Math Awareness Month - April 2001
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MATHEMATICS AND THE OCEAN
by Barry A. Cipra and Katherine Socha
- Planet Ocean
The single most striking fact about the Earth is that it's awash with water.
Dominating our planet's surface and affecting the lives of everyone, even
those who live far inland, the Earth's ocean -- the vast expanse of water
circling the globe and comprising the Atlantic and Pacific Oceans and
numerous smaller seas -- has long been a source of wonder and awe. From the
earliest recorded times, men and women have sought to understand the
behavior of the ocean and of the life within it. Our knowledge of the
ocean is far from complete, but is steadily advancing -- thanks in great part
to new developments in mathematics.
Millennia of trial-and-error experience led to practical and sometimes
elegant solutions to problems in ship-building, navigation, fishing
strategy, and the anticipation of oceanic activity ranging from rough seas
to the rhythm of tides. During the last few centuries, our understanding of
the ocean has become increasingly scientific. The observations and
accumulated wisdom of mariners throughout the ages have been augmented by
detailed measurements of water temperature and salinity and by greater
physical understanding of the watery forces that cause waves and currents.
The scientific approach brought with it the need for mathematical analysis.
Oceanography today uses mathematical equations to describe fundamental ocean
processes and requires mathematical theories to understand their
implications. Researchers use statistics and signal processing to weave
together the many separate strands of data from sonar buoys, shipboard
instruments, and satellites. Partial differential equations describe the
"mechanics" of fluid motion, from the surface waves that rock sea-going
ships to the deep currents that sweep around the globe. Numerical analysis
has made it possible to obtain increasingly accurate solutions to these
equations; dynamical systems theory and statistics have provided additional
insights. Today's oceanographers are really mathematicians, in the best
tradition of Galileo and Newton. Mathematics, you might say, is the
"salty language" of modern oceanography.
- What's math got to do with it?
At its most elemental, any ocean process is all about change.
Measurable quantities may change as time passes (for example, tidelines on a
beach move from low to high twice a day) or may change from location to
location (for example, pressure on a submarine increases as it dives deeper
into the sea), but most quantities such as temperature and salinity change
based on both position and time. The areas of mathematics which are
critical to the description of changing processes are calculus and
differential equations. In particular, partial differential
equations (PDEs, for short) are used to describe quantities that change
continuously in time and space. All areas of oceanography rely heavily on
Marine geology and marine geophysics, for example, study the
structure of the Earth as a whole and the changes it has undergone through
time. Seismic studies for oil exploration, predictions of tsunamis
(devastating waves created by deep sea earthquakes), and investigations into
the formation of the largest mountain ranges on our planet (the oceanic
ridges) are among the interests of marine geologists and geophysicists.
Chemical oceanography studies the chemistry of aquatic environments, with
special attention to interactions between the Earth's crust, the so-called
biota (micro-organisms, plants, and animals), and the atmosphere. Marine
chemists are particularly interested in understanding both the natural
phenomena and the human-generated changes affecting the chemistry of the
world's oceans, rivers, and lakes.
Similarly, biological oceanography studies how marine lifeforms
interact with each other and with their ocean environment. Marine
biologists chart the populations of biota in estuaries, in coastal zones,
and in the open sea, with the ultimate goal of mathematically modeling and
predicting their growth and migration patterns. Both biological and
chemical oceanographers are concerned with ecosystem modeling:
mathematical representations of interactions between the ocean's biological
and chemical constituents, such as plants and animals and the nutrients they
feed on. Can you imagine learning about whales, for example, without asking
about what they eat? Ecosystem modeling, especially in coastal contexts, is
concerned with immediate practical issues such as how to predict the amount
of biological productivity, the fate of pollutants, and the appearance of
harmful algal blooms.
Marine geology, chemistry, and biology occur within the context of the
dynamical behavior of the ocean, which is the province of physical
oceanography. Physical oceanographers study the full spectrum of
circulation patterns of the ocean, from breaking waves on stormy beaches to
the great currents and eddies that transport mass and energy (mostly in the
form of heat) around the globe and that interact with atmospheric dynamics
to drive the weekly weather and the Earth's long-term climate. Physical
oceanographers rely on geophysical fluid dynamics to characterize the
behavior of fluids (such as ocean waters) on a rotating globe (the Earth).
