of convection in the equatorial
Pacific during an El Ni\~{n}o, set up a pattern of pressure
anomalies\index{anomalies!atmospheric pressure} in the atmosphere (solid lines) which influence
the extratropical atmosphere. After Rasmusson and Wallace (1983).
\label{fig:pressureanomaly}
\vspace{-3ex}
\end{figure}
\paragraph{An Example: Variability of Texas Rainfall}\index{rainfall!Texas}
Figure 14.11 shows a global view of teleconnections. Let's zoom in to one
region, Texas, that I chose only because I live there. The global figure shows
that the region should have higher than normal rainfall in the winter season
after El Ni\~{n}o begins. I therefore correlated yearly averaged rainfall for the
state of Texas to the Southern Oscillation Index\index{Southern Oscillation!Index} (figure
14.13). Wet years correspond to El Ni\~{n}o years in the equatorial Pacific. During El
Ni\~{n}o, convection normally found in the western equatorial Pacific moved east into the
central equatorial Pacific. The subtropical jet also moves east, carrying tropical
moisture across Mexico to Texas and the Mississippi Valley. Cold fronts in winter
interact with the upper level moisture to produce abundant winter rains from Texas
eastward.
\begin{figure}[t!]
\centering
%\vspace{-1ex}
\makebox[120mm] [c]{\includegraphics{texasrain}}
\footnotesize
Figure 14.13 Correlation of \rule{0mm}{4ex}yearly averaged rainfall averaged over Texas plotted as a function of the Southern Oscillation Index\index{Southern Oscillation!Index} averaged for the year. From Stewart (1995).
\label{fig:texasrain}
\vspace{-4ex}
\end{figure}
\section{Observing El Ni\~{n}o}
\index{El Ni\~{n}o!observing}\index{La Ni\~{n}a!observing}The tropical and equatorial Pacific is a vast, remote area seldom visited by ships. To observe the region \textsc{noaa}'s Pacific Marine Environmental Laboratory in Seattle deployed an array of buoys to measure oceanographic and meteorological variables (figure 14.14). The first buoy was successfully deployed in 1976 by David Halpern. Since then, new moorings have been added to the array, new instruments have been added to the moorings, and the moorings have been improved. The program has now evolved into the Tropical Atmosphere Ocean \textsc{tao} array of approximately 70 deep-ocean moorings spanning the equatorial Pacific Ocean between 8\degrees N and 8\degrees S from 95\degrees W to 137\degrees E (McPhaden et al, 1998).
The array began full operation in December 1994, and it continues to evolve. The work necessary to design and calibrate instruments, deploy moorings, and process data is coordinated through the \textsc{tao} Project. It is a multi-national effort involving the United States, Japan, Korea, Taiwan, and France with a project office at the Pacific Marine Environmental Laboratory.
The \textsc{tao} moorings measure air temperature, relative humidity, surface wind velocity, sea-surface temperatures, and subsurface temperatures from 10 meters down to 500 meters. Five moorings located on the equator at 110\degrees W, 140\degrees W, 170\degrees W, 165\degrees E, and 147\degrees E also carry upward-looking Acoustic Doppler Current Profilers \textsc{adcp} to measure upper-ocean currents between 10 m and 250 m. Data are sent back through the Argos system\index{Argos system}, and data are processed and made available in near real time. The moorings are recovered and replaced yearly. All sensors are calibrated prior to deployment and after recovery.
\begin{figure}[t!]
%\vspace{-3ex}
\makebox[121mm] [c]{\includegraphics{TaoArray}}
\footnotesize
Figure 14.14 Tropical Atmosphere \rule{0mm}{4ex}Ocean \textsc{tao} array of
moored buoys operated by the \textsc{noaa} Pacific Marine Environmental
Laboratory with help from Japan, Korea, Taiwan, and France. Figure from \textsc{noaa} Pacific
Marine Environmental Laboratory.
\label{fig:TaoArray}
\vspace{-3ex}
\end{figure}
Data from \textsc{tao} are merged with altimeter data from Jasin, and \textsc{ers}-2\index{ERS satellites} to obtain a more comprehensive measurement of El Ni\~{n}o. Jasin and Topex/Poseidon\index{Topex/Poseidon} data have been especially useful because they could be used to produce accurate maps of sea level every ten days. The maps provided detailed views of the development of the 1997--1998 El Ni\~{n}o in near real time that were widely reproduced throughout the world. The observations (figure 10.6) show high sea level propagating from west to east, peaking in the eastern equatorial Pacific in November 1997. In addition, satellite data extended beyond the \textsc{tao} data region to include the entire tropical Pacific. This allowed oceanographers to look for extra-tropical influences on El Ni\~{n}o.
Rain rates\index{rainfall!rates} are measured by \textsc{nasa}'s Tropical Rainfall Measuring Mission
which was specially designed for this purpose. It was launched on 27
November 1997, and it carries five instruments: the first spaceborne
precipitation radar, a five-frequency microwave radiometer, a visible and
infrared scanner, a cloud and earth radiant energy system, and a lightning
imaging sensor. Working together, the instruments provide data necessary to
produce monthly maps of tropical rainfall\index{rainfall!tropical} averaged over 500 km by 500 km areas
with 15\% accuracy\index{accuracy!rainfall}. The grid is global between $\pm$35\degrees\
latitude. In addition, the satellite data are used to measure latent heat released to the
atmosphere by rain, thus providing continuous monitoring of heating of the
atmosphere in the tropics.
\section{Forecasting El Ni\~{n}o}
\index{El Ni\~{n}o!forecasting}\index{La Ni\~{n}a!forecasting}The importance of El Ni\~{n}o to
global weather patterns has led to many schemes for forecasting events in the equatorial
Pacific. Several generations of models have been produced, but the skill of the forecasts has
not always increased. Models worked well for a few years, then failed. Failure was followed by
improved models, and the cycle continued. Thus, the best models in 1991 failed to predict weak
El Ni\~{n}os in 1993 and 1994 (Ji, Leetmaa, and Kousky, 1996). The best model of the mid 1990s
failed to predict the onset of the strong El Ni\~{n}o of 1997-1998 although a new model
developed by the National Centers for Environmental Prediction made the best forecast of the
development of the event. In general, the more sophisticated the model, the better the
forecasts (Kerr, 1998).
The following recounts some of the more recent work to improve the forecasts.
For simplicity, I describe the technique used by the National Centers for
Environmental Prediction (Ji, Behringer, and Leetmaa, 1998). But Chen et al.
(1995), Latif et al. (1993), and Barnett et al. (1993), among others, have all
developed useful prediction models.
\textbf{Atmospheric Models} How well can we model atmospheric processes over the \index{El
Ni\~{n}o!forecasting!atmospheric models}\index{La Ni\~{n}a!forecasting!atmospheric
models}\index{numerical models!atmospheric} Pacific? To help answer the question, the World
Climate Research Program's Atmospheric Model Intercomparison Project (Gates, 1992) compared
output from 30 different atmospheric numerical models for 1979 to 1988. \textit{The
Variability in the Tropics: Synoptic to Intraseasonal Timescales} subproject is especially
important because it documents the ability of 15 atmospheric general-circulation models to
simulate the observed variability in the tropical atmosphere (Slingo et al. 1995). The models
included several operated by government weather forecasting centers, including the model used
for day-to-day forecasts by the European Center for Medium-Range Weather Forecasts.
The first results indicate that none of the models were able to duplicate all
important interseasonal variability of the tropical atmosphere on timescales of
2 to 80 days. Models with weak intraseasonal activity tended to have a weak
annual cycle. Most models seemed to simulate some important aspects of the
interannual variability including El Ni\~{n}o. The length of the time series
was, however, too short to provide conclusive results on interannual
variability.
The results of the substudy imply that numerical models of the atmospheric
general circulation need to be improved if they are to be used to study tropical
variability and the response of the atmosphere to changes in the tropical ocean.
\textbf{Oceanic Models} Our ability to understand El Ni\~{n}o also depends on \index{El
Ni\~{n}o!forecasting!oceanic models }\index{La Ni\~{n}a!forecasting!oceanic
models}\index{numerical models!oceanic}our ability to model the oceanic circulation in the
equatorial Pacific. Because the models provide the initial conditions used for the forecasts,
they must be able to assimilate up-to-date measurements of the Pacific along with heat
fluxes\index{heat flux} and surface winds calculated from the atmospheric models. The
measurements include sea-surface winds from
scatterometers\index{scatterometers}\index{wind!from scatterometers} and moored buoys, surface
temperature from the optimal-interpolation data set (see \S6.6), subsurface temperatures from
buoys and \textsc{xbt}s, and sea level from altimetry and tide-gauges on islands.
Ji, Behringer, and Leetmaa (1998) at the National Centers for Environmental Prediction have modified the Geophysical Fluid Dynamics Laboratory's Modular Ocean Model for use in the tropical Pacific (see \S15.3 for more information about this model). It's domain is the Pacific between 45\degrees{S} and 55\degrees{N} and between 120\degrees{E} and 70\degrees{W}. The zonal resolution is 1.5\degrees . The meridional resolution is {\footnotesize 1/3}\degrees\ within 10\degrees\ of the equator, increasing smoothly to 1\degrees\ poleward of 20\degrees\ latitude. It has 28 vertical levels, with 18 in the upper 400 m to resolve the mixed layer\index{mixed layer!in numerical models} and thermocline\index{thermocline!in numerical models}. The model is driven by mean winds from Hellerman and Rosenstein (1983), anomalies\index{anomalies!wind} in the wind field from Florida State University, and mean heat fluxes\index{heat flux!Oberhuber atlas} from Oberhuber (1988). It assimilates subsurface temperature from the \textsc{tao} array and \textsc{xbt}s, and surface temperatures from the monthly optimal-interpolation data set (Reynolds and Smith, 1994).
The output of the model is an ocean analysis, the density and current field that best fits the data used in the analysis (figures 14.3 and 14.4). This is used to drive a coupled ocean-atmosphere model to produce forecasts.
\textbf{Coupled Models} Coupled models are separate \index{El Ni\~{n}o!forecasting!coupled models}\index{La Ni\~{n}a!forecasting!coupled models}\index{numerical models!coupled}atmospheric and oceanic models that pass information through their common boundary at the sea surface, thus coupling the two calculations. The coupling can be one way, from the atmosphere, or two way, into and out of the ocean. In the scheme used by the \textsc{noaa} National Centers for Environmental Prediction the ocean model is the same Modular Ocean Model described above. It is coupled to a low-resolution version of the global, medium-range forecast model operated by the National Centers (Kumar, Leetmaa, and Ji, 1994). Anomalies\index{anomalies!wind stress} of wind stress\index{wind stress!anomalies}, heat, and
fresh-water fluxes calculated from the atmospheric model are added to the mean annual values of the fluxes, and the sums are used to drive the ocean model. Sea-surface temperature calculated from the ocean model is used to drive the atmospheric model from 15\degrees{N} to 15\degrees{S}.
As computer power decreases in cost, models are becoming ever more complex. The trend is to global coupled models able to include other coupled ocean-atmosphere systems in addition to \textsc{enso}\index{Southern Oscillation!El Ni\~{n}o Southern Oscillation (ENSO)}. I return to the problem in \S 15.6 where I describe global coupled models.
\textbf{Statistical Models} Statistical models are based on an analysis of weather patterns in the Pacific using data going back many decades. The basic idea is that if weather patterns today are similar to patterns at some time in the past, then todays patterns will evolve as they did at that past time. For example, if winds and temperatures in the tropical Pacific today are similar to wind and temperatures just before the 1976 El Ni\~{n}o, then we might expect a similar El Ni\~{n}o to start in the near future.
\textbf{Forecasts} In general, the coupled ocean-atmosphere models produce \index{El Ni\~{n}o!forecasting}\index{La Ni\~{n}a!forecasting}forecasts that are no better than the statistical forecasts (Jan van Oldenborgh, 2005). The forecasts include not only events in the Pacific but also the global consequences of El Ni\~{n}o. The forecasts are judged two ways:
\begin{enumerate}
\vitem
Using the correlation between the area-averaged sea-surface-temperature
anomalies\index{anomalies!sea-surface temperature} from the model and the observed temperature
anomalies in the eastern equatorial Pacific. The area is usually from 170\degrees{W} to
120\degrees{W} between 5\degrees{S} and 5\degrees{N}. Useful forecasts have
correlations exceeding 0.6.
\vitem
Using the root-mean-square difference between the observed and predicted sea-surface temperature in the same area.
\end{enumerate}
The forecasts of the very strong 1997 El Ni\~{n}o have been carefully studied. Jan van Oldenborg et al (2005) and Barnston et al (1999) found no models successfully forecast the earliest onset of the El Ni\~{n}o in late 1996 and early 1997. The first formal announcements of the El Ni\~{n}o were made in May 1997. Nor did any model forecast the large temperature anomalies observed in the eastern equatorial Pacific until the area had already warmed. There was no clear distinction between the accuracy of the dynamical or statistical forecasts.
\section{Important Concepts}
\begin{enumerate}
\item
Equatorial processes are important because heat released by rain in the equatorial region helps drives much of the atmospheric circulation.
\item
Solar energy absorbed by the Pacific is the most important driver of atmospheric circulation. Solar energy is lost from the ocean mainly by evaporation. The heat warms the atmosphere and drives the circulation when the latent heat of evaporation is released in rainy areas, primarily in the western tropical Pacific and the Intertropical Convergence Zone.
\vitem
The interannual variability of currents and temperatures in the equatorial Pacific modulates the oceanic forcing of the atmosphere. This interannual variability is associated with El Ni\~{n}o/La Ni\~{n}a.
\vitem
Changes in equatorial dynamics cause changes in atmospheric circulation by changing the location of rain in tropical Pacific and therefore the location of the major heat source driving the atmospheric circulation.
\vitem
El Ni\~{n}o causes the biggest changes in equatorial dynamics. During El Ni\~{n}o, trade-winds
weaken in the western Pacific, the thermocline\index{thermocline!equatorial} becomes less deep
in the west. This drives a Kelvin\index{waves!Kelvin} wave eastward along the
equator, which deepens the thermocline in the eastern Pacific. The warm pool in the west moves
eastward toward the central Pacific, and the intense tropical rain areas move with the warm
pool.
\vitem
El Ni\~[n]o is the largest source of year-to-year fluctuations in global weather patterns.
\vitem
As a result of El Ni\~{n}o, drought occurs in the Indonesian area and northern Australia, and
floods occur in western, tropical South America. Variations in the atmospheric circulation
influence more distant areas through teleconnections.
\vitem
Forecasts of El Ni\~{n}o are made using coupled ocean-atmospheric numerical models. Forecasts
appear to have useful accuracy\index{accuracy!El Ni\~{n}o forecasts} for 3--6 months in
advance, mostly after the onset of El Ni\~{n}o.
\end{enumerate}
\chapter{Numerical Models}
\addtocounter{figure}{1}
We saw earlier that analytic solutions of the equations of motion are impossible to obtain for typical oceanic flows. The problem is due to non-linear terms in the equations of motion, turbulence\index{turbulence!in numerical models}, and the need for realistic shapes for the sea floor and coastlines. We have also seen how difficult it is to describe the ocean from measurements. Satellites can observe some processes almost everywhere every few days. But they observe only some processes, and only near or at the surface. Ships and floats can measure more variables, and deeper into the water, but the measurements are sparse. Hence, numerical models provide the only useful, global view of ocean currents. Let's look at the accuracy\index{accuracy!numerical models} and validity of the models, keeping in mind that although they are only models, they provide a remarkably detailed and realistic view of the ocean.
\section{Introduction--Some Words of Caution}
\index{numerical models!limitations of}Numerical models of ocean currents have many
advantages. They simulate flows in realistic ocean basins with a realistic sea
floor. They include the influence of viscosity and non-linear dynamics. And they can
calculate possible future flows in the ocean. Perhaps, most important, they
interpolate between sparse observations of the ocean produced by ships,
drifters\index{drifters!and numerical models}, and satellites.
Numerical models are not without problems. ``There is a world of difference
between the character of the fundamental laws, on the one hand, and the nature
of the computations required to breathe life into them, on the
other''---Berlinski (1996). The models can never give complete descriptions of
the oceanic flows even if the equations are integrated accurately. The problems
arise from several sources.
\textit{Discrete equations are not the same as continuous equations.} In Chapter 7 we wrote down the differential equations describing the motion of a continuous fluid. Numerical models use algebraic approximations to the differential equations. We assume that the ocean basins are filled with a grid of points, and time moves forward in tiny steps. The value of the current, pressure, temperature, and salinity are calculated from their values at nearby points and previous times. Ian Stewart (1992), a noted mathematician, points out that
\begin{quote} \small
Discretization is essential for computer implementation and cannot be dispensed
with. The essence of the difficulty is that the dynamics of discrete systems is
only loosely related to that of continuous systems---indeed the dynamics of
discrete systems is far richer than that of their continuous counterparts---and
the approximations involved can create spurious solutions.
\end{quote}
\textit{Calculations of turbulence\index{turbulence!calculation of} are difficult.} Numerical
models provide information only at grid points of the model. They provide no information about
the flow between the points. Yet, the ocean is turbulent, and any oceanic model
capable of resolving the turbulence needs grid points spaced millimeters apart,
with time steps of milliseconds.
Practical ocean models have grid points spaced tens to hundreds of kilometers
apart in the horizontal, and tens to hundreds of meters apart in the vertical.
This means that turbulence\index{turbulence!calculation of} cannot be calculated directly, and
the influence of turbulence must be parameterized. Holloway (1994) states the
problem succinctly:
\begin{quotation} \small
Ocean models retain fewer degrees of freedom than the actual ocean (by about 20
orders of magnitude). We compensate by applying `eddy-viscous goo' to squash
motion at all but the smallest retained scales. (We also use non-conservative
numerics.) This is analogous to placing a partition in a box to prevent gas
molecules from invading another region of the box. Our oceanic models cannot
invade most of the real oceanic degrees of freedom simply because the models do
not include them.
Given that we cannot do things `right', is it better to do nothing? That is not
an option. `Nothing' means applying viscous goo and wishing for the ever bigger
computer. Can we do better? For example, can we guess a higher entropy
configuration toward which the eddies tend to drive the ocean (that tendency
to compete with the imposed forcing and dissipation)?
\end{quotation}
By ``degrees of freedom'' Holloway means all possible motions from the smallest
waves and turbulence\index{turbulence} to the largest currents. Let's do a calculation.
We know that the ocean is turbulent with eddies as small as a few millimeters. To
completely describe the ocean we need a model with grid points spaced 1 mm apart and
time steps of about 1 ms. The model must therefore have 360\degrees $\times$
180\degrees $\times$ (111 km/degree)$^2 \times 10^{12}$ (mm/km)$^2 \times$ 3 km
$\times 10^6$ (mm/km) $= 2.4 \times 10^{27}$ data points for a 3 km deep ocean
covering the globe. The global Parallel Ocean Program Model described in the next
section has
$2.2
\times 10^7$ points. So we need $10^{20}$ times more points to describe the real
ocean. These are the missing $10^{20}$ degrees of freedom.
\textit{Practical models must be simpler than the real ocean.} Models
of the ocean must run on available computers. This means oceanographers further
simplify their models. We use the hydrostatic and Boussinesq approximations\index{Boussinesq
approximation}, and we often use equations integrated in the vertical, the shallow-water
equations (Haidvogel and Beckmann, 1999: 37). We do this because we cannot yet run the
most detailed models of oceanic circulation for thousands of years to understand
the role of the ocean in climate.
\textit{Numerical code has errors.} Do you know of any software without
bugs? Numerical models use many subroutines each with many lines of code
which are converted into instructions understood by processors using other
software called a compiler. Eliminating all software errors is impossible. With
careful testing, the output may be correct, but the accuracy\index{accuracy!numerical models}
cannot be guaranteed. Plus, numerical calculations cannot be more accurate than the
accuracy of the floating-point numbers and integers used by the computer.
Round-off errors cannot be ignored. Lawrence et al (1999), examining the output
of an atmospheric numerical model found an error in the code produced by the
\textsc{fortran-90} compiler used on the \textsc{cray} Research
supercomputer used to run the code. They also found round-off errors in the
concentration of tracers calculated from the model. Both errors produced
important errors in the output of the model.
Most models are not well verified or validated (Post \& Votta, 2005). Yet, without adequate verification and validation, output from numerical models is not credible.
\paragraph{Summary}
Despite these many sources of error, most are small in practice. Numerical models
of the ocean are giving the most detailed and complete views of the
circulation available to oceanographers. Some of the simulations contain
unprecedented details of the flow. I included the words of warning not to lead you to believe
the models are wrong, but to lead you to accept the output with a grain of salt.
\section{Numerical Models in Oceanography}
Numerical models are very widely used for many purposes in oceanography. For our purpose we
can divide models into two classes:
\paragraph{Mechanistic models} are simplified models used for studying \index{numerical
models!mechanistic models|textbf}processes. Because the models are simplified, the
output is easier to interpret than output from more complex models. Many different
types of simplified models have been developed, including models for describing
planetary waves, the interaction of the flow with sea-floor features, or the
response of the upper ocean to the wind. These are perhaps the most useful of all
models because they provide insight into the physical mechanisms influencing the
ocean. The development and use of mechanistic models is, unfortunately, beyond the
scope of this book.
\paragraph{Simulation models} are used for calculating realistic circulation
\index{numerical models!simulation models|textbf}of oceanic regions. The models are
often very complex because all important processes are included, and the output is
difficult to interpret.
\index{numerical models!simulation models}The first simulation model was developed by Kirk Bryan and Michael Cox (Bryan, 1969) at the Geophysical Fluid Dynamics laboratory in Princeton. They calculated the 3-dimensional flow in the ocean using the continuity and momentum equation with the hydrostatic and Boussinesq approximations\index{Boussinesq approximation} and a simple equation of state. Such models are called \textit{primitive equation} models because they use the basic, or primitive form of the equations of motion. The equation of state allows the model to calculate changes in density due to fluxes of heat and water through the surface, so the model includes thermodynamic processes.
The Bryan-Cox model used large horizontal and vertical viscosity and diffusion to eliminate turbulent eddies having diameters smaller about 500 km, which is a few grid points in the model. It had complex coastlines, smoothed sea-floor features, and a rigid lid. The rigid lid was needed to eliminate ocean-surface waves, such as tides and tsunamis,\index{tsunami} that move far too fast for the coarse time steps used by all simulation models. The rigid lid had, however, disadvantages. Islands substantially slowed the computation, and the sea-floor features were smoothed to eliminate steep gradients.
The first simulation model was regional. It was quickly followed by a global model (Cox, 1975) with a horizontal resolution of 2\degrees\ and with 12 levels in the vertical. The model ran far too slowly even on the fastest computers of the day, but it laid the foundation for more recent models. The coarse spatial resolution required that the model have large values for viscosity, and even regional models were too viscous to have realistic western boundary currents or mesoscale eddies\index{mesoscale eddies}.
Since those times, the goal has been to produce models with ever finer resolution, more realistic modeling of physical processes, and better numerical schemes. Computer technology is changing rapidly, and models are evolving rapidly. The output from the most recent models of the north Atlantic, which have resolution of 0.03\degrees\ look very much like the real ocean. Models of other areas show previously unknown currents near Australia and in the south Atlantic.
\paragraph{Ocean and Atmosphere Models} use very different spacing of grid points. As a result, ocean modeling lags about a decade behind atmosphere modeling. Dominant ocean eddies are 1/30 the size of dominant atmosphere eddies (storms). But, ocean features evolve at a rate that is 1/30 the rate in the atmosphere. Thus ocean models running for say one year have $(30 \times 30 )$ more horizontal grid points than the atmosphere, but they have 1/30 the number of time steps. Both have about the same number of grid points in the vertical. As a result, ocean models run 30 times slower than atmosphere models of the same complexity.
\section{Global Ocean Models}
Several types of global models are widely used in oceanography. Most have grid points about one tenth of a degree apart, which is sufficient to resolve mesoscale eddies,\index{mesoscale eddies} such as those seen in figures 11.10, 11.11, and 15.2, that have a diameter larger than two to three times the distance between grid points. Vertical resolution is typically around 30 vertical levels. Models include: i) realistic coasts and bottom features; ii) heat and water fluxes though the surface; iii) eddy dynamics; and iv) the meridional-overturning\index{circulation!meridional overturning} circulation. Many assimilate satellite and float data using techniques described in \S 15.5. The models range in complexity from those that can run on desktop workstations to those that require the world's fastest computers.
All models must be be run to calculate one to two decades of variability before they can be used to simulate the ocean. This is called \textit{spin-up}.\index{numerical models!spin-up|textbf} Spin-up is needed because initial conditions for density, fluxes of momentum and heat through the sea-surface, and the equations of motion are not all consistent. Models are started from rest with values of density from the Levitus (1982) atlas and integrated for a decade using mean-annual wind stress\index{wind stress!mean annual}, heat fluxes\index{heat flux}, and water flux. The model may be integrated for several more years using monthly wind stress, heat fluxes\index{heat flux}, and water fluxes.
\index{numerical models!primitive-equation}The Bryan-Cox models evolved into several widely used
models which are providing impressive views of the global ocean circulation.
\paragraph{Geophysical Fluid Dynamics Laboratory Modular Ocean Model MOM} consists\index{numerical models!primitive-equation!Geophysical Fluid Dynamics Laboratory Modular Ocean Model (MOM)|textbf}of a large set of modules that can be configured to run on many different computersto model many different aspects of the circulation. The source code is open and free, and it is in the public domain. The model is widely use for climate studies and for studying the ocean's circulation over a wide range of space and time scales (Pacanowski and Griffies, 1999).
\begin{quote} \small
Because \textsc{mom} is used to investigate processes which cover a wide range of time and space scales, the code and manual are lengthy. However, it is far from necessary for the typical ocean modeler to become acquainted with all of its aspects. Indeed, \textsc{mom} can be likened to a growing city with many different neighborhoods. Some of the neighborhoods communicate with one another, some are mutually incompatible, and others are basically independent. This diversity is quite a challenge to coordinate and support. Indeed, over the years certain ``neighborhoods'' have been jettisoned or greatly renovated for various reasons.---Pacanowski and Griffies.
\end{quote}
The model uses the momentum equations, equation of state, and the hydrostatic and Boussinesq approximations\index{Boussinesq approximation}. Subgrid-scale motions are reduced by use of eddy viscosity. Version 4 of the model has improved numerical schemes, a free surface, realistic bottom features, and many types of mixing\index{mixing!in numerical models} including horizontal mixing\index{mixing!on surfaces of constant density} along surfaces of constant density. Plus,
it can be coupled to atmospheric models.
\paragraph{Parallel Ocean Program Model} produced by Smith and colleagues at \index{numerical models!primitive-equation!Parallel Ocean Program Model|textbf}Los Alamos National Laboratory (Maltrud et al, 1998) is another widely used model growing out of the original Bryan-Cox code. The model includes improved numerical algorithms, realistic coasts, islands, and unsmoothed bottom features. It has model has $1280 \times 896$ equally spaced grid points on a Mercator projection extending from 77\degrees S to 77\degrees N, and 20 levels in the vertical. Thus it has $2.2 \times 10^{7}$ points giving a resolution of $0.28^{\circ} \times 0.28^{\circ} \cos \theta $, which varies from 0.28\degrees\ (31.25 km) at the equator to 0.06\degrees\ (6.5 km) at the highest latitudes. The average resolution is about $0.2^{\circ}$. The model was is forced by \textsc{ecmwf} wind stress\index{wind stress!and numerical models} and surface heat and water fluxes (Barnier et al, 1995).
\begin{figure}[t!]
\makebox[121mm][c] {\includegraphics{model_out}}
\footnotesize
Figure 15.1 Near-surface \rule{0mm}{1ex}geostrophic currents\index{geostrophic
currents!calculated by numerical model} on October 1, 1995 calculated by the Parallel Ocean
Program numerical model developed at the Los Alamos National Laboratory. The length of the
vector is the mean speed in the upper 50 m of the ocean. The direction is the mean direction
of the current. From Richard Smith, Los Alamos National Laboratory.
\vspace{-4ex}
\label{fig:model_out}
\end{figure}
\paragraph{Hybrid Coordinate Ocean Model} \textsc{hycom} All the models \index{numerical models!primitive-equation!Hybrid Coordinate Ocean Model|textbf}just described use $x, y, z$ coordinates. Such a coordinate system has both advantages and disadvantages. It can have high resolution in the surface mixed layer and in shallower regions. But it is less useful in the interior of the ocean. Below the mixed layer, mixing\index{mixing!on surfaces of constant density} in the ocean is easy along surfaces of constant density, and difficult across such surfaces. A more natural coordinate system in the interior of the ocean uses $x, y, \rho$, where $\rho$ is density. Such a model is called an \textit{isopycnal model}\index{isopycnal model|textbf}\index{numerical models!isopycnal|textbf}\index{numerical models!isopycnal|textbf}. Essentially, $\rho (z)$ is replaced with $z (\rho )$. Because isopycnal surfaces are surfaces of constant density, horizontal mixing\index{mixing!in numerical models} is always on constant-density surfaces in this model.
The Hybrid Coordinate Ocean Model \textsc{hycom} model uses different vertical coordinates in different regions of the ocean, combining the best aspects of $z$-coordinate model and isopycnal-coordinate model (Bleck, 2002). The hybrid model has evolved from the Miami Isopycnic-Coordinate Ocean Model (figure 15.2). It is a primitive-equation model driven by wind stress\index{wind stress!and numerical models} and heat fluxes\index{heat flux}. It has realistic mixed layer and improved horizontal and vertical mixing schemes that include the influences of internal waves, shear instability, and double-diffusion (see \S 8.5). The model results from collaborative work among investigators at many oceanographic laboratories.
\paragraph{Regional Oceanic Modeling System} \textsc{roms} is a regional model that can be imbedded in models of much larger regions. It is widely used for studying coastal current systems closely tied to flow further offshore, for example, the California Current. \textsc{roms} is a hydrostatic, primitive equation, terrain-following model using stretched vertical coordinates, driven by surface fluxes of momentum, heat, and water. It has improved surface and bottom boundary layers (Shchepetkin and McWilliams, 2004).
\begin{figure}[t!]
%\centering
\makebox [121mm][c]{\includegraphics{blecksgulfstream}}
\footnotesize
Figure 15.2 Output of\rule{0mm}{4ex} Bleck's Miami Isopycnal Coordinate Ocean
Model \textsc{micom}. It is a high-resolution model of the Atlantic showing the
Gulf Stream\index{Gulf Stream!calculated by MICOM numerical model}, its variability, and the
circulation of the north Atlantic. From Bleck.
\vspace{-3ex}
\label{fig:blecksgulfstream}
\end{figure}
\paragraph{Climate models} are used for studies of large-scale \index{numerical models!primitive-equation!climate models}hydrographic structure, climate dynamics, and water-mass formation. These models are the same as the eddy-admitting, primitive equation models I have just described except the horizontal resolution is much coarser because they must simulate ocean processes for decades or centuries. As a result, they must have high dissipation for numerical stability, and they cannot simulate mesoscale eddies\index{mesoscale eddies}. Typical horizontal resolutions are 2\degrees\ to 4\degrees. The models tend, however, to have high vertical resolution necessary for describing the deep circulation important for climate.
\section{Coastal Models}
\index{numerical models!coastal}The great economic importance of the coastal zone has led to
the development of many different numerical models for describing coastal currents, tides, and
storm surges. The models extend from the beach to the continental slope, and they can include
a free surface, realistic coasts and bottom features, river runoff, and atmospheric forcing.
Because the models don't extend very far into deep water, they need additional information
about deep-water currents or conditions at the shelf break.
The many different coastal models have many different goals, and many different implementations. Several of the models described above, including \textsc{mom} and \textsc{rom}, have been used to model coastal processes. But many other specialized models have also been developed. Heaps (1987), Lynch et al (1996), and Haidvogel and Beckman (1998) provide good overviews of the subject. Rather than look at a menu of models, let's look at two typical models.
\textit{Princeton Ocean Model} developed by Blumberg and Mellor (1987, and Mellor, 1998) and is widely \index{numerical models!coastal!Princeton Ocean Model|textbf}used for describing coastal currents. It includes thermodynamic processes, turbulent mixing\index{mixing!in numerical models}, and the Boussinesq and hydrostatic approximations\index{Boussinesq approximation}. The Coriolis parameter\index{B-plane@$\beta$-plane} is allowed to vary using a beta-plane approximation. Because the model must include a wide range of depths, Blumberg and Mellor used a vertical coordinate $\sigma$ scaled by the depth of the water:
\begin{equation}
\sigma = \frac{z-\eta}{H+\eta}
\end{equation}
where $z=\eta(x, y, t)$ is the sea surface, and $z=-H(x,y)$ is the bottom.
Sub-grid turbulence\index{turbulence!subgrid} is parameterized using a closure scheme proposed by Mellor and Yamada (1982) whereby eddy diffusion coefficients vary with the size of the eddies producing the mixing\index{mixing!in numerical models!Mellor and Yamada scheme} and the shear of the flow.
The model is driven by wind stress\index{wind stress!and numerical models} and heat and water
fluxes from meteorological models. The model uses known geostrophic, tidal, and Ekman currents
at the outer boundary.
The model has been used to calculate the three-dimensional distribution of
velocity, salinity, sea level, temperature, and turbulence\index{turbulence!calculation of} for
up to 30 days over a region roughly 100--1000 km on a side with grid spacing of 1--50 km.
\textit{Dartmouth Gulf of Maine Model} developed by Lynch et al (1996) is a \index{numerical
models!coastal!Dartmouth Gulf of Maine Model|textbf} 3-dimensional model of the circulation
using a triangular, finite-element grid. The size of the triangles is proportional to both
depth and the rate of change of depth. The triangles are small in regions where the bottom
slopes are large and the depth is shallow, and they are large in deep water. The variable mesh
is especially useful in coastal regions where the depth of water varies greatly. Thus the
variable grid gives highest resolution where it is most needed.
\begin{figure}[t!]
\makebox[121mm][c] {\includegraphics{GulfofMaine}}
\footnotesize
Figure 15.3 \textbf{Top}: Topographic \rule{0mm}{3ex}map of the Gulf of Maine showing
important features. \textbf{Inset}: Triangular, finite-element grid used
to compute flow in the gulf. The size of the triangles varies
with depth and rate of change of depth. After Lynch et al, (1996).
\vspace{-3ex}
\label{fig:GulfofMaine}
\end{figure}
The model uses roughly 13,000 triangles to cover the Gulf of Maine and nearby waters of the
north Atlantic (figures 15.3). Minimum size of the elements is roughly one kilometer. The
model has 10 to 40 horizontal layers. The vertical spacing of the layers is not uniform.
Layers are closer together near the top and bottom and they are more widely spaced in the
interior. Minimum spacing is roughly one meter in the bottom boundary layer.
The model integrates the three-dimensional, primitive equations in
shallow-water form. The model has a simplified equation of state and a
depth-averaged continuity equation, and it uses the hydrostatic and Boussinesq
assumptions\index{Boussinesq approximation}. Sub-grid mixing\index{mixing!in numerical
models!Mellor and Yamada scheme} of momentum, heat and mass is parameterized using the Mellor
and Yamada (1982) turbulence-closure scheme\index{turbulence!closure problem} which gives
vertical mixing\index{mixing!in numerical models} coefficients that vary with stratification
and velocity shear. Horizontal mixing\index{mixing!in numerical models!Smagorinski scheme}
coefficients were calculated from Smagorinski (1963). A carefully chosen, turbulent, eddy
viscosity is used in the bottom boundary layer. The model is forced by wind, heating, and
tidal forcing from the deep ocean.
The model is spun up from rest for a few days using a specified density field at
all grid points, usually from a combination of \textsc{ctd}\index{CTD} data plus historical
data. This gives a velocity field consistent with the density field. The
model is then forced with local winds and heat fluxes to calculate the evolution
of the density and velocity fields.
\paragraph{Comments on Coastal Models} Roed et al. (1995) examined the
accuracy\index{accuracy!numerical models!coastal} of coastal models by comparing the ability of
five models, including Blumberg and Mellor's to describe the flow in typical cases. They found
that the models produced very different results, but that after the models were adjusted, the
differences were reduced. The differences were due to differences in vertical and
horizontal mixing\index{mixing!in numerical models!coastal} and spatial and temporal
resolution.
Hackett et al. (1995) compared the ability of two of the five models to describe
observed flow on the Norwegian shelf. They conclude that
\begin{quote} \small
\ldots both models are able to qualitatively
generate many of the observed features of the flow, but neither is able to
quantitatively reproduce detailed currents \ldots [Differences] are primarily
attributable to inadequate parameterizations of subgrid scale turbulent mixing\index{mixing!in
numerical models}, to lack of horizontal resolution and to imperfect initial and boundary
conditions.
\end{quote}
\paragraph{Storm-Surge Models} Storms coming ashore across wide, shallow, \index{numerical models!storm-surge}continental shelves drive large changes of sea level at the coast called storm surges (see \S 17.3 for a description of surges and processes influencing surges). The surges can cause great damage to coasts and coastal structures. Intense storms in the Bay of Bengal have killed hundreds of thousands in a few days in Bangladesh. Because surges are so important, government agencies in many countries have developed models to predict the changes of sea level and the extent of coastal flooding.
Calculating storm surges is not easy. Here are some reasons, in a rough order of importance.
\begin{enumerate}
\vitem The distribution of wind over the ocean is not well known. Numerical weather models calculate wind speed at a constant pressure surface, storm-surge models need wind at a constant height of 10 m. Winds in bays and lagoons tend to be weaker than winds just offshore because nearby land distorts the airflow, and this is not included in the weather models.
\vitem The shoreward extent of the model's domain changes with time. For example, if sea level rises, water will flood inland, and the boundary between water and sea moves inland with the water.
\vitem The drag coefficient\index{drag!coefficient} of wind on water is not well known for hurricane force winds.
\vitem The drag coefficient\index{drag!coefficient} of water on the seafloor is also not well known.
\vitem The models must include waves and tides which influence sea level in shallow waters.
\vitem Storm surge models must include the currents generated in a stratified, shallow sea by wind.
\end{enumerate}
To reduce errors, models are tuned to give results that match conditions seen in past storms. Unfortunately, those past conditions are not well known. Changes in sea level and wind speed are rarely recorded accurately in storms except at a few, widely paced locations. Yet storm-surge heights can change by more than a meter over distances of tens of kilometers.
Despite these problems, models give very useful results. Let's look at the official \textsc{noaa} model, and a new experimental model developed by the Corps of Engineers.
\textit{Sea, Lake, and Overland Surges Model} \textsc{slosh} is used by \textsc{noaa}
\index{numerical models!storm-surge!Sea, Lake, and Overland Surges Model|textbf}for
forecasting storm surges produced by hurricanes coming ashore along the Atlantic and Gulf
coasts of the United States (Jelesnianski, Chen, and Shaffer, 1992).