The Earth's rotation "pushes" on large-scale fluid flows in much the same
manner as the rotation of a merry-go-round "pushes" on a person walking a
radial line from its center to its rim. This phenomenon, called the
Coriolis effect, must be included in any description of large-scale ocean
phenomena. (The Coriolis 'force' is not strong enough to affect
small-scale fluid behavior such as water draining from an ordinary household
The notion that partial differential equations may be used to describe the
motion of physical fluids goes back at least to the Swiss mathematician
Leonhard Euler. In 1755, he gave the first physically and mathematically
successful description of the behavior of an idealized fluid. The Euler
equations, as they're called today, are a set of nonlinear PDEs which
express Newton's law of "force equals mass times acceleration" for a
non-viscous fluid -- the watery equivalent of a frictionless mechanical
In 1821, Claude Navier improved on Euler's equations by including the
effects of viscosity. Oddly enough, the equations he obtained are correct,
even though the physical assumptions on which he based his derivation were
wrong! In 1845, George Gabriel Stokes rederived the same set of equations,
but on a more sound theoretical basis. The result, known as the
Navier-Stokes equations, forms the starting point for all modern fluid
dynamics studies. Together with the laws of thermodynamics, which were
developed in the latter half of the nineteenth century, they are the basis
for modern physical oceanography.
The study of nonlinear PDEs is a huge field that underlies much of applied
mathematics. With certain notable exceptions, the presence of nonlinearity
makes it virtually impossible to obtain exact solutions to these equations.
This is certainly true of the Navier-Stokes equations. Consequently, much
work is being carried out in computational fluid dynamics, with the goal of
using computers to approximate numerically the solution of the
Navier-Stokes (and Euler) equations. Researchers also attempt to simplify
the equations in order to emphasize key physical features and to reduce the
computational problem to a manageable size. An ongoing challenge for
oceanographers and mathematicians is to understand enough about the physical
meaning of the Navier-Stokes equations to make sensible simplifications.
The goal is to work with simplified versions that still provide useful
approximate descriptions and predictions.
What could be so difficult about simplifying the equations of fluid
dynamics? There are two major obstacles which any study of ocean behavior
must overcome: the vast range of temporal and spatial scales present in the
ocean and the tendency of fluid flows to be unstable. Physical oceanography
must contend with turbulent eddies that span mere centimeters and last mere
seconds; traveling surface gravity waves with wavelengths of kilometers and
periods of minutes to hours; ocean tides with wavelengths of thousands of
kilometers and periods of half a day; and ocean currents with spatial
extents of thousands of kilometers and lifetimes measured in centuries. The
computation of ocean circulation on these scales, from a millimeter up to
the size of the Earth, is an enormous problem. Current theory and
technology cannot approximate behavior over such a wide scope.
Similarly, the tendency toward instability complicates the prediction of
fluid behavior. Even in a stable flow, the trajectory of an idealized fluid
particle can be unpredictable. The eventual path of a fluid particle, or
some object carried by the flow, can be highly sensitive to its initial
position. Put two floating objects -- say Tom Hanks and a volleyball -- side
by side in the ocean, wait a few days, and the chance of finding them still
together is a Hollywood coincidence. Instability makes matters that much
The basic problem is that small disturbances to a flow may, if they have the
right structure, draw energy from the flow and grow rapidly until they are
so large as to alter the flow in fundamental ways. This kind of
instability can lead to turbulence; one atmospheric example is gusts of wind
on a breezy day. The mathematical and physical elements of oceanic
instabilities are similar to those that operate in the atmosphere and make
the prediction of storms so very difficult for meteorologists. In some ways
the surprising fact is that large-scale patterns, such as the Gulf Stream,
are so long-lived despite the ocean's tendency toward instability.
- Aspects of physical oceanography
Physical oceanography has many subdisciplines, including planetary-scale
circulation and climate, coastal oceanography, equatorial oceanography,
internal waves and turbulence, and surface waves and air-sea interaction.