The model is the result of a lifetime of work by Chester Jelesnianski. In
developing the model, Jelesnianski paid careful attention to the relative
importance of errors in the model. He worked to reduce the largest errors, and
ignored the smaller ones. For example, the distribution of winds in a hurricane
is not well known, so it makes little sense to use a spatially
varying drag coefficient\index{drag!coefficient} for the wind. Thus, Jelesnianski used a
constant drag coefficient\index{drag!coefficient} in the air, and a constant eddy stress
coefficient in the water.
\textsc{slosh} calculates water level from depth-integrated, quasi-linear,
shallow-water equations. Thus it ignores stratification. It also ignores river
inflow, rain, and tides. The latter may seem strange, but the model is designed
for forecasting. The time of landfall cannot be forecast accurately, and hence the
height of the tides is mostly unknown. Tides \index{tides!and storm surges}can be added to the
calculated surge, but the nonlinear interaction of tides and surge is ignored.
The model is forced by idealized hurricane winds. It needs only atmospheric
pressure at the center of the storm, the distance from the center to the area of
maximum winds, the forecast storm track and speed along the track.
In preparation for hurricanes coming ashore near populated areas, the model has
been adapted for 27 basins from Boston Harbor Massachusetts to Laguna Madre
Texas. The model uses a fixed polar mesh. Mesh spacing begins with a fine mesh
near the pole, which is located near the coastal city for which the model is
adapted. The grid stretches continuously to a coarse mesh at distant boundaries
of a large basin. Such a mesh gives high resolution in bays and near the coast
where resolution is most needed. Using measured depths at sea and elevations on
land, the model allows flooding of land, overtopping of levees and dunes, and
sub-grid flow through channels between offshore islands.
Sea level calculated from the model has been compared with heights measured by
tide gauges for 13 storms, including Betsy: 1965, Camile: 1969, Donna: 1960,
and Carla: 1961. The overall accuracy\index{accuracy!storm surge} is $\pm 20$\% .
\textit{Advanced Circulation Model} \textsc{adcirc}\index{numerical models!storm-surge!Advanced Circulation Model|textbf} is an experimental model for forecasting storm surges produced by hurricanes coming ashore along the Atlantic and Gulf coasts of the United States (Graber et al, 2006). The model uses a finite-element grid, the Boussinesq approximation, quadratic bottom friction, and vertically integrated continuity and momentum equations for flow on a rotating earth. It can be run as either a two-dimensional, depth-integrated model, or as a three-dimensional model. Because waves contribute to storm surges, the model includes waves calculated from the \textsc{wam} third-geneation wave model (see \S 16.5).
The model is forced by:
\begin{enumerate}
\vitem
High resolution winds and surface pressure obtained by combining weather forecasts from the \textsc{noaa} National Weather Service and the National Hurricane Center along the official and alternate forecast storm tracks.
\vitem
Tides at the open-ocean boundaries of the model.
\vitem
Sea-surface height and currents at the open-ocean boundaries of the model.
\end{enumerate}
The model successfully forecast the Hurricane Katrina storm surge, giving values in excess of 6.1 m near New Orleans.
\section{Assimilation Models}
Many of the models I have described so far have output, such as current velocity or surface topography, constrained by oceanic observations of the variables they calculate. Such models are called \textit{assimilation models}\index{numerical models!assimilation|textbf}\index{data assimilation}. In this section, I will consider how data can be assimilated into numerical models.
Let's begin with a primitive-equation, eddy-admitting numerical model used to calculate the position of the Gulf Stream\index{Gulf Stream!calculation of}. Let's assume that the model is driven with real-time surface winds from the \textsc{ecmwf} weather model. Using the model, we can calculate the position of the current and also the sea-surface topography associated with the current. We find that the position of the Gulf Stream\index{Gulf Stream!wiggles} wiggles offshore of Cape Hatteras due to instabilities, and the position calculated by the model is just one of many possible positions for the same wind forcing. Which position is correct, that is, what is the position of the current today? We know, from satellite altimetry, the position of the current at a few points a few days ago. Can we use this information to calculate the current's position today? How do we assimilate this information into the model?
Many different approaches are being explored (Malanotte-Rizzoli, 1996). Roger Daley (1991) gives a complete description of how data are used with atmospheric models. Andrew Bennet (1992) and Carl Wunsch (1996) describe oceanic applications.
The different approaches are necessary because assimilation\index{data assimilation} of data into models is not easy.
\begin{enumerate}
\vitem Data assimilation\index{data assimilation} is an \textit{inverse problem}\index{inverse
problem|textbf}: A finite number of observations are used to estimate a continuous field---a
function, which has an infinite number of points. The calculated fields, the solution to the
inverse problem, are completely under-determined. There are many fields that fit the
observations and the model precisely, and the solutions are not unique. In our
example, the position of the Gulf Stream\index{Gulf Stream!position of} is a function. We may
not need an infinite number of values to specify the position of the stream if we assume the
position is somewhat smooth in space. But we certainly need hundreds of values along the
stream's axis. Yet, we have only a few satellite points to constrain the position of
the Stream.
To learn more about inverse problems and their solution, read Parker (1994) who
gives a very good introduction based on geophysical examples.
\vitem Ocean dynamics are non-linear, while most methods for calculating solutions
to inverse problems depend on linear approximations. For example the position of
the Gulf Stream\index{Gulf Stream!position of} is a very nonlinear function of the forcing by
wind and heat fluxes\index{heat flux} over the north Atlantic.
\vitem Both the model and the data are incomplete and both have errors. For example, we have altimeter measurements only along the tracks such as those shown in figure 2.6, and the measurements have errors of $\pm 2$ cm.
\vitem Most data available for assimilation\index{data assimilation} into data comes from the
surface, such as \textsc{avhrr} \index{Advanced Very High Resolution Radiometer (AVHRR)}and
altimeter data. Surface data obviously constrain the surface geostrophic
velocity\index{geostrophic currents!assimilated into numerical models}, and surface velocity is
related to deeper velocities. The trick is to couple the surface observations to deeper
currents.
\end{enumerate}
While various techniques are used to constrain numerical models in oceanography, perhaps the most practical are techniques borrowed from meteorology.
\begin{quote} \small
Most major ocean currents have dynamics which are significantly nonlinear. This precludes the ready development of inverse methods\dots Accord\-ing\-ly, most attempts to combine ocean models and
measurements have followed the practice in operational meteorology: measurements are used to prepare initial conditions for the model, which is then integrated forward in time until further measurements are available. The model is thereupon re-initialized. Such a strategy may be described as sequential\index{sequential estimation techniques}.---Bennet (1992).
\end{quote}
Let's see how Professor Allan Robinson and colleagues at Harvard University used sequential estimation techniques\index{sequential estimation techniques} to make the first forecasts of the Gulf Stream\index{Gulf Stream!forecasts} using a very simple model.
\textit{The Harvard Open-Ocean Model} was an eddy-admitting, quasi-geostrop\-ic
\index{numerical models!assimilation!Harvard Open-Ocean Model|textbf}mod\-el of the
Gulf Stream\index{Gulf Stream!forecasts} east of Cape Hatteras (Robinson et al. 1989). It had
six levels in the vertical, 15 km resolution, and one-hour time steps. It used a filter
to smooth high-frequency variability and to damp grid-scale variability.
\begin{figure}[t!]
\includegraphics{harvardmodelR}
\footnotesize
Figure 15.4 Output \rule{0mm}{4ex} from the Harvard Open-Ocean Model: \textbf{A} the
initial state of the model, the analysis, and \textbf{B} Data used to produce the
analysis for 2 March 1988. \textbf{C} The forecast for 9 March 1988. \textbf{D}
The analysis for 9 March. Although the Gulf Stream\index{Gulf Stream!forecasts} changed
substantially in one week, the model forecasts the changes well. After Robinson et al. (1989).
\label{fig:harvardmodel}
\vspace{-3ex}
\end{figure}
By \textit{quasi-geostrophic}\index{quasi-geostrophic|textbf} we mean that the flow
field is close to geostrophic balance. The equations of motion include the
acceleration terms $D/Dt$, where $D/Dt$ is the substantial derivative and $t$ is
time. The flow can be stratified, but there is no change in density due to heat
fluxes\index{heat flux} or vertical mixing\index{mixing!in numerical
models!quasi-geostrophic}. Thus the quasi-geostrophic equations are simpler than the primitive
equations, and they could be integrated much faster. Cushman-Roisin (1994: 204) gives a good
description of the development of quasi-geostrophic equations of motion.
The model reproduces the important features of the Gulf Stream\index{Gulf Stream!calculation
of} and it's extension, including meanders, cold- and warm-core rings, the interaction of
rings with the stream, and baroclinic instability (figure 15.4). Because the model was designed
to forecast the dynamics of the Gulf Stream, it must be constrained by oceanic
measurements:
\begin{enumerate}
\vitem Data provide the initial conditions for the model. Satellite measurements
of sea-surface temperature from the \textsc{avhrr} \index{Advanced Very High Resolution
Radiometer (AVHRR)}and topography from an
altimeter are used to determine the location of features in the region.
Expendable bathythermograph\index{bathythermograph (BT)!expendable
(XBT)}, \textsc{axbt} measurements of subsurface temperature,
and historical measurements of internal density are also used. The
features are represented by an analytic functions in the model.
\vitem The data are introduced into the numerical model, which interpolates and
smoothes the data to produce the best estimate of the initial fields of
density and velocity. The resulting fields are called an \textit{analysis}.
\vitem The model is integrated forward for one week, when new data are available,
to produce a forecast.
\vitem Finally, the new data are introduced into the model as in the first step
above, and the processes is repeated.
\end{enumerate}
The model made useful, one-week forecasts of the Gulf
Stream\index{Gulf Stream!forecasts} region. Much more advanced models with much higher resolution are now being used to make global forecasts of ocean currents up to one month in advance in support of the Global Ocean Data Assimilation Experiment\index{data assimilation}
\textsc{godae} \index{Global Ocean Data Assimilation Experiment!products} that started in
2003. The goal of \textsc{godae} is produce routine oceanic forecasts similar to todays weather
forecasts.
An example of a \textsc{godae} model is the global US Navy Layered Ocean
Model\index{numerical models!assimilation!NLOM}. It is a primitive equation model with
1/32\degrees\ resolution in the horizontal and seven layers in the vertical. It assimilates
altimeter data from Jason\index{Jason}, Geosat Follow-on\index{Geosat!Follow-On mission}
(\textsc{gfo}), and
\textsc{ers}-2\index{ERS satellites} satellites and sea-surface temperature from
\textsc{avhrr}\index{temperature!measurement at surface!Advanced Very High Resolution
Radiometer (AVHRR)} on \textsc{noaa} satellites. The model is forced with winds and heat
fluxes for up to five days in the future using output from the Navy Operational Global
Atmospheric Prediction System. Beyond five days, seasonal mean winds and fluxes are used. The
model is run daily (figure 15.5) and produces forecasts for up to one month in the future. The
model has useful skill out to about 20 days.
\begin{figure}[h!]
\centering
\vspace{-1ex}
\makebox[121mm][c] {\includegraphics{nlom-gulfstream}}
\footnotesize
Figure 15.5 Analysis \rule{0mm}{4ex}of the Gulf Stream\index{Gulf Stream!calculation of}
region from the Navy Layered Ocean Model.\\From the U.S. Naval Oceanographic Office.
\label{fig:nlom-gulfstream}
\vspace{-3ex}
\end{figure}
A group of French laboratories and agencies operates a similar operational forecasting system,
Mercator,\index{numerical models!assimilation!Mercator} based on assimilation of altimeter
measurements of sea-surface height, satellite measurements of sea-surface temperature, and
internal density fields in the ocean, and currents at 1000 m from thousands of Argo
floats\index{floats!Argo}. Their model has 1/15\degrees\ resolution in the Atlantic and
2\degrees\ globally.
\section{Coupled Ocean and Atmosphere Models}
\index{numerical models!coupled}Coupled numerical models of the atmosphere and ocean are used to study the climate, its variability, and its response to external forcing. The most important use of the models has been to study how earth's climate might respond to a doubling of $CO_{2}$ in the atmosphere. Much of the literature on climate change is based on studies using such models. Other important uses of coupled models include studies of El Ni\~{n}o and the meridional overturning circulation\index{circulation!meridional overturning}. The former varies over a few years, the latter varies over a few centuries.
Development of the coupled models tends to be coordinated through the World Climate Research Program of the World Meteorological Organization \textsc{wcrp/wmo}, and recent progress is summarized in Chapter 8 of the \textit{Climate Change 2001: The Scientific Basis} report by the Intergovernmental Panel on Climate Change (\textsc{ipcc}, 2007).
Many coupled ocean and atmosphere models have been developed. Some include only physical processes in the ocean, atmosphere, and the ice-covered polar seas. Others add the influence of land and biological activity in the ocean. Let's look at the oceanic components of a few models.
\paragraph{Climate System Model} The Climate System Model developed by the National
\index{numerical models!coupled!Climate System Model|textbf}Center for Atmospheric
Research \textsc{ncar} includes physical and biogeochemical influence on the climate
system (Boville and Gent, 1998). It has atmosphere, ocean, land-surface, and sea-ice
components coupled by fluxes between components. The atmospheric component is the
\textsc{ncar} Community Climate Model, the oceanic component is a modified version
of the Princeton Modular Ocean Model, using the Gent and McWilliams (1990) scheme
for parameterizing mesoscale eddies\index{mesoscale eddies}. Resolution is approximately
2\degrees\ $\times$ 2\degrees\ with 45 vertical levels in the ocean.
The model has been spun up and integrated for 300 years, the results are
realistic, and there is no need for a flux adjustment\index{numerical models!coupled!flux
adjustments in}. (See the special issue of
\textit{Journal of Climate}, June 1998).
\paragraph{Princeton Coupled Model} The model consists of an atmospheric model with
\index{numerical models!coupled!Princeton Coupled Model|textbf}a horizontal
resolution of 7.5\degrees\ longitude by 4.5\degrees\ latitude and 9 levels in the
vertical, an ocean model with a horizontal resolution of 4\degrees\ and 12 levels in
the vertical, and a land-surface model. The ocean and atmosphere are coupled through
heat, water, and momentum fluxes. Land and ocean are coupled through river runoff.
And land and atmosphere are coupled through water and heat fluxes\index{heat flux}.
\paragraph{Hadley Center Model} This is an atmosphere-ocean-ice model that \index{numerical
models!coupled!Hadley Center Model|textbf}minimizes the need for
flux adjustments\index{numerical models!coupled!flux adjustments in} (Johns et
al, 1997). The ocean component is based on the Bryan-Cox primitive equation model, with
realistic bottom features, vertical mixing \index{mixing!in numerical
models!Pacanowski and Philander scheme}coefficients from Pacanowski and Philander (1981). Both
the ocean and the atmospheric component have a horizontal resolution of 96
$\times$ 73 grid points, the ocean has 20 levels in the vertical.
In contrast to most coupled models, this one is spun up as a coupled system with flux
adjustments \index{numerical models!coupled!flux
adjustments in}during spin up to keep sea surface temperature and salinity close to
observed mean values. The coupled model was integrated from rest using Levitus
values for temperature and salinity for September. The initial integration was from
1850 to 1940. The model was then integrated for another 1000 years. No flux
adjustment \index{numerical models!coupled!flux
adjustments in}was necessary after the initial 140-year integration because drift of
global-averaged air temperature was $\le 0.016$ K/century.
\paragraph{Comments on Accuracy of Coupled Models}\index{accuracy!numerical models!coupled}
\index{numerical models!coupled!accuracy of}Models of the
coupled, land-air-ice-ocean climate system must simulate hundreds to thousands of years. Yet,
\begin{quote} \small
It will be very hard to establish an integration framework, particularly on a
global scale, as present capabilities for modelling the earth system are rather
limited. A dual approach is planned. On the one hand, the relatively conventional
approach of improving coupled atmosphere-ocean-land-ice models will be pursued.
Ingenuity aside, the computational demands are extreme, as is borne out by the
Earth System Simulator --- 640 linked supercomputers providing 40 teraflops
[$10^{12}$ floating-point operations per second] and a cooling system from hell
under one roof --- to be built in Japan by 2003.--- Newton, 1999.
\end{quote}
Because models must be simplified to run on existing computers, the models must be
simpler than models that simulate flow for a few years (\textsc{wcrp}, 1995).
In addition, the coupled model must be integrated for many years for the ocean and
atmosphere to approach equilibrium. As the integration proceeds, the coupled system
tends to drift away from reality due to errors in calculating fluxes of heat and
momentum between the ocean and atmosphere. For example, very small errors in
precipitation over the Antarctic Circumpolar Current\index{Antarctic Circumpolar
Current} leads to small changes the salinity of the current, which leads to large changes in
deep convection in the Weddell Sea, which greatly influences the volume of deep water masses.
Some modelers allow the system to drift, others adjust sea-surface temperature and the
calculated fluxes between the ocean and atmosphere. Returning to the example, the flux of
fresh water in the circumpolar current could be adjusted to keep salinity close to the
observed value in the current. There is no good scientific basis for the adjustments except
the desire to produce a ``good'' coupled model. Hence, the adjustments are ad hoc and
controversial. Such adjustments are called \textit{flux adjustments}\index{flux
adjustments|textbf}\index{numerical models!coupled!flux
adjustments in} or \textit{flux corrections}\index{flux corrections|textbf}.
Fortunately, as models have improved, the need for adjustment or the magnitude
of the adjustment has been reduced. For example, using the Gent-McWilliams scheme
for mixing\index{mixing!in numerical models!Gent-McWilliams scheme} along constant-density
surfaces in a coupled ocean-atmosphere model greatly reduced climate drift in a coupled
ocean-atmos\-phere model because the mixing\index{mixing!in Circumpolar Current} scheme reduced
deep convection in the Antarctic Circumpolar Current\index{Antarctic Circumpolar
Current!calculations of} and elsewhere (Hirst, O'Farrell, and Gordon, 2000).
Grassl (2000) lists four capabilities of a credible coupled general
circulation model:
\begin{enumerate}
\vitem
``Adequate representation of the present climate.
\vitem
``Reproduction (within typical interannual and decades time-scale climate
variability) of the changes since the start of the instrumental record for a
given history of external forcing;
\vitem
``Reproduction of a different climate episode in the past as derived from paleoclimate
records for given estimates of the history of external forcing; and
\vitem
``Successful simulation of the gross features of an abrupt climate change event
from the past.''
\end{enumerate}
McAvaney et al. (2001) compared the oceanic component of twenty-four coupled models, including models with and without flux adjustments\index{numerical models!coupled!flux
adjustments in}. They found substantial differences among the models. For example, only five models calculated a meridional overturning circulation\index{circulation!meridional overturning} within 10\% the observed value of 20 Sv\index{circulation!meridional overturning}. Some had values as low as 3 Sv, others had values as large as 36 Sv. Most models could not calculate a realistic transport\index{transport!by Antarctic Circumpolar Current} for the Antarctic
Circumpolar Current\index{Antarctic Circumpolar Current!calculations of}.
Grassl (2000) found that many coupled climate models, including models with and
without flux adjustment\index{numerical models!coupled!flux
adjustments in}, meet the first criterion. Some models meet the second
criterion (Smith et al. 2002, Stott et al. 2000), but external solar forcing is still not well known and more work
is needed. And a few models are starting to reproduce some aspects of the warm event
of 6,000 years ago.
\begin{quote} \small
But how useful are these models in making projections of future climate? Opinion
is polarized. At one extreme are those who take the model results as gospel. At
the other are those who denigrate results simply because they distrust models, or
on the grounds that the model performance is obviously wrong in some respects or
that a process is not adequately included. The truth lies in between. All models
are of course wrong because, by design, they give a simplified view of the
system being modelled. Nevertheless, many---but not all---models are very
useful.---Trenberth, 1997.