While the phenomena studied by these subdisciplines certainly interact in
complicated ways, most oceanographers specialize in one. A comprehensive
account of all these areas would fill many, many volumes of an oceanic
encyclopedia, but here are a few examples to suggest the tang of modern
Planetary-scale circulation and climate
During 1982-83, an environmental condition called El Niño was
blamed for a multitude of natural disasters: severe damage to the Pacific
Ocean's coral populations; droughts in Indonesia and the Amazon rain forests
that led to destructive wildfires; and the loss of over 2000 lives in the
United States due to great storms that caused floods in the Gulf states and
torrential rains and high tides in California. In 1998, the return of
El Niño led to the death by starvation of thousands of seals and sea
lions in the California channel islands because the fish on which they
normally feed were driven away by atmospheric and oceanic conditions.
What is El Niño? Basically, it's a warming of the upper layers of
the tropical Pacific Ocean, caused by interaction with the atmosphere.
Normally the winds over the Pacific form a circular pattern above the
equator: near the sea surface, the trade winds blow west across the Pacific,
from South America to Indonesia, where they cycle up through the atmosphere
to form the Upper Westerlies, blowing east back across the Pacific. These
strong winds drive the ocean to create an upwelling of cooler, nutrient-rich
waters along the tropical coast of South America and along the equator.
During El Niño years, the trade winds weaken and upwelling is
reduced. This causes surface temperature to rise over a vast area of the
ocean, and these temperature changes greatly affect the local climate.
Normally, high rainfall occurs north of the equator and in the tropical
southwest Pacific area. In El Niño years, the areas of high
rainfall are over the ocean, rather than over Indonesia and Australia. The
weakened trade winds and reduced upwelling reduce the nutrients available to
the phyto- and zoo-plankton that form the foundation of the marine food
chain. This has proved disastrous for Peruvian fisheries, and has
necessitated a ban on fishing off the coast of Peru during these years. In
normal years, about 20 percent (by weight) of the entire world's fish
harvest has been caught there! El Niño effects can lead to more
hurricanes in the Pacific, fewer hurricanes in the Gulf of Mexico, and
droughts and floods throughout the world.
A related effect of El Niño is a dramatic increase of surface
atmospheric pressure over Indonesia and Australia. This atmospheric portion
of the El Niño effects is called the Southern Oscillation. The
El Niño Southern Oscillation (ENSO) pattern can occur two or three
times a decade.
Modelling the ENSO phenomenon has been a great challenge for oceanographers,
requiring the use of sophisticated mathematical techniques. While
researchers have a pretty good understanding of the physical dynamics that
cause El Niño Southern Oscillation, accurate predictions are very
hard to make. Oceanographers and meteorologists find it difficult
predicting even when an El Niño year will occur -- let alone
predicting the number and intensity of hurricanes that may form during that
A central enigma to physical oceanographers is the structure of the
"thermocline," the distribution of water temperatures throughout the
ocean. Due to variations in solar and other incoming thermal energy, the
ocean is not heated uniformly at the surface. This variable heating
contributes to the existence of ocean currents, which in turn lead to
variations in water temperatures throughout the full depth of the ocean.
The temperature variations in the surface waters can have an enormous and
immediate impact on all life in and out of the ocean, especially through
their influence on climate, as observed in the studies of El Niño.
During the last twenty years, several breakthroughs in physical and
mathematical understanding of the thermocline have been achieved, through
the work of a group of geophysical fluid dynamicists including Joseph
Pedlosky, at the Woods Hole Oceanographic Institution, Peter Rhines, now at
the University of Washington, and William Young, now at the Scripps
Institution of Oceanography.
Internal waves and turbulence
In c. 600 B.C., the despot Periander sent off, by ship, the sons of certain
noble families with orders that the boys be castrated. Though under full
sail, the ship suddenly halted dead in the water. According to the
historian Pliny, the cause was a kind of mollusk which attached itself to
the ship's hull, preventing its progress and thus rescuing the boys. Pliny
provides other accounts of ships under full power being suddenly held fast
in the water, often blaming not a mollusk but a small clinging fish called a
Remora. Even one Remora could, it was supposed, halt an entire ship!
Becalmed ships continued to trouble navigators of coastal and polar waters
through the centuries. In fact, Norwegian sailors encountered it so
frequently in their fjords that their word now defines the effect:
dödvand, in English "dead water." Eventually, mariners recognized that
dead water appears where there is a great influx of fresh, cold water
forming a layer over the salty sea.
In old mariners' lore, ships were held by fresh water sticking to the hull.
Sailors tried many ways to get out of dead water: pouring oil on the waters
in front of the ship; running the entire crew up and down the ship; working
the rudder; drawing a heavy rope under the ship, stem to stern; banishing
monks from the ship; and even firing guns into the water or using oars and
handspikes to beat the water.
The phenomenon of dead water was finally explained scientifically when the
theory of "internal waves" was developed. These are waves that can occur
at the boundary between two fluids of different densities. For example, in
1762, Benjamin Franklin described how swinging a suspended glass containing
oil on water created a "great commotion" at the water -- oil interface,
"tho' the surface of the oil was perfectly tranquil." However, two fluids
need not be as different as oil and water for internal waves to form. In
1904, the noted oceanographer V. Walfrid Ekman confirmed mathematically that
the passage of a sufficiently large ship through a layered region (fresh,
lower-density water atop salty, higher-density water) generates great waves
at the interface between the fresh and salt waters. This causes drag on the
vessel, as the momentum of the ship is transferred to the waves that its
entry to the two-layer region initiated. The mathematics required to study
this phenomenon comes from what are called eigenvalue problems; that is, the
motion may be modeled by a collection of "modes" (for example,
corresponding to different frequencies), and the fluid state is computed by
adding together the contributions from each mode.
The ocean is rich with eddies: tiny short-lived swirls near rocky
coastlines; fascinating vortex rings which "pinch off" from the Gulf
Stream; and gigantic ocean gyres which span thousands of kilometers and last
for decades. Their presence has implications for all areas of oceanographic
research, because eddies (at all space and time scales) are responsible for
transporting and mixing different waters.
One intriguing area of study for oceanographers is the formation and
properties of eddies that are 50 to 200 kilometers in size and have
rotational periods of one to a few months. These are called "mesoscale
eddies," meaning they are of intermediate size and lifespan. Mesoscale
eddies are the oceanic equivalent of hurricanes.
One example of mesoscale eddies is given by the eddy rings that pinch off
from the Gulf Stream. The rings that form on the continental side of the
Gulf Stream typically consist of a core of warm, biologically unproductive
water from the Sargasso Sea surrounded by a ring of colder Gulf Stream
water. Similarly, cold core "Gulf rings" may pinch off from the opposite
side of the Gulf Stream and wander into the warm Sargasso Sea. Physical
oceanographers study the formation and evolution of Gulf rings. Marine
biologists and marine geochemists study these eddies because they exchange
heat, nutrients, and chemical elements such as salt between the Sargasso Sea
and the cold, nutrient-rich waters off the Atlantic coast of the United
The importance of mesoscale eddies was unsuspected until the early 1970s. At
that time, a massive experiment called the Mid-Ocean Dynamics Experiment
(MODE) was conducted in the Atlantic Ocean east of the Gulf Stream. MODE
gathered data about the ocean dynamics on space and time scales far smaller
than general circulation scales. Mathematical analysis of the MODE results
revealed the astonishing conclusion that water motions at intermediate
scales were almost entirely driven and dominated by mesoscale eddies. This
led to intense experimental, numerical, and mathematical studies of the
formation and behavior of these eddies. Researchers have discovered that
mesoscale eddies are often created from instabilities at boundaries between
ocean regions having different densities. This corresponds,
atmospherically, to the creation of storms at "fronts."
Despite the remarkable success of MODE, the region of the North Atlantic it
studied was in fact very small: the physical challenges of gathering and
analyzing enough data at a scale which permits recognizing and tracking
eddies are enormous. Similarly, numerically simulating the mathematical
models in enough detail to analyze eddy behavior requires so many data
points that only recently have computers grown powerful enough to carry out
For many years, unrecognized eddies posed great challenges to studying
oceanic circulation, due primarily to how fluid motion was measured. The
original approach, now called the Eulerian description, relied on anchored
buoys to gather current data. This provides information about the water
flow at one fixed point of latitude and longitude. However, a second,
complementary approach is particularly effective at describing eddies. It
is called the Lagrangian description, in honor of the eighteenth century
French mathematician J.L. Lagrange who studied many problems of fluid
dynamics. (Ironically, the Lagrangian description is actually also due to
Euler, not Lagrange!) The Lagrangian approach is to use freely drifting
floats which track the movement of a small parcel of water and is quite
similar to tossing the proverbial 'message in a bottle' into the ocean and
waiting to see where it travels. This method of data collection initially
had its own difficulties, because the nature of ocean flows causes floats to
get lost as they wander in seemingly random fashion -- aside from the
volleyball, very little of the cargo made it to the island with Tom Hanks.