\end{quote}
\section{Important Concepts}
\begin{enumerate}
\item
Numerical models are used to simulate oceanic flows with realistic and useful
results. The most recent models include heat fluxes\index{heat flux} through the surface, wind
forcing, mesoscale eddies\index{mesoscale eddies}, realistic coasts and sea-floor features,
and more than 20 levels in the vertical.
\vitem
Recent models are now so good, with resolution near 0.1\degrees, that they show previously
unknown aspects of the ocean circulation,
\vitem
Numerical models are not perfect. They solve discrete equations, which
are not the same as the equations of motion described in earlier chapters. And,
\vitem
Numerical models cannot reproduce all turbulence\index{turbulence!subgrid} of the ocean because
the grid points are tens to hundreds of kilometers apart. The influence of turbulent motion
over smaller distances must be calculated from theory, and this introduces errors.
\vitem
Numerical models can be forced by real-time oceanographic data from ships
and satellites to produce forecasts of oceanic conditions, including El Ni\~{n}o
in the Pacific, and the position of the Gulf Stream\index{Gulf Stream!forecasts of} in the
Atlantic.
\vitem
Coupled ocean-atmosphere models have much coarser spatial resolution so that that
they can be integrated for hundreds of years to simulate the natural variability
of the climate system and its response to increased CO$_2$ in the atmosphere.
\end{enumerate}
\chapter{Ocean Waves}
Looking out to sea from the shore, we can see waves on the sea surface.
Looking carefully, we notice the waves are undulations of the sea surface with a
height of around a meter, where height is the vertical distance between the
bottom of a trough and the top of a nearby crest. The wavelength, which we
might take to be the distance between prominent crests, is around 50-100 meters.
Watching the waves for a few minutes, we notice that wave height and wave length
are not constant. The heights vary randomly in time and space, and the
statistical properties of the waves, such as the mean height averaged for a few
hundred waves, change from day to day. These prominent offshore waves are
generated by wind. Sometimes the local wind generates the waves, other times
distant storms generate waves which ultimately reach the coast. For example,
waves breaking on the Southern California coast on a summer day may come from
vast storms offshore of Antarctica 10,000 km away.
If we watch closely for a long time, we notice that sea level changes from hour
to hour. Over a period of a day, sea level increases and decreases relative to a
point on the shore by about a meter. The slow rise and fall of sea level is due
to the tides, another type of wave on the sea surface. Tides\index{tides} have wavelengths of
thousands of kilometers, and they are generated by the slow, very small changes
in gravity due to the motion of the sun\index{sun} and the moon\index{moon} relative to earth.
In this chapter you will learn how to describe ocean-surface waves
quantitatively. In the next chapter I will describe tides and waves along
coasts.
\section{Linear Theory of Ocean Surface Waves}
\index{waves!linear theory}Surface waves are inherently nonlinear: The solution of the
equations of motion depends on the surface boundary conditions, but the surface boundary
conditions are the waves we wish to calculate. How can we proceed?
We begin by assuming that the amplitude of waves on the water surface is
infinitely small so the surface is almost exactly a plane. To simplify the
mathematics, we can also assume that the flow is 2-dimensional with waves
traveling in the $x$-direction. We also assume that the Coriolis
force and viscosity can be neglected. If we retain rotation, we get
Kelvin\index{waves!Kelvin} waves discussed in \S 14.2.
With these assumptions, the sea-surface elevation $\zeta$ of a wave traveling in the
$x$ direction is:
\begin{equation}
\zeta = a \sin (k \, x - \omega \, t)
\end{equation}
with
\begin{eqnarray}
\omega = 2 \pi f = \frac{2 \pi}{T}; & & k = \frac{2 \pi}{L}
\end{eqnarray}
where $\omega$ is wave frequency in radians per second, $f$ is the wave
frequency in Hertz (Hz), $k$ is wave number, $T$ is wave period, $L$ is wave
length, and where we assume, as stated above, that $ka = O(0)$.
The \textit{wave period}\index{waves!period|textbf} $T$ is the time it takes two successive
wave crests or troughs to pass a fixed point. The \textit{wave
length}\index{waves!length|textbf} $L$ is the distance between two successive wave crests or
troughs at a fixed time.
\paragraph{Dispersion Relation}
\index{waves!dispersion relation|textbf}\index{dispersion relation|textbf}Wave frequency
$\omega$ is related to wave number $k$ by the \textit{dispersion relation} (Lamb, 1945
\S{228}):
\begin{equation}
\omega ^{2} = g \, k \tanh (k d)
\end{equation}
where $d$ is the water depth and $g$ is the acceleration of gravity.
Two approximations are especially useful.
\begin{enumerate}
\item \textit{Deep-water approximation} is valid if the water depth $d$ is much
greater than the wave length $L$. In this case, $d \gg L$, $kd \gg 1$, and
$\tanh (kd) = 1$.
\item \textit{Shallow-water approximation} is valid if the water depth is much
less than a wavelength. In this case, $d \ll L$, $kd \ll 1$, and $\tanh (kd) =
kd$.
\end{enumerate}
For these two limits of water depth compared with wavelength the dispersion
relation reduces to:
\begin{align}
\omega ^2 &= g \, k & & \text{Deep-water dispersion relation}\\
d &> L/4 & & \nonumber \\
& & \nonumber \\
\omega ^2 &= g \, k^{2} \, d & & \text{Shallow-water dispersion relation}\\
d &< L/11 & & \nonumber
\end{align}
The stated limits for $d/L$ give a dispersion relation accurate within 10\%.
Because many wave properties can be measured with accuracies of 5--10\%, the
approximations are useful for calculating wave properties. Later we will
learn to calculate wave properties as the waves propagate from deep to
shallow water.
\paragraph{Phase Velocity}
\index{waves!phase velocity}\index{phase velocity}The phase velocity $c$ is the speed at which
a particular phase of the wave propagates, for example, the speed of propagation of the wave
crest. In one wave period $T$ the crest advances one wave length $L$ and the phase speed is
$c=L/T= \omega /k$. Thus, the definition of phase speed is:
\begin{equation}
c \equiv \frac {\omega}{k}
\end{equation}
The direction of propagation is perpendicular to the wave crest and toward the
positive $x$ direction.
The deep- and shallow-water approximations for the dispersion
relation give:
\begin{align}
c &= \sqrt{\frac{g}{k}} = \frac{g}{\omega} & & \text{ Deep-water phase
velocity} \\
& & & \nonumber \\
c &=\sqrt{g\,d} & & \text{ Shallow-water phase velocity}
\end{align}
The approximations are accurate to about 5\% for limits stated in (16.4, 16.5).
In deep water, the phase speed depends on wave length or wave frequency. Longer
waves travel faster. Thus, deep-water waves are said to be dispersive. In
shallow water, the phase speed is independent of the wave; it depends only on the
depth of the water. Shallow-water waves are non-dispersive.
\paragraph{Group Velocity}
\index{waves!group velocity}\index{group velocity}The concept of group velocity $c_{g}$ is
fundamental for understanding the propagation of linear and nonlinear waves. First, it is the
velocity at which a group of waves travels across the ocean. More importantly, it is also the
propagation velocity of wave energy. Whitham (1974, \S 1.3 and \S 11.6) gives a clear
derivation of the concept and the fundamental equation (16.9).
The definition of group velocity in two dimensions is:
\begin{equation}
c_{g} \equiv \frac {\partial \omega }{\partial k}
\end{equation}
Using the approximations for the dispersion relation:
\begin{align}
c_{g} &= \frac{g}{2\omega} = \frac{c}{2} & & \text{ Deep-water group velocity} \\
& & & \nonumber \\
c_{g} &= \sqrt{g\,d} \, = c & & \text{ Shallow-water group velocity}
\end{align}
For ocean-surface waves, the direction of propagation is perpendicular to the
wave crests in the positive $x$ direction. In the more general case of other
types of waves, such as Kelvin\index{waves!Kelvin} and
Rossby\index{waves!Rossby} waves that we met in \S14.2, the group velocity
is not necessarily in the direction perpendicular to wave crests.
Notice that a group of deep-water waves moves at half the phase speed of the
waves making up the group. How can this happen? If we could watch closely a
group of waves crossing the sea, we would see waves crests appear at the back of
the wave train, move through the train, and disappear at the leading edge of the
group. Each wave crest moves at twice the speed of the group.
Do real ocean waves move in groups governed by the dispersion relation? Yes.
Walter Munk and colleagues (1963) in a remarkable series of experiments in the
1960s showed that ocean waves propagating over great distances are dispersive, and
that the dispersion could be used to track storms. They recorded waves for many
days using an array of three pressure gauges just offshore of San Clemente
Island, 60 miles due west of San Diego, California. Wave spectra were
calculated for each day's data. (The concept of a spectra is discussed below.) From
the spectra, the amplitudes and frequencies of the low-frequency waves and the
propagation direction of the waves were calculated. Finally, they plotted contours
of wave energy on a frequency-time diagram (figure 16.1).
To understand the figure, consider a distant storm that produces waves of many
frequencies. The lowest-frequency waves (smallest $\omega$) travel the fastest
(16.11), and they arrive before other, higher-frequency waves. The further
away the storm, the longer the delay between arrivals of waves of different
frequencies. The ridges of high wave energy seen in the figure are produced by
individual storms. The slope of the ridge gives the distance to the storm in
degrees $\Delta$ along a great circle; and the phase information from the array
gives the angle to the storm $\theta$. The two angles give the storm's location
relative to San Clemente. Thus waves arriving from 15 to 18 September produce a
ridge indicating the storm was 115\degrees\ away at an angle of 205\degrees\ which
is south of new Zealand near Antarctica.
The locations of the storms producing the waves recorded from June through
October 1959 were compared with the location of storms plotted on weather maps and in
most cases the two agreed well.
\begin{figure}[t!]
\makebox[121mm][c]{\includegraphics{dispersedwaves}}
\footnotesize
Figure 16.1 Contours \rule{0mm}{4ex}of wave energy on a frequency-time plot
calculated from spectra of waves measured by pressure gauges offshore of
southern California. The ridges of high wave energy show the arrival of
dispersed wave trains from distant storms. The slope of the ridge is inversely
proportional to distance to the storm.
$\Delta$ is distance in degrees, $\theta$ is direction of arrival of waves at
California. After Munk et al. (1963).
\label{fig:dispersedwaves}
\vspace{-3ex}
\end{figure}
\paragraph{Wave Energy}
\index{waves!energy}Wave energy $E$ in Joules per square meter is related to the variance of
sea-surface displacement
$\zeta$ by:
\begin{equation}
E = \rho _{w} g \, \left< \zeta ^{2} \right>
\end{equation}
where $\rho _{w}$ is water density, $g$ is gravity, and the brackets denote a
time or space average.
\begin{figure}[h!]
\vspace{1ex}
\makebox[121mm][c]{\includegraphics{waveheight}}
\centering
\footnotesize
Figure 16.2 A short record \rule{0mm}{3ex}of wave amplitude measured\\
by a wave buoy in the north Atlantic.
\label{fig:waveheight}
\vspace{-1ex}
\end{figure}
\paragraph{Significant Wave Height}
\index{waves!significant height}What do we mean by wave height? If we look at a
wind-driven sea, we see waves of various heights. Some are much larger than most, others are
much smaller (figure 16.2). A practical definition that is often used is the height of the
highest 1/3 of the waves, $H_{1/3}$. The height is computed as follows: measure wave height
for a few minutes, pick out say 120 wave crests and record their heights. Pick the 40 largest
waves and calculate the average height of the 40 values. This is $H_{1/3}$ for the wave record.
%The quantity $H_{1/3}$ was chosen because it
%corresponds closely to wave height estimated by a trained observer looking at the
%sea.
The concept of significant wave height was developed during the World War II
as part of a project to forecast ocean wave heights and periods. Wiegel (1964: p.
198) reports that work at the Scripps Institution of Oceanography showed
\begin{quote} \small
\ldots wave height estimated by observers corresponds to the average of the highest
20 to 40 per cent of waves\ldots Originally, the term significant wave height was
attached to the average of these observations, the highest 30 per cent of the
waves, but has evolved to become the average of the highest one-third of the waves,
(designated $H_S$ or $H_{1/3}$)
\end{quote}
More recently, significant wave height is calculated from measured wave
displacement. If the sea contains a narrow range of wave frequencies, $H_{1/3}$ is
related to the standard deviation of sea-surface displacement (\textsc{nas} , 1963:
22; Hoffman and Karst, 1975)
\begin{equation}
H_{1/3} = 4 \left< \zeta ^{2}\right>^{1/2}
\end{equation}
where $ \left< \zeta ^{2}\right>^{1/2}$ is the standard deviation of surface
displacement. This relationship is much more useful, and it is now the accepted
way to calculate wave height from wave measurements.
\section{Nonlinear waves}
\index{waves!nonlinear}We derived the properties of an ocean surface wave assuming the wave
amplitude was infinitely small $ka = O(0)$. If $ka \ll 1$ but not
infinitely small, the wave properties can be expanded in a power series of $ka$ (Stokes,
1847). He calculated the properties of a wave of finite amplitude and found:
\begin{equation}
\zeta = a \cos(kx - \omega t) + \frac{1}{2} k a^{2}\cos 2(kx-\omega t) +
\frac{3}{8} k^{2} a^{3} \cos 3(k x - \omega t) + \cdots
\end{equation}
The phases of the components for the Fourier series expansion of $\zeta$ in
(16.14) are such that non-linear waves have sharpened crests and flattened
troughs. The maximum amplitude of the Stokes wave is $a_{max} = 0.07 L$ ($ ka =
0.44$). Such steep waves in deep water are called Stokes waves (See also Lamb, 1945,
\S 250).
Knowledge of non-linear waves came slowly until Hasselmann (1961, 1963a, 1963b,
1966), using the tools of high-energy particle physics, worked out to 6th order
the interactions of three or more waves on the sea surface. He, Phillips (1960),
and Longuet-Higgins and Phillips (1962) showed that $n$ free waves on the sea
surface can interact to produce another free wave only if the frequencies and wave
numbers of the interacting waves sum to zero:
\begin{subequations}
\begin{eqnarray}
& \omega _{1} \pm \omega _{2} \pm \omega _{3} \pm \cdots \omega _{n} = 0
& \\
& \mathbf{k_{1}} \pm \mathbf{k_{2}} \pm \mathbf{k_{3}} \pm \cdots
\mathbf{k_{n}} = 0 &
\\
& \omega_{i}^{2} = g \, k_{i} &
\end{eqnarray}
\end{subequations}
where we allow waves to travel in any direction, and $\mathbf{k_{i}}$ is the
vector wave number giving wave length and direction. (16.15) are general
requirements for any interacting waves. The fewest number of waves that meet the
conditions of (16.15) are three waves which interact to produce a fourth. The
interaction is weak; waves must interact for hundreds of wave lengths and periods
to produce a fourth wave with amplitude comparable to the interacting waves. The
Stokes wave does not meet the criteria of (16.15) and the wave components are
not free waves; the higher harmonics are bound to the primary wave.
\paragraph{Wave Momentum}
\index{waves!momentum}The concept of wave momentum has caused considerable confusion
(McIntyre, 1981). In general, waves do not have momentum, a mass flux, but they do
have a momentum flux. This is true for waves on the sea surface. Ursell (1950)
showed that ocean swell on a rotating earth has no mass transport\index{transport!by waves}.
His proof seems to contradict the usual textbook discussions of steep, non-linear waves such as
Stokes waves. Water particles in a Stokes wave move along paths that are
nearly circular, but the paths fail to close, and the particles move slowly in the
direction of wave propagation. This is a mass transport, and the phenomena is called
Stokes drift. But the forward transport near the surface is balanced by an equal
transport in the opposite direction at depth, and there is no net mass flux.
\section{Waves and the Concept of a Wave Spectrum}
\index{waves!spectra!concept}If we look out to sea, we notice that waves on the sea surface are
not sinusoids. The surface appears to be composed of random waves of various lengths
and periods. How can we describe this surface? The answer is, Not very easily. We can
however, with some simplifications, come close to describing the surface. The simplifications
lead to the concept of the spectrum of ocean waves. The spectrum gives the distribution of
wave energy among different wave frequencies or wave lengths on the sea surface.
The concept of a spectrum is based on work by Joseph Fourier (1768--1830), who
showed that almost any function $\zeta (t)$ (or $\zeta (x)$ if you like), can be
represented over the interval $-T/2 \le t \le T/2$ as the sum of an infinite
series of sine and cosine functions with harmonic wave frequencies:
\begin{equation}
\zeta (t) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos 2\pi nft + b_n \sin
2\pi nft)
\end{equation}
where
\begin{subequations}
\begin{align}
a_n &= \frac{2}{T} \int_{-T/2}^{T/2} \zeta (t) \cos 2\pi nft \, dt, \qquad
(n=0,1,2,\ldots) \\
b_n &= \frac{2}{T} \int_{-T/2}^{T/2} \zeta (t) \sin 2\pi nft \, dt, \qquad
(n=0,1,2,\ldots)
\end{align}
\end{subequations}
$f = 1/ T$ is the fundamental frequency, and $nf$ are harmonics of the fundamental frequency. This form of $\zeta (t)$ is called a \textit{Fourier series}\index{waves!Fourier series|textbf}\index{Fourier series|textbf} (Bracewell, 1986: 204; Whittaker and Watson, 1963: \S 9.1). Notice that $a_0$ is the mean value of $\zeta (t)$ over the interval.
Equations (16.18 and 16.19) can be simplified using
\begin{equation}
\exp (2\pi inft) = \cos (2\pi nft) + i \sin (2\pi nft)
\end{equation}
where $i = \sqrt{-1}$. Equations (16.18 and 16.19) then become:
\begin{equation}
\zeta (t) = \sum_{n=-\infty}^{\infty} Z_n \exp ^{i2\pi nft}
\end{equation}
where
\begin{equation}
Z_n = \frac{1}{T} \int_{-T/2}^{T/2} \zeta (t) \exp ^{-i2\pi nft} \, dt, \qquad
(n=0,1,2,\ldots)
\end{equation}
$Z_n$ is called the \textit{Fourier
transform}\index{waves!Fourier transform|textbf} \index{Fourier transform|textbf} of $\zeta (t)$.
The spectrum $S(f)$ of $\zeta (t)$ is:
\begin{equation}
S(nf) = Z_n Z^*_n
\end{equation}
where $Z^*$ is the complex conjugate of $Z$. We will use these forms for the
Fourier series and spectra when we describing the computation of ocean wave
spectra.
We can expand the idea of a Fourier series to include series that represent
surfaces $\zeta (x,y)$ using similar techniques. Thus, any surface can be
represented as an infinite series of sine and cosine functions oriented in all
possible directions.
Now, let's apply these ideas to the sea surface. Suppose for a moment that the
sea surface were frozen in time. Using the Fourier expansion, the frozen surface
can be represented as an infinite series of sine and cosine functions of
different wave numbers oriented in all possible directions. If we unfreeze the
surface and let it move, we can represent the sea surface as an
infinite series of sine and cosine functions of different wave lengths
moving in all directions. Because wave lengths and wave frequencies are related
through the dispersion relation, we can also represent the sea surface as an
infinite sum of sine and cosine functions of different frequencies moving in
all directions.
Note in our discussion of Fourier series that we assume the coefficients
$(a_n, b_n, Z_n)$ are constant. For times of perhaps an hour, and distances of
perhaps tens of kilometers, the waves on the sea surface are sufficiently
fixed that the assumption is true. Furthermore, non-linear interactions among
waves are very weak. Therefore, we can represent a local sea surface by a linear
superposition of real, sine waves having many different wave lengths or
frequencies and different phases traveling in many different directions. The
Fourier series in not just a convenient mathematical expression, it states that
the sea surface is really, truly composed of sine waves, each one propagating
according to the equations I wrote down in \S 16.1.
The concept of the sea surface being composed of independent waves can be carried further. Suppose I throw a rock into a calm ocean, making a big splash. According to Fourier, the splash can be represented as a superposition of cosine waves all of nearly zero phase so the waves add up to a big splash at the origin. Each individual Fourier wave begins to travel away from the splash. The longest waves travel fastest, and eventually, far from the splash, the sea consists of a dispersed train of waves with the longest waves further from the splash and the shortest waves closest. This is exactly what we see in figure 16.1. The storm makes the splash, and the waves disperse as seen in the figure.
\paragraph{Sampling the Sea Surface}
Calculating the Fourier series that represents the sea surface is perhaps
impossible. It requires that we measure the height of the sea surface
$\zeta (x,y,t)$ everywhere in an area perhaps ten kilometers on a side for
perhaps an hour. So, let's simplify. Suppose we install a wave staff somewhere
in the ocean and record the height of the sea surface as a function of time
$\zeta (t)$. We would obtain a record like that in figure 16.2. All waves on
the sea surface will be measured, but we will know nothing about the direction
of the waves. This is a much more practical measurement, and it will give the
frequency spectrum of the waves on the sea surface.
Working with a trace of wave height on say a piece of paper is difficult, so
let's digitize the output of the wave staff to obtain
\begin{align}
\zeta _{j} \equiv \zeta (t_{j}), \qquad t_{j} &\equiv j \Delta \\
j &= 0, 1, 2, \cdots , N-1 \nonumber
\end{align}
where $\Delta $ is the time interval between the samples, and $N$ is
the total number of samples. The length $T$ of the record is $T = N \, \Delta
$. Figure 16.3 shows the first 20 seconds of wave height from figure 16.2
digitized at intervals of $\Delta = 0.32$ s.
\begin{figure}[t!]
\centering
\makebox[121mm][c]{\includegraphics{wavepts}}
\footnotesize
Figure 16.3 The first 20 seconds of \rule{0mm}{3ex}digitized data from figure 16.2.
$\Delta = 0.32$ s.
\label{fi2:wavepts}
\vspace{-3ex}
\end{figure}
Notice that $\zeta _{j}$ is not the same as $\zeta (t)$. We have absolutely no
information about the height of the sea surface between samples. Thus we have
converted from an infinite set of numbers which describes $\zeta (t)$ to a
finite set of numbers which describe $\zeta _{j}$. By converting from a
continuous function to a digitized function, we have given up an infinite
amount of information about the surface.
The sampling interval $\Delta $ defines a \textit{Nyquist critical frequency}\index{Nyquist
critical frequency|textbf}\index{waves!Nyquist critical frequency|textbf} (Press et al, 1992:
494)
\begin{equation}
Ny \equiv 1/( 2 \Delta )
\end{equation}
\begin{quotation} \small
The Nyquist critical frequency is important for two related, but distinct,
reasons. One is good news, the other is bad news. First the good news. It is
the remarkable fact known as the \textit{sampling theorem}: If a continuous
function $\zeta(t)$, sampled at an interval $\Delta $, happens to be
\textit{bandwidth limited} to frequencies smaller in magnitude than $Ny$, i.e.,
if $S(nf)=0$ for all $|nf| \geq Ny$, then the function $\zeta(t)$ is
\textit{completely determined} by its samples $\zeta _j$\dots This is a
remarkable theorem for many reasons, among them that it shows that the
``information content'' of a bandwidth limited function is, in some sense,
infinitely smaller than that of a general continuous function\dots
Now the bad news. The bad news concerns the effect of sampling a continuous
function that is \textit{not} bandwidth limited to less than the Nyquist
critical frequency. In that case, it turns out that all of the power spectral
density that lies outside the frequency range $-Ny \le nf \le Ny$ is spuriously
moved into that range. This phenomenon is called \textit{aliasing}. Any
frequency component outside of the range $(-Ny, Ny)$ is \textit{aliased} (falsely
translated) into that range by the very act of discrete sampling\dots There is
little that you can do to remove aliased power once you have discretely sampled
a signal. The way to overcome aliasing is to (i) know the natural bandwidth
limit of the signal --- or else enforce a known limit by analog filtering of
the continuous signal, and then (ii) sample at a rate sufficiently rapid to
give at least two points per cycle of the highest frequency present. ---Press
et al 1992, but with notation changed to our notation.
\end{quotation}
Figure 16.4 illustrates the aliasing problem. Notice how a high frequency
signal is aliased into a lower frequency if the higher frequency is above the
critical frequency. Fortunately, we can can easily avoid the
problem: (i) use instruments that do not respond to very short,
high frequency waves if we are interested in the bigger waves; and (ii) chose
$\Delta t$ small enough that we lose little useful information. In the example
shown in figure 16.3, there are no waves in the signal to be digitized with
frequencies higher than $Ny = 1.5625$ Hz.
\begin{figure}[t!]
\makebox[121mm] [c]{\includegraphics{aliasplot}}
\footnotesize
\centering
Figure 16.4 Sampling a 4 Hz \rule{0mm}{4ex}sine wave (heavy line) every 0.2 s
aliases the frequency to 1 Hz (light line). The critical frequency is 1/(2
$\times$ 0.2 s) = 2.5 Hz, which is less than 4 Hz.
\label{fig:aliasplot}
\vspace{-3ex}
\end{figure}
Let's summarize. Digitized signals from a wave staff cannot be used to study
waves with frequencies above the Nyquist critical frequency. Nor can the signal
be used to study waves with frequencies less than the fundamental frequency
determined by the duration $T$ of the wave record. The digitized wave record
contains information about waves in the frequency range:
\begin{equation}
\frac{1}{T} < f < \frac{1}{2 \Delta}
\end{equation}
where $T = N \Delta $ is the length of the time series, and $f$
is the frequency in Hertz.
\paragraph{Calculating The Wave Spectrum}
\index{waves!spectra!calculating}The digital Fourier transform $Z_n$ of a wave record $\zeta
_j$ equivalent to (16.21 and 16.22) is:
\begin{subequations}
\begin{align}
Z_{n} &= \frac{1}{N} \sum_{j=0}^{N-1} \zeta_{j} \exp [-i2 \pi j n /N] \\
\zeta_{n} &= \sum_{n=0}^{N-1} Z_{j} \exp [i 2 \pi j n /N]
\end{align}
\end{subequations}
for $j=0,1,\cdots, N-1$; $n= 0, 1, \cdots , N-1$. These equations can be summed
very quickly using the Fast Fourier Transform, especially if $N$ is a power of
2 (Cooley, Lewis, and Welch, 1970; Press et al. 1992: 542).
This spectrum $S_{n}$ of $\zeta $, which is called the
\textit{periodogram}\index{periodogram|textbf}\index{waves!periodogram|textbf}, is:
\begin{align}
S_{n} &= \frac{1}{N^{2}} \left[ |Z_{n}|^{2} + |Z_{N-n}|^{2} \right] ; \qquad n =
1, 2, \cdots , ( N/2 - 1 ) \\
S_{0} &= \frac{1}{N^{2}} |Z_{0}|^{2} \notag \\
S_{N/2} &= \frac{1}{N^{2}} |Z_{N/2}|^{2} \notag
\end{align}
where $ S_{N} $ is normalized such that:
\begin{equation}
\sum_{j=0}^{N-1} |\zeta_{j}|^2 = \sum_{n=0}^{N/2} S_{n}
\end{equation}
The variance of $\zeta_{j}$ is the sum of the $(N/2 + 1)$ terms in the
periodogram. Note, the terms of $S_{n}$ above the frequency $(N/2)$ are symmetric
about that frequency. Figure 16.5 shows the periodogram of the time series shown
in figure 16.2.
\begin{figure}[t!]
\makebox[121mm][c]{\includegraphics{periodogram}}
\footnotesize
\centering
Figure 16.5 The periodogram calculated \rule{0mm}{3ex}from the first 164 s of
data\\from figure 16.2. The Nyquist frequency is 1.5625 Hz.
\label{fig:periodogram}
\vspace{-3ex}
\end{figure}
The periodogram is a very noisy function. The variance of each point is equal to the expected
value at the point. By averaging together 10--30 periodograms we can reduce the uncertainty in
the value at each frequency. The averaged periodogram is called the spectrum of the wave
height (figure 16.6). It gives the distribution of the variance of sea-surface height at the
wave staff as a function of frequency. Because wave energy is proportional to the variance
(16.12) the spectrum is called the \textit{energy spectrum}\index{waves!spectra!energy|textbf}
or the
\textit{wave-height spectrum}\index{waves!spectra!wave-height|textbf}. Typically three hours of
wave staff data are used to compute a spectrum of wave height.
\begin{figure}[t!]
\makebox[121mm][c]{\includegraphics{wavespectrum}}
\footnotesize
Figure 16.6 The spectrum of \rule{0mm}{4ex}waves calculated from 11 minutes of
data shown in figure 7.2 by averaging four periodograms to reduce uncertainty in
the spectral values. Spectral values below 0.04 Hz are in error due to noise.
\label{fig:wavespectrum}
\vspace{-3ex}
\end{figure}
\paragraph{Summary}
We can summarize the calculation of a spectrum into the following steps:
\begin{enumerate}
\vitem
Digitize a segment of wave-height data to obtain useful limits according to
(16.26). For example, use 1024 samples from 8.53 minutes of data sampled at
the rate of 2 samples/second.
\vitem
Calculate the digital, fast Fourier transform $Z_{n}$ of the time series.
\vitem
Calculate the periodogram $S_{n}$ from the sum of the squares of the real and
imaginary parts of the Fourier transform.
\vitem
Repeat to produce $M=20$ periodograms.
\vitem
Average the 20 periodograms to produce an averaged spectrum $S_{M}$.
\vitem
$S_{M}$ has values that are $\chi ^{2}$ distributed with $2 M$ degrees of
freedom.
\end{enumerate}
This outline of the calculation of a spectrum ignores many details. For more
complete information see, for example, Percival and Walden (1993), Press et al.
(1992: \S 12), Oppenheim and Schafer (1975), or other texts on digital signal
processing.
\section{Ocean-Wave Spectra}
\index{waves!spectra}Ocean waves are produced by the wind. The faster the wind, the longer the
wind blows, and the bigger the area over which the wind blows, the bigger the waves. In
designing ships or offshore structures we wish to know the biggest waves produced by a given
wind speed. Suppose the wind blows at 20 m/s for many days over a large area of the North
Atlantic. What will be the spectrum of ocean waves at the downwind side of the area?
\paragraph{Pierson-Moskowitz Spectrum}
\index{waves!spectra!Pierson-Moskowitz}Various idealized spectra are used to answer the
question in ocean\-ography and ocean engineering. Perhaps the simplest is that proposed by
Pierson and Moskowitz (1964). They assumed that if the wind blew steadily for a long time over
a large area, the waves would come into equilibrium with the wind. This is the concept of a
\textit{fully developed sea}\index{fully developed sea|textbf}. Here, a ``long time'' is
roughly ten-thousand wave periods, and a ``large area'' is roughly five-thousand wave lengths
on a side.
\begin{figure}[t!]
\makebox[121mm][c]{\includegraphics{PMSpectra}}
\footnotesize
\centering
Figure 16.7 Wave spectra of a fully developed \rule{0mm}{4ex}sea for different\\wind speeds
according to Moskowitz (1964).
\label{fig:PMSpectra}
\vspace{-2ex}
\end{figure}
To obtain a spectrum of a fully developed sea, they used measurements of waves
made by accelerometers on British weather ships in the north Atlantic. First, they
selected wave data for times when the wind had blown steadily for long times over
large areas of the north Atlantic. Then they calculated the wave spectra for various
wind speeds (figure 16.7), and they found that the function
\begin{equation}
S(\omega) = \frac{\alpha g^{2}}{\omega ^{5}} \exp \left[ - \beta \left(
\frac{\omega _{0}}{\omega } \right) ^{4} \right]
\end{equation}
was a good fit to the observed spectra, where $\omega = 2\pi f$, $f$ is the wave frequency in
Hertz,
$\alpha = 8.1
\times 10^{-3}$,
$\beta = 0.74
$ ,
$\omega _{0} = g/U_{19.5}$ and $U_{19.5}$ is the wind speed at a height of 19.5
m above the sea surface, the height of the anemometers on the weather ships
used by Pierson and Moskowitz (1964).
For most airflow over the sea the atmospheric boundary layer has nearly neutral
stability, and
\begin{equation}
U_{19.5}\approx 1.026\, U_{10}
\end{equation}
assuming a drag coefficient\index{drag!coefficient} of $1.3 \times 10^{-3}$.
The frequency of the peak of the Pierson-Moskowitz spectrum is calculated
by solving
$dS/d\omega = 0$ for $\omega _{p}$, to obtain
\begin{equation}
\omega _{p} = 0.877 \,g/U_{19.5}.
\end{equation}
The speed of waves at the peak is calculated from (16.7), which gives:
\begin{equation}
c_{p} = \frac{g}{\omega _{p}} = 1.14 \, U_{19.5} \approx 1.17\,U_{10}
\end{equation}
Hence waves with frequency $\omega _{p}$ travel 14\% faster than the wind at a height of 19.5 m or 17\% faster than the wind at a height of 10 m. This poses a difficult problem: How can the wind produce waves traveling faster than the wind? I will return to the problem after I discuss the \textsc{jonswap}
spectrum and the influence of nonlinear interactions among wind-generated waves.
\begin{figure}[b!]
\vspace{-2ex}
\makebox[121mm][c]{\includegraphics{wavehtperiod}}
\footnotesize
\centering
Figure 16.8 Significant \rule{0mm}{3ex}wave height and period at the peak of the
spectrum of a fully developed sea calculated from the Pierson-Moskowitz spectrum
using (16.35 and 16.32).
\label{fig:wavehtperiod}
%\vspace{-2ex}
\end{figure}
By integrating $S(\omega)$ over all $\omega$ we get the variance of surface elevation:
\begin{equation}
\left<\zeta ^{2}\right> = \int_{0}^{\infty} S(\omega )\, d \omega = 2.74 \times
10^{-3}
\,\frac{\left(U_{19.5}
\right)^4}{g^2}
\end{equation}
Remembering that $H_{1/3} = 4 \left<\zeta ^{2}\right>^{1/2}$, the significant wave
height is:
\begin{equation}
H_{1/3} = 0.21 \, \frac{\left(U_{19.5} \right)^2}{g}\approx 0.22 \,
\frac{\left(U_{10} \right)^2}{g}
\end{equation}
Figure 16.8 gives significant wave heights and periods for the Pierson-Moskowitz spectrum.
\paragraph{JONSWAP Spectrum}Hasselmann et al. (1973), after analyzing data
\index{waves!spectra!JONSWAP}collected during the Joint North Sea Wave Observation Project
\textsc{jonswap}, found that the wave spectrum is never fully developed (figure 16.9). It
continues to develop through non-linear, wave-wave interactions even for very long times and
distances. They therefore proposed the spectrum:
\begin{figure}[t!]
%\vspace{-3ex}
\makebox[121mm][c]{\includegraphics{hasselmannspect}}
\footnotesize
\centering
Figure 16.9 Wave spectra of a developing \rule{0mm}{3ex}sea for different
fetches\\measured at \textsc{jonswap}. After Hasselmann et al. (1973).
\label{fig:hasselmannspect}
\vspace{-3ex}
\end{figure}
\begin{subequations}
\begin{align}
S(\omega) &= \frac{\alpha g^{2}}{\omega ^{5}} \exp \left[ - \frac{5}{4}
\left(
\frac{\omega _{p}}{\omega } \right) ^{4} \right] \gamma ^{r} \\
r &= \exp \biggl[ - \frac{\left(\omega - \omega _{p}\right)^{2}}{2\, \sigma ^{2}
\,\omega _{p}^{2}}
\biggr]
\end{align}
\end{subequations}
Wave data collected during the \textsc{jonswap} experiment were used to
determine the values for the constants in (16.36):
\begin{subequations}
\begin{align}
\alpha &= 0.076 \, \left( \frac{U_{10}^{2}}{F \, g} \right)^{0.22} \\
\omega_p &= 22 \left(\frac{g^2}{U_{10}F}\right)^{1/3} \\
\gamma &= 3.3 \\
\sigma &= \left\{ \begin{array}{ll}
0.07 & \omega \leq \omega _{p} \\
0.09 & \omega > \omega _{p}
\end{array}
\right.
\end{align}
\end{subequations}
where $F$ is the distance from a lee shore, called the \textit{fetch}\index{fetch|textbf}\index{waves!fetch|textbf},
or the distance over which the wind blows with constant velocity.
The energy of the waves increases with fetch:
\begin{equation}
\left< \zeta ^{2}\right> = 1.67 \times 10^{-7}\, \frac{\left( U_{10} \right)^{2} }{g} \, F
\end{equation}
where $F$ is fetch.
The \textsc{jonswap} spectrum is similar to the Pierson-Moskowitz spectrum
except that waves continues to grow with distance (or time) as specified by the
$\alpha$ term, and the peak in the spectrum is more pronounced, as specified by the
$\gamma$ term. The latter turns out to be particularly important because it
leads to enhanced non-linear interactions and a spectrum that changes in time
according to the theory of Hasselmann (1966).
\paragraph{Generation of Waves by Wind}
\index{waves!generation by wind}\index{wind!generation of waves}We have seen in the
last few paragraphs that waves are related to the wind. We have, however, put off
until now just how they are generated by the wind. Suppose we begin with a
mirror-smooth sea (Beaufort Number 0). What happens if the wind suddenly begins to
blow steadily at say 8 m/s? Three different physical processes begin:
\begin{enumerate}
\vitem
The turbulence\index{turbulence!atmospheric} in the wind produces random pressure fluctuations
at the sea surface, which produces small waves with wavelengths of a few
centimeters (Phillips, 1957).
\vitem
Next, the wind acts on the small waves, causing them to become larger. Wind
blowing over the wave produces pressure differences along the wave profile
causing the wave to grow. The process is unstable because, as the wave gets
bigger, the pressure differences get bigger, and the wave grows faster. The
instability causes the wave to grow exponentially (Miles, 1957).
\vitem
Finally, the waves begin to interact among themselves to produce longer
waves (Hasselmann et al. 1973). The interaction transfers wave energy
from short waves generated by Miles' mechanism to waves with frequencies slightly
lower than the frequency of waves at the peak of the spectrum.
Eventually, this leads to waves going faster than the wind, as noted by Pierson
and Moskowitz.
\end{enumerate}
\section{Wave Forecasting}
\index{waves!forecasting}Our understanding of ocean waves, their spectra, their generation by
the wind, and their interactions are now sufficiently well understood that the wave spectrum
can be forecast using winds calculated from numerical weather models. If we observe some small
ocean area, or some area just offshore, we can see waves generated by the local wind, the
\textit{wind sea}\index{wind sea|textbf}, plus waves that were generated in other areas at
other times and that have propagated into the area we are observing, the \textit{swell}.
Forecasts of local wave conditions must include both sea and swell, hence wave forecasting is
not a local problem. We saw, for example, that waves off California can be generated by storms
more than 10,000 km away.