However, technological developments have improved scientists' ability to
track drifting floats, making the Lagrangian description practical. The
tendency of floats to meander is, in fact, an advantage, because it gives
researchers access to more of the ocean and tells them more about the
formation and dissipation of eddies and other processes.
Modern oceanographers use information from both the Eulerian and the
Lagrangian descriptions in order to gain a complete picture of the ocean's
dynamics. For example, researcher Amy Bower of the Woods Hole Oceanographic
Institution uses the Lagrangian approach to study so-called 'meddies,' which
are Mediterranean eddies: their westward flow is considered essential in
maintaining the Mediterranean salt tongue in the Atlantic ocean. She also
studies a large-scale circulation phenomenon called the Conveyor Belt,
seeking good observations in order to check the validity of current
mathematical models. Similarly, every oceanographer who uses satellite data
necessarily is using information from an Eulerian description.
Mesoscale eddies appear in all areas of the world's seas, acting to stir the
oceanic soup. In fact, some physical oceanographers believe that most of
the ocean's kinetic energy resides in these eddies; however, much
mathematical work remains to understand how eddies interact with the general
- A fluids future
One of the most exciting things about becoming an oceanographer or an
applied mathematician today is how rapidly technological and theoretical
advances are being made. Spectacular technological improvements have allowed
the gathering and analysis of quantities of data that would have been
unimaginable to early oceanographers. The vastly superior computing power
available now (compared to even ten years ago) has enabled researchers to
compute very high resolution numerical results of large-scale ocean models,
providing for the first time enough theoretical results to compare with the
wealth of actual ocean data. Future computational improvements may yield
better resolution of model outcomes, providing accurate predictions of local
Improvements in laboratory equipment and in data analysis techniques mean
that much work can profitably (and less expensively) be carried out in the
laboratory and yet provide useful insights into the nature of the real
ocean. For example, scientists observed centuries ago that the Earth's
rotation substantially affects ocean currents and circulation. This
Coriolis effect is now being modeled in various laboratory settings,
including labs at Woods Hole. Oceanographer Lawrence Pratt and his student
Heather Deese study the behavior of large-scale currents affected by the
Coriolis 'force.' (Pratt, who combines theory and experiment, is generally
interested in understanding the manifestation of theoretically predicted
structures involving chaotic advection.) Their equipment includes a large,
fluid-filled cylinder that has a bottom carefully slanted to provide a
laboratory-style Coriolis effect. As the cylinder rotates, dye is injected
into the water, which provides a visual track of the currents and eddies
formed by the fluid motion. The analysis of lab results may improve
understanding of the flow of cold water from north to south.
Similarly, Karl Helfrich, another Woods Hole researcher, uses both
laboratory experiments and theoretical work to study the physics of
nonlinear waves and the hydraulics of rotating flows. He is particularly
interested in rotating but restricted flows (as in the strait of Gibraltar,
in deep ocean sub-basins, and in regions around islands). The mathematics
in his work includes using statistical techniques to draw inferences from
data and studying numerical analysis to validate model results.
Despite recent technological advances, there are still many long-standing
theoretical problems for applied mathematicians and oceanographers to study
analytically. For example, in May 2000, the Clay Mathematics Institute of
Cambridge, Massachusetts announced seven "Millennium Prize Problems."