Various techniques have been used to forecast waves. The earliest attempts were
based on empirical relationships between wave height and wave length and wind
speed, duration, and fetch. The development of the wave spectrum allowed evolution
of individual wave components with frequency $f$ travelling in direction $\theta $
of the directional wave spectrum
$\psi (f, \theta )$ using
\begin{equation}
\frac{\partial \psi_0 }{\partial t} + \mathbf{c_g \cdot \nabla }\psi_0 = S_{i}
+ S_{nl} + S_{d}
\end{equation}
where $\psi_0 = \psi _0 (f, \theta ; \mathbf{x},t)$ varies in space
($\mathbf x$) and time $t$, $S_{i}$ is input from the wind given by the
Phillips (1957) and Miles (1957) mechanisms, $S_{nl}$ is the transfer among wave
components due to nonlinear interactions, and $S_{d}$ is dissipation.
The third-generation wave-forecasting models now used by meteorological agencies throughout the world are based on integrations of (16.39) using many individual wave components (The \textsc{swamp} Group 1985; The \textsc{wamdi} Group, 1988; Komen et al, 1996). The forecasts follow individual components of the wave spectrum in space and time, allowing each component to grow or decay depending on local winds, and allowing wave components to interact according to Hasselmann's theory. Typically the sea is represented by 300 components: 25 wavelengths going in 12 directions (30\degrees ). To reduce computing time, the models use a nested grid of points: the grid has a high density of points in storms and near coasts and a low density in other regions. Typically, grid points in the open ocean are
3\degrees\ apart.
Some recent experimental models take the wave-forecasting process one step further by assimilating altimeter and scatterometer\index{scatterometers}\index{wind!from scatterometers} observations of wind speed and wave height into the model. Forecasts of waves using assimilated satellite data are available from the European Centre for Medium-Range Weather Forecasts.
\textsc{Noaa}'s Ocean Modeling Branch at the National Centers for Environmental Predictions also produces regional and global forecasts of waves. The Branch use a third-generation model based on the Cycle-4 \textsc{wam} model. It accommodates ever-changing ice edge, and it assimilates buoy and
satellite altimeter wave data. The model calculates directional frequency spectra in 12 directions and 25 frequencies at 3-hour intervals up to 72 hours in advance. The lowest frequency is 0.04177 Hz and higher frequencies are spaced logarithmically at increments of 0.1 times the lowest frequency. Wave spectral data are available on a 2.5\degrees $\times$ 2.5\degrees\ grid for deep-water points between 67.5\degrees S and 77.5\degrees N. The model is driven using 10-meter winds calculated from the lowest winds from the Centers weather model adjusted to a height of 10 meter by using a logarithmic profile (8.20). The Branch is testing an improved forecast with 1\degrees $\times$ 1.25\degrees\ resolution (figure 16.10).
\begin{figure} [t!]
\makebox[121 mm] [c] {\includegraphics{NoaaWaves}}
%\centering
\footnotesize
Figure 16.10 Output of a third-generation \rule{0pt}{4ex}wave forecast model
produced by
\textsc{Noaa}'s Ocean Modeling Branch for 20 August 1998. Contours are
significant wave height in meters, arrows give direction of waves at the peak
of the wave spectrum, and barbs give wind speed in m/s and direction. From
\textsc{noaa} Ocean Modeling Branch.
\label{fig:noaa.waves}
\vspace{-4ex}
\end{figure}
\section{Measurement of Waves}
\index{waves!measurement of|(}Because waves influence so many processes and operations at sea, many techniques have been invented for measuring waves. Here are a few of the more commonly used. Stewart (1980) gives a more complete description of wave measurement techniques, including methods for measuring the directional distribution of waves.
\paragraph{Sea State Estimated by Observers at Sea} This is perhaps the most common observation included in early tabulations of wave heights. These are the significant wave heights summarized in the U.S. Navy's \textit{Marine Climatological Atlas} and other such reports printed before the age of satellites.
\paragraph{Satellite Altimeters} The satellite altimeters used to measure \index{waves!measurement of!satellite altimeters}\index{satellite altimetry}surface geo\-strophic currents also measure wave height. Altimeters were flown on Seasat in 1978, Geosat\index{Geosat} from 1985 to 1988, \textsc{ers--1 \&2} from 1991, Topex/Poseidon\index{Topex/Poseidon} from 1992, and Jason\index{Jason} from 2001. Altimeter data have been used to produce monthly mean maps of wave heights and the variability of wave energy density in time and space. The next step, just begun, is to use altimeter observation with wave forecasting programs, to increase the accuracy\index{accuracy!wave height} of wave forecasts.
The altimeter technique works as follows. Radio pulse from a satellite altimeter reflect first from the wave crests, later from the wave troughs. The reflection stretches the altimeter pulse in time, and the stretching is measured and used to calculate wave height (figure 16.11). Accuracy is $\pm 10$\%.
\begin{figure}[t!]
\makebox[121 mm] [c] {\includegraphics{altimeterpulse}}
\footnotesize
Figure 16.11 Shape of \rule{0mm}{4ex}radio pulse received by the Seasat altimeter, showing the influence of ocean waves. The shape of the pulse is used to calculate significant wave height. After Stewart (1985: 264).
\label{fig:altimeterpulse}
\vspace{-3ex}
\end{figure}
\paragraph{Accelerometer Mounted on Meteorological or Other Buoy} This is a less common measurement, although it is often used for measuring waves during short experiments at sea. For example, accelerometers on weather ships measured wave height used by Pierson \& Moskowitz and the waves shown in figure 16.2. The most accurate measurements are made using an accelerometer stabilized by a gyro so the axis of the accelerometer is always vertical.
Double integration of vertical acceleration gives displacement. The double integration, however, amplifies low-frequency noise, leading to the low frequency signals seen in figures 16.5 and 16.6. In addition, the buoy's heave is not sensitive to wavelengths less than the buoy's diameter, and buoys measure only waves having wavelengths greater than the diameter of the buoy. Overall, careful measurements are accurate to $\pm \,$10\% or better.
\paragraph{Wave Gages} Gauges may be mounted on platforms or on the sea floor in \index{waves!measurement of!gages}shallow water. Many different types of sensors are used to measure the height of the wave or subsurface pressure which is related to
wave height. Sound, infrared beams, and radio waves can be used to determine the distance from the sensor to the sea surface provided the sensor can be mounted on a stable platform that does not interfere with the waves. Pressure gauges described in
\S 6.8 can be used to measure the depth from the sea surface to the gauge. Arrays of bottom-mounted pressure gauges are useful for determining wave directions. Thus arrays are widely used just offshore of the surf zone to determine offshore wave directions.
Pressure gauge must be located within a quarter of a wavelength of the surface because wave-induced pressure fluctuations decrease exponentially with depth. Thus, both gauges and pressure sensors are restricted to shallow water or to large platforms on the continental shelf. Again, accuracy\index{accuracy!wave height} is $\pm \,$10\% or better.
\paragraph{Synthetic Aperture Radars on Satellites} These radars map the \index{waves!measurement of!synthetic aperture radars}radar reflectivity of the sea surface with spatial resolution of 6--25 m. Maps of reflectivity often show wave-like features related
to the real waves on the sea surface. I say `wave-like' because there is not an exact one-to-one relationship between wave height and image density. Some waves are clearly mapped, others less so. The maps, however, can be used to obtain additional information about waves, especially the spatial distribution of wave directions in shallow water (Vesecky and Stewart, 1982). Because the directional information can be calculated directly from the radar data without the need to calculate an image (Hasselmann, 1991), data from radars and altimeters on \textsc{ers}--1 \& 2\index{ERS satellites} are being used to determine if the radar and altimeter observations can be used directly in wave forecast programs\index{waves!measurement
of|)}.
\section{Important Concepts}
\begin{enumerate}
\item Wavelength and frequency of waves are related through the dispersion relation.
\vitem The velocity of a wave phase can differ from the velocity at which wave energy propagates.
\vitem Waves in deep water are dispersive, longer wavelengths travel faster than shorter wavelengths. Waves in shallow water are not dispersive.
\vitem The dispersion of ocean waves has been accurately measured, and observations of dispersed waves can be used to track distant storms.
\vitem The shape of the sea surface results from a linear superposition of waves of all possible wavelengths or frequencies travelling in all possible directions.
\vitem The spectrum gives the contributions by wavelength or frequency to the variance of surface displacement.
\vitem Wave energy is proportional to variance of surface displacement.
\vitem Digital spectra are band limited, and they contain no information about waves with frequencies higher than the Nyquist frequency.
\vitem Waves are generated by wind. Strong winds of long duration generate the largest waves.
\vitem Various idealized forms of the wave spectrum generated by steady, homogeneous winds have been proposed. Two important ones are the Pierson-Moskowitz and \textsc{jonswap} spectra.
\vitem Observations by mariners on ships and by satellite altimeters have been used to make global maps of wave height. Wave gauges are used on platforms in shallow water and on the continental shelf to measure waves. Bottom-mounted pressure gauges are used to measure waves just offshore of beaches. And synthetic-aperture radars are used to obtain information about wave
directions.
\end{enumerate}
\chapter{Coastal Processes and Tides}
\addtocounter{figure}{1}
In the last chapter I described waves on the sea surface. Now we can consider several special and important cases: the transformation of waves as they come ashore and break; the currents and edge waves generated by the interaction of waves with coasts; tsunamis\index{tsunami}; storm surges; and tides, especially tides along coasts and in the deep ocean.
\section{Shoaling Waves and Coastal Processes}
\index{waves!shoaling}Wave phase and group velocities are a function of depth when the depth is less than about one-quarter wavelength in deep water. Wave period and frequency are invariant (don't change as the wave comes ashore); and this is used to compute the properties of shoaling waves. Wave height increases as wave group velocity slows. Wave length decreases. Waves change direction due to refraction. Finally, waves break if the water is sufficiently shallow; and broken waves pour water into the surf zone, creating long-shore and rip currents\index{rip currents}.
\paragraph{Shoaling Waves}
The dispersion relation (16.3) is used to calculate wave properties as the waves propagate from deep water offshore to shallow water just outside the surf zone. Because $\omega$ is constant, (16.3) leads to:
\begin{equation}
\frac{L}{L_{0}} = \frac{c}{c_{0}}=\frac{\sin \alpha }{\sin \alpha _{0}} = \tanh
\frac{2 \pi d}{L} \label{eq:intermed}
\end{equation}
where
\begin{equation}
L_{0} = \frac{g T^{2}}{2 \pi }, \qquad c_{0} = \frac{g T}{2 \pi } \label{eq:Lzero}
\end{equation}
and $L$ is wave length, $c$ is phase velocity, $\alpha $ is the angle of the crest relative to contours of constant depth, and $d$ is water depth. The subscript $0$ indicates values in deep water.
The quantity $d/L$ is calculated from the solution of
\begin{equation}
\frac{d}{L_{0}} = \frac{d}{L} \tanh \frac{2 \pi d}{L} \label{eq:intermediate}
\end{equation}
using an iterative technique, or from figure 17.1, or from Wiegel (1964: A1).
\begin{figure}[t!]
%\centering
\makebox[121 mm] [c]{\includegraphics{wiegelgraph}}
\footnotesize
Figure 17.1 Properties of \rule{0mm}{3ex}waves in intermediate depths between deep and shallow water. $d=$ depth, $L=$ wavelength, $C=$ phase velocity, $C_g=$ group velocity, $\alpha = $ angle of crests relative to contours of constant depth, $H = $ wave height. Subscript $0$ refers to properties in deep water. From values in Wiegel (1964: A1).
\label{wiegelgraph}
\vspace{-4ex}
\end{figure}
\begin{figure}[b!]
\vspace{-1ex}
\centering
\makebox[121 mm] [c]{\includegraphics{wavefocusing}}
\footnotesize
Figure 17.2 sub-sea \rule{0mm}{4ex}features, such as submarine canyons and ridges, offshore of coasts can greatly influence the height of breakers\index{breakers!height of} inshore of the features. After Thurman (1985: 229).
\label{wavefocusing}
%\vspace{-2ex}
\end{figure}
Because wave velocity is a function of depth in shallow water, variations in offshore water depth can focus or defocus wave energy reaching the shore. Consider the simple case of waves with deep-water crests almost parallel to a straight beach with two ridges each extending seaward from a headland (figure 17.2). Wave group velocity is faster in the deeper water between the ridges, and the wave crests become progressively deformed as the wave propagates toward the beach. Wave energy, which propagates perpendicular to wave crests, is refracted out of the region between the headland. As a result, wave energy is focused into the headlands, and breakers\index{breakers!height of} there are much larger than breakers in the bay. The difference in wave height can be surprisingly large. On a calm day, breakers can be knee high shoreward of a submarine canyon at La Jolla Shores, California, just south of the Scripps Institution of Oceanography. At the same time, waves just north of the canyon can be high enough to attract surfers.
\begin{figure}[t!]
\centering
\makebox[121 mm] [c]{\includegraphics{breakers}}
\footnotesize
Figure 17.3 \textbf{Left}: \rule{0mm}{3ex}Classification\index{breakers!types of} of breaking waves as a function of beach steepness and wave steepness offshore. \textbf{Right}: Sketch of types of breaking waves. After Horikawa (1988: 79, 81).
\label{fig:breakers}
\vspace{-3ex}
\end{figure}
\paragraph{Breaking Waves}
\index{waves!breaking}As waves move into shallow water, the group velocity becomes small, wave energy per square meter of sea surface increases, and non-linear terms in the wave equations become important. These processes cause waves to steepen, with
short steep crests and broad shallow troughs. When wave slope at the crest becomes sufficiently steep, the wave breaks (figure 17.3 Right). The shape of the breaking wave depends on the slope of the bottom, and the steepness of waves offshore (figure 17.3 Left).
\begin{enumerate}
\vitem
Steep waves tend to lose energy slowly as the waves moves into shallower water through water spilling down the front of the wave. These are spilling breakers\index{breakers!spilling}.
\vitem
Less steep waves on steep beaches tend to steepen so quickly that the crest of the wave moves much faster than the trough, and the crest, racing ahead of the trough, plunges\index{breakers!plunging} into the trough (figure 17.4).
\vitem
If the beach is sufficiently steep, the wave can surge\index{breakers!surging} up the face of the beach without breaking in the sense that white water is formed. Or if it is formed, it is at the leading edge of the water as it surges up the beach. An extreme example would be a wave incident on a vertical breakwater.
\end{enumerate}
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{wavecropped}}
\centering
\footnotesize
Figure 17.4 Steep, plunging \rule{0pt}{4ex}breakers\index{breakers!plunging} are the archetypical breaker. The edge\\of such breakers are ideal for surfing. From photo by Jeff Devine.
\label{fig:wavecropped}
\vspace{-3ex}
\end{figure}
\paragraph{Wave-Driven Currents}
\index{currents!wave-driven}\index{waves!currents}Waves break in a narrow band of shallow water along the beach, the \textit{surf zone}\index{surf zone|textbf}. After breaking, waves continues as a near-vertical wall of turbulent water called a \textit{bore}\index{bore|textbf} which carries water to the beach. At first, the bore surges up the beach, then retreats. The water carried by the bore is left in the shallow waters inside the breaker\index{breakers!and long-shore currents} zone.
\begin{figure}[b!]
\vspace{-2ex}
\makebox[121mm][c]{\includegraphics{rips}}
\centering
\footnotesize
Figure 17.5 Sketch of rip currents\index{rip currents} \rule{0mm}{4ex}generated by water carried to the beach by breaking waves.
\label{fig:rips}
%\vspace{-3ex}
\end{figure}
Water dumped inside the breaker\index{breakers!and long-shore currents} zone must return offshore. It does this by first moving parallel to the beach as an \textit{along-shore current}\index{currents!along shore|textbf}. Then it turns and flows offshore perpendicular to the beach in a narrow, swift \textit{rip current}\index{currents!rip|textbf}\index{rip currents}. The rips are usually spaced hundreds of meters apart (figure 17.5). Usually there is a band of deeper water between the breaker zone and the beach, and the long-shore current runs in this channel. The strength of a rip current\index{rip currents} depends on the height and frequency of breaking waves and the strength of the onshore wind. Rips are a danger to unwary swimmers, especially poor swimmers bobbing along in the waves inside the breaker zone. They are carried along by the along-shore current until they are suddenly carried out to sea by the rip. Swimming against the rip is futile, but swimmers can escape by swimming parallel to the beach.
\textit{Edge waves}\index{waves!edge|textbf} are produced by the variability of wave energy reaching shore. Waves tend to come in groups, especially when waves come from distant storms. For several minutes breakers\index{breakers!and edge waves} may be smaller than average, then a few very large waves will break. The minute-to-minute variation in the height of breakers produces low-frequency variability in the along-shore current. This, in turn, drives a low-frequency wave attached to the beach, an edge wave. The waves have periods of a few minutes, a long-shore wave length of around a kilometer, and an amplitude that decays exponentially offshore (figure 17.6).
\begin{figure}[h!]
\vspace{-2ex}
\makebox[121mm][c]{\includegraphics{edgewave}}
\centering
\footnotesize
Figure 17.6 Computer-assisted sketch of \rule{0mm}{4ex}an edge wave. Such waves
exist in the breaker\\zone near the beach and on the continental shelf. After
Cutchin and Smith (1973).
\label{fig:edgewave}
\vspace{-4ex}
\end{figure}
\section{Tsunamis}
\index{tsunamis|textbf}Tsunamis are low-frequency ocean waves generated by submarine earthquakes. The sudden motion of sea floor over distances of a hundred or more kilometers generates waves with periods of 15--40 minutes (figure 17.7). A quick calculation shows that such waves must be shallow-water waves, propagating at a speed of 180 m/s and having a wavelength of 130 km in water 3.6 km deep (figure 17.8). The waves are not noticeable at sea, but after slowing on approach to the coast, and
after refraction by sub-sea features, they can come ashore and surge to heights ten or more meters above sea level. In an extreme example, the great 2004 Indian Ocean tsunami\index{tsunami!Indian Ocean} destroyed hundreds of villages, killing at least 200,000 people.
\begin{figure}[t!]
\makebox[121mm][c]{\includegraphics{tsunami}}
\footnotesize Figure 17.7 \textbf{Left} Hourly positions \rule{0mm}{3ex}of leading
edge of tsunami\index{tsunami!Hawaiian} generated by the large earthquake in the Aleutian
Trench on April 1, 1946 at 1:59 AM Hawaiian time (12:59 GMT).
\textbf{Right: top} Sealevel recorded by a river gauge in the estuary of the Waimea
River.
\textbf{Right: lower} Map of Kauai showing the heights reached by the water (in meters above
lower low water) during the tsunami, wave fronts, orthogonals, and submarine contours. Times
refer to the computed arrival time of the first wave. After Shepard, MacDonald, and Cox (1950).
\label{fig:tsunami}
\vspace{-2ex}
\end{figure}
Shepard (1963, Chapter 4) summarized the influence of tsunamis\index{tsunami!characteristics}
based on his studies in the Pacific.
\begin{enumerate}
\vitem
Tsunamis appear to be produced by movement (an earthquake) along a linear fault.
\vitem
Tsunamis can travel thousands of kilometers and still do serious damage.
\vitem
The first wave of a tsunami is not likely to be the biggest.
\vitem
Wave amplitudes are relatively large shoreward of submarine ridges. They are relatively low shoreward of submarine valleys, provided the features extend into deep water.
\vitem
Wave amplitudes are decreased by the presence of coral reefs bordering the coast.
\vitem
Some bays have a funneling effect, but long estuaries attenuate waves.
\vitem
Waves can bend around circular islands without great loss of energy, but they are considerably smaller on the backsides of elongated, angular islands.
\end{enumerate}
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{tsunamiwave}}
%\centering
\footnotesize Figure 17.8 Tsunami \rule{0mm}{3ex}waves\index{tsunami!Cascadia 1700} four hours after the great M9 Cascadia earthquake off the coast of Washington on 26 January 1700 calculated by a finite-element, numerical model. Maximum open-ocean wave height, about one meter, is north of Hawaii. After Satake et al. (1996).
\label{fig:tsunamiwave}
\vspace{-3ex}
\end{figure}
Numerical models are used to forecast tsunami heights throughout ocean basins and the inundation of coasts. For example, \textsc{noaa}'s Center for Tsunami Research uses the Method of Splitting Tsunami \textsc{most} model (Titov and Gonzalez, 1997). The model uses nested grids to resolve the tsunami wavelength, it propagates the wave across ocean basins, and then calculates run-up when the wave comes ashore. It is initialized from a ground deformation model that uses measured earthquake magnitude and location to calculate vertical displacement of the sea floor. The forcing is modified once waves are measured near the earthquake by seafloor observing stations.
\section{Storm Surges}
\index{storm surges}Storm winds blowing over shallow, continental shelves pile water against the coast. The increase in sea level is known as a storm surge. Several processes are important:
\begin{enumerate}
\vitem
Ekman transport\index{Ekman transport} by winds parallel to the coast transports water toward the coast causing a rise in sea level.
\vitem
Winds blowing toward the coast push water directly toward the coast.
\vitem
Wave run-up and other wave interactions transport\index{transport!mass and storm surges} water toward the coast adding to the first two processes.
\vitem
Edge waves generated by the wind travel along the coast.
\vitem
The low pressure inside the storm raises sea level by one centimeter for each millibar decrease in pressure through the inverted-barometer effect.
\vitem
Finally, the storm surge adds to the tides, and high tides can change a relative weak surge into a much more dangerous one.
\end{enumerate}
See Graber et al (2006) and \S 15.5 for a description of Advanced Circulation Model used by the National Hurricane Center for predicting storm-surges.
To a crude first approximation, wind blowing over shallow water causes a slope in the sea surface proportional to wind stress\index{wind stress!and storm surges}.
\begin{equation}
\frac{\partial \zeta }{\partial x}= \frac{\tau _{0}}{\rho g H}
\end{equation}
where $\zeta $ is sea level, $x$ is horizontal distance, $H$ is water depth, $T_{0}$ is wind stress at the sea surface, $\rho $ is water density; and $g$ is gravitational acceleration.
If $x=100$ km, $U=40$ m/s, and $H=20$ m, values typical of a hurricane offshore of the Texas Gulf Coast, then $T= 2.7$ Pa, and $\zeta = 1.3$ m at the shore. Figure 17.9 shows the frequency of surges at the Netherlands and a graphical method for estimating the probability of extreme events using the probability of weaker events.
\begin{figure}[t!]
%\vspace{-3ex}
\makebox[121mm][c]{\includegraphics{surgeprob}}
\footnotesize Figure 17.9 Probability (per year) \rule{0mm}{3ex}density distribution of vertical height of storm surges in the Hook of Holland in the Netherlands. The distribution function is Rayleigh, and the probability of large surges is estimated by extrapolating the observed frequency of smaller, more common surges. After Wiegel (1964: 113).
\label{fig:surgeprob}
\vspace{-3ex}
\end{figure}
\section{Theory of Ocean Tides}
\index{tides!theory of|(}
Tides have been so important for commerce and science for so many thousands of years that tides have entered our everyday language: \textit{time and tide wait for no one}, \textit{the ebb and flow of events}, \textit{a high-water mark}, and \textit{turn the tide of battle}.
\begin{enumerate}
\vitem
Tides produce strong currents in many parts of the ocean. Tidal currents\index{tidal!currents}\index{currents!tidal} can have speeds of up to 5 m/s in coastal waters, impeding navigation and mixing coastal waters\index{mixing!tidal}.
\vitem
Tidal currents\index{tidal!currents}\index{currents!tidal} generate internal waves over seamounts, continental slopes, and mid-ocean ridges. The waves dissipate tidal energy. Breaking internal waves and tidal currents are a major force driving oceanic mixing\index{mixing!tidal}.
\vitem
Tidal mixing helps drive the deep circulation, and it influences climate and abrupt climate change.
\vitem
Tidal currents\index{tidal!currents}\index{currents!tidal} can suspend bottom sediments, even in the deep ocean.
\vitem
Earth's crust is elastic. It bends under the influence of the tidal potential. It also bends under the weight of oceanic tides. As a result, the sea floor, and the continents move up and down by about 10 cm in response to the tides. The deformation of the solid earth influence almost all precise geodetic measurements.
\vitem
Oceanic tides lag behind the tide-generating potential. This produces forces that transfer angular momentum between earth and the tide produc\-ing body, especially the moon\index{moon}. As a result of tidal forces, earth's rotation about it's axis slows, increasing the length of day; the rotation of the moon about earth slows, causing the moon to move slowly away from earth; and moon's rotation about it's axis slows, causing the moon to keep the same side facing earth as the moon rotates about earth.
\vitem
Tides influence the orbits of satellites. Accurate knowledge of tides is needed for computing the orbit of altimetric satellites and for correcting altimeter measurements of oceanic topography.
\vitem
Tidal forces on other planets and stars are important for understanding many aspects of solar-system dynamics and even galactic dynamics. For example, the rotation rate of Mercury, Venus, and Io result from tidal forces.
\end{enumerate}
Mariners have known for at least four thousand years that tides are related to the
phase of the moon\index{moon}. The exact relationship, however, is hidden behind many
complicating factors, and some of the greatest scientific minds of the last four
centuries worked to understand, calculate, and predict tides. Galileo, Descartes,
Kepler, Newton, Euler, Bernoulli, Kant, Laplace, Airy, Lord Kelvin, Jeffreys, Munk
and many others contributed. Some of the first computers were developed to
compute and predict tides. Ferrel built a tide-predicting machine in 1880 that
was used by the U.S. Coast Survey to predict nineteen tidal
constituents. In 1901, Harris extended the capacity to 37 constituents.
Despite all this work important questions remained: What is the amplitude
and phase of the tides at any place on the ocean or along the coast? What is the
speed and direction of tidal currents? What is the shape of the tides on the ocean?
Where is tidal energy dissipated? Finding answers to these simple questions is
difficult, and the first, accurate, global maps of deep-sea tides were only
published in 1994 (LeProvost et al. 1994). The problem is hard because the tides
are a self-gravitating, near-resonant, sloshing of water in a rotating, elastic,
ocean basin with ridges, mountains, and submarine basins.
Predicting tides along coasts and at ports is much easier. Data from a tide gauge
plus the theory of tidal forcing gives an accurate description of tides near the
tide gauge.
\paragraph{Tidal Potential}
\index{tidal!potential}Tides are calculated from the hydrodynamic equations for a self-gravitating ocean on a rotating, elastic earth. The driving force is the gradient of the gravity field of the moon\index{moon} and sun. If the earth were an ocean planet with no land, and if we ignore the influence of inertia and currents, the gravity gradient produces a pair of bulges of water on earth, one on the side facing the moon or sun, one on the side away from the moon or sun. A clear derivation of the forces is given by Pugh (1987) and by Dietrich, Kalle, Krauss, and Siedler (1980). Here I follow the discussion in Pugh (1987: \S 3.2).
Note that many oceanographic books state that the tide is produced by two processes: i) the centripetal acceleration at earth's surface as the earth and moon circle around a common center of mass, and ii) the gravitational attraction of mass on earth and the moon. However, the derivation of the tidal potential does not involve centripetal acceleration, and the concept is not used by the astronomical or geodetic communities.
\begin{figure}[h!]
\makebox[121 mm] [c]{\includegraphics{tidesketch}}
\footnotesize
\centering
Figure 17.10 Sketch of \rule{0mm}{4ex}coordinates for determining the tide-generating potential\index{tidal!potential}.
\label{fig:tidesketch}
\vspace{-1ex}
\end{figure}
To calculate the amplitude and phase of the tide on an ocean planet, we begin by calculating the tide-generating potential\index{tidal!potential}. This is much easier than calculating the forces. Ignoring for now earth's rotation, the rotation of moon\index{moon} about earth produces a potential $V_M$ at any point on earth's surface
\begin{equation}
V_{M} = -\frac{\gamma M}{r_{1}}
\end{equation}
where the geometry is sketched in figure 17.10, $\gamma $ is the gravitational constant, and $M$ is moon's mass. From the triangle $OPA$ in the figure,
\begin{equation}
r_{1}^{2} = r^{2} + R^{2} - 2 r R \cos \varphi
\end{equation}
Using this in (17.5) gives
\begin{equation}
V_{M} = -\frac{\gamma M}{R} \left\{ 1 - 2 \left(\frac{r}{R}\right) \cos \varphi +
\left(\frac{r}{R}\right)^{2}\right\}^{-1/2}
\end{equation}
$r/R \approx 1/60$, and (17.7) may be expanded in powers of $r/R$ using
Legendre polynomials (Whittaker and Watson, 1963: \S 15.1):
\begin{equation}
V_M = -\frac{\gamma M}{R} \left\{1+\left(\frac{r}{R}\right) \cos \varphi +
\left(\frac{r}{R}\right)^2 \left(\frac{1}{2}\right) (3\cos ^2 \varphi - 1) + \cdots
\right\}
\end{equation}
The tidal forces are calculated from the spatial gradient of the potential. The first term in (17.8) produces no force. The second term, when differentiated with respect to ($r \cos \varphi $) produces a constant force $\gamma M/R^{2}$ parallel to OA that keeps earth in orbit around the center of mass of the earth-moon system. The third term produces the tides, assuming the higher-order terms can be ignored. The tide-generating potential is therefore:
\begin{equation}
V= -\frac{\gamma M r^{2}}{2 R^{3}} (3 \cos ^2 \varphi - 1)
\end{equation}
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{horiztideforce}}
\footnotesize
\centering
Figure 17.11 The horizontal \rule{0mm}{4ex}component of the tidal force on earth when the\\tide-generating body is above the Equator at $Z$. After Dietrich et al. (1980: 413).
\label{fig:horiztideforce}
\vspace{-3ex}
\end{figure}
The tide-generating force can be decomposed into components perpendicular $P$ and
parallel $H$ to the sea surface. Tides are produced by the horizontal component. ``The
vertical component is balanced by pressure on the sea bed, but the ratio of the horizontal
force per unit mass to vertical gravity has to be balanced by an opposing slope of the sea
surface, as well as by possible changes in current momentum'' (Cartwright,
1999: 39, 45). The horizontal component, shown in figure 17.11, is:
\begin{equation}
H = - \frac{1}{r} \frac{\partial V}{\partial\varphi} = \frac{2 G}{r} \sin
2\varphi
\end{equation}
where
\begin{equation}
G = \frac{3}{4} \gamma M \left( \frac{r^{2}}{R^{3}} \right)
\end{equation}
The tidal potential is symmetric about the earth-moon\index{moon} line, and it produces
symmetric bulges.
If we allow our ocean-covered earth to rotate, an observer in space sees the two
bulges fixed relative to the earth-moon line as earth rotates. To an observer on
earth, the two tidal bulges seems to rotate around earth because moon appears to
move around the sky at nearly one cycle per day. Moon produces high tides every 12
hours and 25.23 minutes on the equator if the moon is above the equator. Notice that
high tides are not exactly twice per day because the moon is also rotating around
earth. Of course, the moon is above the equator only twice per lunar month, and this
complicates our simple picture of the tides on an ideal ocean-covered earth.
Furthermore, moon's distance from earth $R$ varies because moon's orbit is
elliptical and because the elliptical orbit is not fixed.
Clearly, the calculation of tides is getting more complicated than we might have
thought. Before continuing on, we note that the solar tidal forces are derived in a
similar way. The relative importance of the sun and moon are nearly the same.
Although the sun is much more massive than moon\index{moon}, it is much further away.
\begin{align}
G_{sun} = G_{S} &= \frac{3}{4} \gamma S \left( \frac{r^{2}}{R_{sun}^{3}} \right) \\
G_{moon} = G_{M} &= \frac{3}{4} \gamma M \left( \frac{r^{2}}{R_{moon}^{3}} \right) \\
\frac{G_{S}}{G_{M}} &= 0.46051
\end{align}
where $R_{sun}$ is the distance to the sun, $S$ is the mass of the sun, $R_{moon}$
is the distance to the moon, and $M$ is the mass of the moon.
\paragraph{Coordinates of Sun and Moon}
\index{coordinate systems!for sun and moon}
\index{sun!coordinates}\index{moon!coordinates}Before we can proceed further we need to know
the position of moon and sun relative to earth. An accurate description of the positions in
three dimensions is very difficult, and it involves learning arcane terms and concepts from
celestial mechanics. Here, I paraphrase a simplified description from Pugh (1987). See also
figure 4.1.
A natural reference system for an observer on earth is the equatorial system described at the
start of Chapter 3. In this system, \textit{declinations}\index{declinations|textbf} $\delta$
of a celestial body are measured north and south of a plane which cuts the earth's equator.
\begin{quotation} \small
Angular distances around the plane are measured relative to
a point on this celestial equator which is fixed with respect to the stars. The
point chosen for this system is the \textit{vernal equinox}, also called the `First
Point of Aries'\dots The angle measured eastward, between Aries and the equatorial
intersection of the meridian through a celestial object is called the \textit{right
ascension} of the object. The declination and the right ascension together define
the position of the object on a celestial background\dots
[Another natural reference] system uses the plane of the earth's revolution around
the sun as a reference. The celestial extension of this plane, which is traced by
the sun's annual apparent movement, is called the \textit{ecliptic}. Conveniently,
the point on this plane which is chosen for a zero reference is also the vernal
equinox, at which the sun crosses the equatorial plane from south to north around 21
March each year. Celestial objects are located by their ecliptic latitude and
ecliptic longitude. The angle between the two planes, of 23.45\degrees, is called
the obliquity of the ecliptic\dots ---Pugh (1987: 72).
\end{quotation}
\paragraph{Tidal Frequencies}
\index{tidal!frequencies|(}
Now, let's allow earth to spin about its polar axis. The
changing potential at a fixed geographic coordinate on earth is:
\begin{equation}
\cos \varphi = \sin \varphi _p \sin \delta + \cos \varphi _p \cos \delta \cos (\tau
_{1} - 180^{\circ})
\end{equation}
where $\varphi _p$ is latitude at which the tidal potential is calculated, $\delta $
is declination of moon\index{moon} or sun north of the equator, and $\tau _{1}$ is the hour angle of moon
or sun. The \textit{hour angle}
\index{tides!hour angle|textbf}
\index{tidal!frequencies!hour angle|textbf}
is the longitude where the imaginary plane containing the sun or moon and
earth's rotation axis crosses the Equator.
The period of the solar hour angle is a solar day of 24 hr 0 m. The period of the
lunar hour angle is a lunar day of 24 hr 50.47 m.
Earth's axis of rotation is inclined 23.45\degrees\ with respect to the plane of earth's orbit
about the sun. This defines the ecliptic, and the sun's declination varies between $\delta =
\pm 23.45$\degrees\ with a period of one solar year. The orientation of earth's rotation axis
precesses with respect to the stars with a period of 26\medspace 000 years. The rotation of
the ecliptic plane causes $\delta$ and the vernal equinox to change slowly, and the movement
called the \textit{precession of the equinoxes}.
\index{tides!and the equinox}
\index{equinox}
\index{earth!equinox}
\index{equinox!precession of|textbf}
Earth's orbit about the sun is elliptical, with the sun in one focus. That point in the orbit
where the distance between the sun and earth is a minimum is called \textit{perigee}.
\index{sun!perigee of|textbf} \index{tides!and perigee} \index{earth!perigee of|textbf} \index{perigee|textbf} The orientation of the ellipse in
the ecliptic plane changes slowly with time, causing perigee to rotate with a period of
20\medspace 942 years. Therefore $R_{sun}$ varies with this period.
Moon's orbit is also elliptical, but a description of moon's orbit is much
more complicated than a description of earth's orbit. Here are the basics. The
moon's orbit lies in a plane inclined at a mean angle of 5.15\degrees\ relative to
the plane of the ecliptic. And lunar declination varies between $\delta = 23.45 \pm
5.15$\degrees\ with a period of one tropical month of 27.32 solar days. The
actual inclination of moon's orbit varies between 4.97\degrees, and
5.32\degrees.
The shape of moon's orbit also varies. First, perigee rotates with a period of
8.85 years. The eccentricity of the orbit has a mean value of 0.0549, and it varies
between 0.044 and 0.067. Second, the plane of moon's orbit rotates around earth's
axis of rotation with a period of 18.613 years. Both processes cause variations in
$R_{moon}$.
Note that I am a little imprecise in defining the position of the sun and moon.
Lang (1980: \S \ 5.1.2) gives much more precise definitions.
Substituting (17.15) into (17.9) gives:
\begin{multline}
V = \frac{\gamma M r^{2}}{R^{3}} \frac{1}{4} \left[ \left( 3 \sin ^{2} \varphi_p - 1
\right) \left( 3 \sin ^{2} \delta -1 \right) \right. \\
+ 3 \sin 2 \varphi_p \,\, \sin 2 \delta \,\, \cos \tau _{1} \\
+ \left. 3 \cos^2 \varphi_p \,\, \cos^2 \delta \,\, \cos 2 \tau _1 \right]
\end{multline}
Equation (17.16) separates the period of the lunar tidal potential into three
terms with periods near 14 days, 24 hours, and 12 hours. Similarly the solar
potential has periods near 180 days, 24 hours, and 12 hours. Thus there are three
distinct groups of tidal frequencies: twice-daily, daily, and long period, having
different latitudinal factors $\sin^2 \theta $, $\sin 2 \theta$, and $(1 - 3 \cos^2 \theta
)/2$, where $\theta $ is the co-latitude $\left( 90^{\circ} - \varphi \right) $.
\begin{table} [t!] \small \centering
%\vspace{-2ex}
\begin{tabular*}{121mm}{@{}crrll@{}}
\multicolumn{5}{@{}l@{}}{\bfseries Table 17.1 Fundamental Tidal Frequencies \rule[-1ex]{0mm}{1ex}}\\
\hline
\rule{0ex}{2.5ex} & Frequency & & Period & Source \\
& \degrees/hour & & & \\[0.5ex]
\hline
$f_1$ & 14.49205211 & 1 & lunar day & Local mean lunar time \rule{0ex}{2.5ex} \\
$f_2$ & 0.54901653 & 1 & month & Moon's mean longitude \\
$f_3$ & 0.04106864 & 1 & year & Sun's mean longitude \\
$f_4$ & 0.00464184 & 8.847 & years & Longitude of moon's
perigee \\
$f_5$ & -0.00220641 & 18.613 & years & Longitude of moon's ascending
node \\
$f_6$ & 0.00000196 & 20,940 & years & Longitude of sun's perigee \\[0.5ex]
\hline
\end{tabular*} \\[0.5ex]
\vspace{-3ex}
\end{table}
Doodson (1922) expanded (17.16) in a Fourier series using the cleverly
chosen frequencies in table 17.1. Other choices of fundamental
frequencies are possible, for example the local, mean, solar time can be used
instead of the local, mean, lunar time. Doodson's expansion, however, leads to an
elegant decomposition of tidal constituents into groups with similar frequencies and
spatial variability.
Using Doodson's expansion, each constituent of the tide has a frequency
\begin{equation}
f = n_1 f_1 + n_2 f_2 + n_3 f_3 + n_4 f_4 + n_5 f_5 + n_6 f_6
\end{equation}
where the integers $n_i$ are the \textit{Doodson numbers}\index{tidal!Doodson
numbers|textbf}\index{Doodson numbers|textbf}. $n_1 = 1, 2, 3 $ and $n_2$--$n_6$ are between
$-5$ and $+5$. To avoid negative numbers, Doodson added five to $n_{2 \cdots 6}$. Each
tidal wave having a particular frequency given by its Doodson number is called a \textit{tidal
constituent}\index{tidal!constituents|textbf}, sometimes called a \textit{partial
tides}\index{tides!partial|textbf}. For example, the principal,
twice-per-day, lunar tide has the number 255.555. Because the very long-term modulation of the
tides by the change in sun's perigee is so small, the last Doodson number
$n_6$ is usually ignored.
\begin{table} [b!] \vspace{-1.5ex} \small{{\textbf{Table 17.2 Principal Tidal
Constituents}}\index{tidal!constituents!principal}
\\[1ex]
\begin{tabular*}{121mm}{@{\extracolsep{\fill}}lcccccccc@{}}
\hline
\rule{0ex}{2.5ex} & & & & & & & Equilibrium & \\
Tidal & & & & & & & Amplitude\dag & Period \\
Species & Name &$n_1$ & $n_2$ & $n_3$ & $n_4$ & $n_5$ & $(m)$ & (hr) \\[0.5ex]
\hline
Semidiurnal &$ n_1 =2$ & & & & & &\rule{0ex}{3ex}& \\[1ex]
Principal lunar & $M_2$ & 2 & 0 & 0 & 0 & 0 & 0.242334 & 12.4206 \\
Principal solar & $S_2$ & 2 & 2 & -2 & 0 & 0 & 0.112841 & 12.0000 \\
Lunar elliptic & $N_2$ & 2 & -1 & 0 & 1 & 0 & 0.046398 & 12.6584 \\
Lunisolar & $K_2$ & 2 & 2 & 0 & 0 & 0 & 0.030704 & 11.9673 \\[0.5ex]
\hline
Diurnal & $n_1 =1$ & & & & & &\rule{0ex}{3ex}& \\[1ex]
Lunisolar & $K_1$ & 1 & 1 & 0 & 0 & 0 & 0.141565 & 23.9344 \\
Principal lunar & $O_1$ & 1 & -1 & 0 & 0 & 0 & 0.100514 & 25.8194 \\
Principal solar & $P_1$ & 1 & 1 & -2 & 0 & 0 & 0.046843 & 24.0659 \\
Elliptic lunar & $Q_1$ & 1 & -2 & 0 & 1 & 0 & 0.019256 & 26.8684 \\[0.5ex]
\hline
Long Period &$ n_1 =0$ & & & & & &\rule{0ex}{3ex}& \\[1ex]
Fortnightly & $Mf$ & 0 & 2 & 0 & 0 & 0 & 0.041742 & 327.85 \\
Monthly & $Mm$ & 0 & 1 & 0 & -1 & 0 & 0.022026 & 661.31 \\
Semiannual & $Ssa$ & 0 & 0 & 2 & 0 & 0 & 0.019446 &4383.05 \\[0.5ex]
\hline
\end{tabular*}
{\footnotesize \raisebox{-3ex}{\dag Amplitudes from Apel (1987)}}}
%\vspace{-3mm}
\end{table}
If the ocean surface is in equilibrium\index{tides!equilibrium|textbf} with the tidal
potential, which means we ignore inertia and currents and assume no land (Cartwright 1999: 274), the
largest tidal constituents would have amplitudes given in table 17.2. Notice that tides with
frequencies near one or two cycles per day are split into closely spaced lines with spacing separated
by a cycle per month. Each of these lines is further split into lines separated by a cycle per year
(figure 17.12). Furthermore, each of these lines is split into lines separated by a cycle per
8.8 yr, and so on. Clearly, there are very many possible tidal constituents.
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{combinedtides}}
\footnotesize
Figure 17.12 \textbf{Upper:} Spectrum \rule{0mm}{3ex}of equilibrium
tides\index{tides!equilibrium} with frequencies near twice per day. The spectrum is split into
groups separated by a cycle per month (0.55 deg/hr). \textbf{Lower:} Expanded spectrum of the
$S_2$ group, showing splitting at a cycle per year (0.04 deg/hr). The finest splitting
in this figure is at a cycle per 8.847 years (0.0046 deg/hr). From Richard Eanes, Center for
Space Research, University of Texas.
\label{combinedtides}
\vspace{-3ex}
\end{figure}
Why is the tide split into the many constituents shown in figure 17.12? To answer the question,
suppose moon's elliptical orbit was in the equatorial plane of earth. Then $\delta = 0$. From
(17.16), the tidal potential on the equator, where
$\varphi_p =0$, is:
\begin{equation}
V = \frac{\gamma M r^{2}}{R^{3}} \frac{1}{4} \cos \left(4 \pi f_1 \right)
\end{equation}
If the ellipticity of the orbit is small, $R= R_0 (1+\epsilon )$, and (17.18) is
approximately
\begin{equation}
V = a (1-3\epsilon) \cos \left(4 \pi f_1 \right)
\end{equation}
where $a=\left( \gamma M r^2 \right)/\left( 4 R^3 \right)$ is a constant.
$\epsilon$ varies with a period of 27.32 days, and we can write $\epsilon = b \cos
(2 \pi f_2)$ where $b$ is a small constant. With these simplifications, (17.19)
can be written:
\begin{subequations}
\begin{align}
V &= a \cos \left(4 \pi f_1 \right) -3 a b \cos \left( 2 \pi f_2 \right) \cos \left(4 \pi f_1 \right) \\
V &= a \cos \left(4\pi f_1 \right) - 3 a b \left[ \cos 2\pi \left( 2f_1 - f_2 \right) + \cos 2\pi \left( 2f_1 + f_2 \right) \right]
\end{align}
\end{subequations}
which has a spectrum with three lines at $2 f_1$ and $ 2 f_1 \pm f_2$. Therefore,
the slow modulation of the amplitude of the tidal potential at two cycles per
lunar day causes the potential to be split into three frequencies. This is the
way amplitude modulated AM radio works. If we add in the slow changes in
the shape of the orbit, we get still more terms even in this very idealized case
of a moon\index{moon} in an equatorial orbit.
If you are very observant, you will have noticed that the tidal spectrum in figure 17.12 does
not look like the ocean-wave spectrum of ocean waves in figure 16.6. Ocean waves have all
possible frequencies, and their spectrum is continuous. Tides have precise frequencies
determined by the orbit of earth and moon\index{moon}, and their spectrum is not continuous. It consists
of discrete lines.
Doodson's expansion included 399 constituents, of which 100 are long period, 160
are daily, 115 are twice per day, and 14 are thrice per day. Most have very small
amplitudes, and only the largest are included in table 17.2. The largest tides
were named by Sir George Darwin (1911) and the names are included in the table.
Thus, for example, the principal, twice-per-day, lunar tide, which has Doodson
number 255.555, is the $M_2$ tide, called the \textit{M-two} tide.
\index{tidal!frequencies|)}
\index{tides!theory of|)}
\section{Tidal Prediction}
\index{tidal!prediction}If tides in the ocean were in equilibrium with the tidal potential,
tidal prediction would be much easier. Unfortunately, tides are far from equilibrium. The
shallow-water wave which is the tide cannot move fast enough to keep up with sun and moon. On
the equator, the tide would need to propagate around the world in one day. This requires a
wave speed of around 460 m/s, which is only possible in an ocean 22 km deep. In addition, the
continents interrupt the propagation of the wave. How to proceed?
We can separate the problem of tidal prediction into two parts. The first deals with
the prediction of tides in ports and shallow water where tides can be measured by
tide gauges. The second deals with the prediction of tides in the deep ocean where
tides are measured by satellite altimeters.
\paragraph{Tidal Prediction for Ports and Shallow Water}
\index{tidal!prediction!shallow water}Two methods are used to predict future tides at a
tide-gauge station using past observations of sea level measured at the gauge.
\textit{The Harmonic Method} This is the traditional method, and it is still
\index{tidal!prediction!harmonic method|textbf}widely used. The method typically uses 19 years
of data from a coastal tide gauge from which the amplitude and phase of each
tidal constituent (the tidal harmonics) in the tide-gage record are calculated. The
frequencies used in the analysis are specified in advance from the basic frequencies given in
table 17.1.
Despite its simplicity, the technique had disadvantages compared with the
response method described below.
\begin{enumerate}
\vitem
More than 18.6 years of data are needed to resolve the modulation of the lunar
tides.
\vitem
Amplitude accuracy\index{accuracy!tides} of $10^{-3}$ of the largest term require that at least
39 frequencies be determined. Doodson found 400 frequencies were needed for an
amplitude accuracy of $10^{-4}$ of the largest term.
\vitem
Non-tidal variability introduces large errors into the calculated amplitudes and
phases of the weaker tidal constituents. The weaker tides have amplitudes smaller
than variability at the same frequency due to other processes such as wind set up
and currents near the tide gauge.
\vitem
At many ports, the tide is non-linear, and many more tidal constituents are
important. For some ports, the number of frequencies is unmanageable. When tides
propagate into very shallow water, especially river estuaries, they steepen and
become non-linear. This generates harmonics of the original frequencies. In extreme
cases, the incoming waves steepens so much the leading edge is nearly vertical, and
the wave propagates as solitary wave\index{waves!solitary}. This is a \textit{tidal
bore}\index{tidal!bore}.
\end{enumerate}
\textit{The Response Method} \index{tidal!prediction!response method|textbf}This
method, developed by Munk and Cartwright (1966), calculates the relationship between the
observed tide at some point and the tidal potential. The relationship is the spectral
admittance between the major tidal constituents and the tidal potential at each station. The
admittance is assumed to be a slowly varying function of frequency so that the admittance of
the major constituents can be used for determining the response at nearby frequencies. Future
tides are calculated by multiplying the tidal potential by the admittance function.
\begin{enumerate}
\vitem
The technique requires only a few months of data.
\vitem
The tidal potential is easily calculated, and a knowledge of the tidal frequencies
is not needed.
\vitem
The admittance is $Z(f) = G(f)/H(f)$. $G(f)$ and $H(f)$ are the Fourier transforms
of the potential and the tide gage data, and $f$ is frequency.
\vitem
The admittance is inverse transformed to obtain the admittance as a function
of time.
\vitem
The technique works only if the waves propagate as linear waves.
\end{enumerate}
\index{tidal!prediction!shallow water}
\paragraph{Tidal Prediction for Deep-Water}
\index{tidal!prediction!deep water}Prediction of deep-ocean tides has been much more difficult than prediction of shallow-water tides because tide gauges were seldom deployed in deep water. All this changed with the launch of Topex/ Poseidon\index{Topex/Poseidon!observations of tides}. The satellite was placed into an
orbit especially designed for observing ocean tides (Parke et al. 1987), and the altimetric
system was sufficiently accurate to measure many tidal constituents\index{tidal!constituents}.
Data from the satellite have now been used to determine deep-ocean tides with an
accuracy\index{accuracy!tides} of $\pm \, 2$ cm. For most practical purposes, the tides are
now known accurately for most of the ocean.
Two avenues led to the new knowledge of deep-water tides using altimetry.
\textit{Prediction Using Hydrodynamic Theory} Purely theoretical calculations of
\index{tidal!prediction!from hydrodynamic theory} tides are not very accurate, especially
because the dissipation of tidal energy is not well known. Nevertheless, theoretical
calculations provided insight into processes influencing ocean tides. Several processes must be
considered:
\begin{enumerate}
\vitem
The tides in one ocean basin perturb earth's gravitational field, and the mass in
the tidal bulge attracts water in other ocean basins. The self-gravitational
attraction of the tides must be included.
\vitem
The weight of the water in the tidal bulge is sufficiently great that it deforms
the sea floor. The earth deforms as an elastic solid, and the deformation extends
thousands of kilometers.
\vitem
The ocean basins have a natural resonance close to the tidal frequencies. The tidal
bulge is a shallow-water wave on a rotating ocean, and it propagates as a high tide
rotating around the edge of the basin. Thus the tides are a nearly resonant
sloshing of water in the ocean basin. The actual tide heights in deep water can be
higher than the equilibrium values noted in table 17.2.
\vitem
Tides are dissipated by bottom friction especially in shallow seas, by the flow over
seamounts and mid-ocean ridges, and by the generation of internal waves over
seamounts and at the edges of continental shelves. If the tidal forcing stopped, the
tides would continue sloshing in the ocean basins for several days.
\vitem
Because the tide is a shallow-water wave everywhere, its velocity depends on depth.
Tides propagate more slowly over mid-ocean ridges and shallow seas. Hence, the
distance between grid points in numerical models must be proportional to depth with
very close spacing on continental shelves (LeProvost et al. 1994).
\vitem
Internal waves generated by the tides produce a small signal at the sea surface
near the tidal frequencies, but not phase-locked to the potential. The noise near
the frequency of the tides causes the spectral cusps in the spectrum of
sea-surface elevation first seen by Munk and Cartwright (1966). The noise is due to
deep-water, tidally generated, internal waves.
\end{enumerate}
\textit{Altimetry Plus Response Method} Several years of altimeter data from
\index{tidal!prediction!altimetry plus response method}
Topex/ Poseidon\index{Topex/Poseidon} have been used with the response method to calculate
deep-sea tides almost everywhere equatorward of 66\degrees\ (Ma et al.\ 1994). The altimeter
measured sea-surface heights in geocentric coordinates at each point along the sub-satellite
track every 9.97 days. The temporal sampling aliased the tides into long frequencies, but the
aliased periods are precisely known and the tides can be recovered (Parke et al.\ 1987).
Because the tidal record is shorter than 8 years, the altimeter data are used with
the response method to obtain predictions for a much longer time.
Recent solutions by ten different groups, have accuracy\index{accuracy!tides} of $\pm \,
$2.8 cm in deep water (Andersen, Woodworth, and Flather, 1995). Work has begun to
improve knowledge of tides in shallow water.
\begin{figure}[t!]
\makebox[121 mm] [c]{\includegraphics{m2_tide}}
\footnotesize
Figure 17.13 Global map of $M_2$ tide \rule{0mm}{4ex}calculated from
Topex/Poseidon\index{Topex/Poseidon!tide map} observations of the height of the sea surface
combined with the response method for extracting tidal information. Full lines are contours of
constant tidal phase, contour interval is 30\degrees. Dashed lines are lines of constant
amplitude, contour interval is 10 cm. From Richard Ray, \textsc{nasa} Goddard Space Flight
Center.
\label{fig:m2_tide}
\vspace{-3ex}
\end{figure}
\textit{Altimetry Plus Numerical Models} Altimeter data can be used directly with
\index{numerical models!tidal prediction}
\index{tidal!prediction!altimetry plus numerical models}
\index{satellite altimetry}
numerical models of the tides to calculate tides in all areas of the ocean from deep water all
the way to the coast. Thus the technique is especially useful for determining tides near
coasts and over sea-floor features where the altimeter ground track is too widely spaced to
sample the tides well in space. Tide models use finite-element grids similar to the one shown
in figure 15.3. Recent numerical calculations by (LeProvost et al. 1994; LeProvost, Bennett,
and Cartwright, 1995) give global tides with
$\pm$ 2--3 cm accuracy\index{accuracy!tides} and full spatial resolution.
Maps produced by this method show the essential features of the deep-ocean tides (figure 17.13). The tide consists of a crest that rotates counterclockwise around the ocean basins in the northern hemisphere, and in the opposite direction in the southern
hemisphere. Points of minimum amplitude are called \textit{amphidromes}\index{tidal!amphidromes|textbf}\index{amphidromes|textbf}. Highest tides tend to be along the coast.
The maps also show the importance of the size of the ocean basins. The semi-diurnal\index{tides!semi-diurnal} (12 hr period) tides are relatively large in all ocean basins. But the diurnal\index{tides!diurnal} (24 hr period) tides are small in the Atlantic and relatively
large in the Pacific and Indian ocean. The Atlantic is too small to have a resonant sloshing with a period near 24 hr.
\paragraph{Tidal Dissipation}
\index{tidal!dissipation}Tides dissipate $3.75\pm0.08$ TW of power (Kantha, 1998), of which 3.5 TW are dissipated in the ocean, and much smaller amounts in the atmosphere and solid earth. The dissipation increases the length of day by about 2.07 milliseconds per century, it causes the semimajor axis of moon's orbit to increase by 3.86 cm/yr, and it mixes water masses in the ocean.
The calculations of dissipation from Topex/Poseidon\index{Topex/Poseidon!observations of dissipation} observations of tides are remarkably close to estimates from lunar-laser ranging, astronomical observations, and ancient eclipse records. The calculations show that roughly two thirds of the M2 tidal energy is dissipated on shelves and in shallow seas, and one third is transferred to internal waves and dissipated in the deep ocean (Egbert and ray, 2000). 85 to 90\% of the energy of the K1 tide is dissipated in shallow water, and only about 10--15\% is transferred to internal waves in the deep ocean (LeProvost 2003, personal communication).
Overall, our knowledge of the tides is now sufficiently good that we can begin to use the information to study mixing\index{mixing!tidal} in the ocean. Recent results show that ``tides are perhaps responsible for a large portion of the vertical mixing in the ocean'' (Jayne et al. 2004). Remember, mixing helps drive the abyssal circulation\index{abyssal circulation}\index{circulation!abyssal}\index{oceanic circulation!abyssal} in the ocean as discussed in \S 13.2 (Munk and Wunsch, 1998). Who would have thought that an understanding of the influence of the ocean on climate would require accurate knowledge of tides\index{tides}?
\section{Important Concepts}
\begin{enumerate}
\item
Waves propagating into shallow water are refracted by features of the seafloor, and they eventually break on the beach. Breaking waves drive near-shore currents including long-shore currents, rip currents\index{rip currents}, and edge waves.
\vitem
Storm surges are driven by strong winds in storms close to shore. The amplitude of the surge is a function of wind speed, the slope of the seafloor, and the propagation of the storm.
\vitem
Tides are important for navigation; they influence accurate geodetic measurements; and they change the orbits and rotation of planets, moons, and stars in galaxies.
\vitem
Tides are produced by a combination of time-varying gravitational potential of the moon and sun and the centrifugal forces generated as earth rotates about the common center of mass of the earth-moon-sun system.
\vitem
Tides have six fundamental frequencies. The tide is the superposition of hundreds of tidal constituents, each having a frequency that is the sum and difference of five fundamental frequencies.
\vitem
Shallow water tides are predicted using tide measurements made in ports and other locations along the coast. Tidal records of just a few months duration can be used to predict tides many years into the future.
\vitem
Tides in deep water are calculated from altimetric measurements, especially Topex/Poseidon\index{Topex/Poseidon} measurements. As a result, deep water tides are known almost everywhere with an accuracy\index{accuracy!tides} approaching $\pm 2$ cm.
\vitem
The dissipation of tidal energy in the ocean transfers angular momentum from moon\index{moon} to earth, causing the day to become longer.
\vitem
Tidal dissipation mixes water masses, and it is a major driver of the deep, meridional-overturning circulation\index{circulation!meridional overturning}. Tides, abyssal circulation, and climate are closely linked.
\end{enumerate}
\index{Rossby wave|see{waves, Rossby}}
\index{Kelvin wave|see{waves, Kelvin}}
\index{Brunt-Vaisala frequency|see{stability, frequency}}
\index{heat flux!latent|see{latent heat flux}}
\index{heat flux!sensible|see{sensible heat flux}}
\index{heat flux!infrared|see{infrared flux}}
\index{heat flux!solar|see{insolation}}
\index{El Ni\~{n}o--Southern Oscillation (ENSO)|see{Southern Oscillation}}
\index{altimeters|see{satellite altimetry}}__