These are old and important mathematical problems, each of which now has a
one-million dollar prize for its solution. Among them is the challenge to
develop a mathematical theory that will determine if smooth, physically
reasonable solutions to the Navier-Stokes equations actually exist. (A
precise statement of the problem can be found at the Clay Institute's
Mathematicians also work directly on developing models of oceanography
problems. One effect of an El Niño year, for example, is an
acceleration of beach erosion, which is particularly troubling to coastal
communities. Beach and coastline erosion occur as a result of complex
interactions between the shore, the incoming waves, and the passing
currents. A mathematical description of these interactions must include
certain features: a PDE to describe the changing sea surface (obtained by
simplifying the Navier-Stokes equations); a "transport equation" to
describe the sediment-laden bottom layer of sea water overlying the beach;
"forcing terms" to describe the effects of wind stress and of incoming
waves; and "initial conditions" to describe the starting state of the
beach-ocean system. Two researchers in this field are mathematicians Jerry
Bona of the University of Texas and Juan Restreppo of the University of
Arizona. Their mathematical analyses of this type of "coupled problem"
may provide further insight into physical mechanisms that could reduce
An exciting new development in mathematical oceanography has grown out of
joint work between applied mathematicians and physical oceanographers:
dynamical systems theory can describe the mixing properties and "Lagrangian
transport" created by certain ocean phenomena. For example, Chris Jones of
Brown University's Division of Applied Mathematics uses dynamical systems to
study the transport of fluid parcels by the Gulf Stream and its associated
eddies. Similar geometric techniques are being applied by Roger Samelson of
Oregon State University, Chad Couliette of the California Institute of
Technology, and Stephen Wiggins of the University of Bristol. These
scientists study localized phenomena such as the transport of fluid in and
out of bays (like Monterey Bay in California) and the transport of fluid by
"meandering jets" (like the Kuroshio, which is the Pacific Ocean
equivalent of the Gulf Stream).
Similarly, mathematical control theory (also called inverse methods or data
assimilation) is having a huge impact on the understanding of ocean
circulation. Mathematical control theory is the result of studying how best
to "drive" a system to achieve some predetermined goal; for example,
mechanical engineers may use control theory to move a robotic arm to a
particular position with a prescribed error tolerance in a pre-specified
amount of time. The physical oceanography analogue of this happens when
researchers attempt to determine the forces such as winds or heat exchanges
which drove the ocean from a previous (observed) state to its current state.
Oceanographic data assimilation is a rapidly expanding area of study. For
example, a research group at Oregon State University led by John Allen and
Robert Miller use these techniques in combining high-frequency radar maps of
coastal surface currents and numerical results of circulation models to
estimate the structure and evolution of the state of the coastal oceans.
Similar work has been carried out by Andrew Bennett of Oregon State
University to model tropical atmosphere-ocean interactions. Substantial
applications of these inverse methods techniques are also being developed by
Carl Wunsch of MIT and collaborators from the Scripps Institution of
Oceanography at UC San Diego and NASA's Jet Propulsion Laboratory.
Another important, ongoing modeling problem is to improve the description
and representation of small-scale processes and their impact on large-scale
features, such as climate change or the frequency of El Niño
years. Related to this is the study of turbulence, often called the most
difficult problem faced by researchers in modern fluid mechanics. Turbulence
and physical instabilities are the primary causes of inaccurate
meteorological and oceanographical forecasting. Turbulence in the atmosphere
causes airplane pilots to flash the 'FASTEN SEATBELTS' sign, warning
passengers of an unpredictable ride. Turbulence in the oceans can create an
equally unpredictable environment for all creatures -- in the sea and on the
- Dreaming of the deeps
Few other aspects of the world around us have evoked such lyricism and
stoicism, rapture and despair, science and superstition, awe and fear as the
ocean. The beauty of the seas has resonated in human consciousness through
the centuries, drawing out of our imagination art, music, poetry, and
The ocean embodies the most fundamental force of nature on planet Earth. It
shapes coastlines, links with the atmosphere to create the climate, and
provides a home to countless creatures. Undoubtedly, mankind will remain
fascinated with the siren song of the seas, using every resource
available -- including mathematics -- to live with the ocean, to understand
it, and to heed its call.
- For further reading
Bascom, Willard. Waves and Beaches: The Dynamics of the Ocean
Surface, Anchor Press/Doubleday, Garden City, New York, 1980.
Earle, Sylvia A. Sea Change: A Message of the Oceans, G. P. Putnam's
Sons, New York, 1995.
Stommel, H. A View of the Sea, Princeton University Press, Princeton,
New Jersey, 1987.
Summerhayes, C.P., and Thorpe, S.A. (editors). Oceanography: An
Illustrated Guide, John Wiley & Sons, New York, 1996.
- Technical references
- Acheson, D.J. Elementary Fluid Dynamics, Oxford University Press,
- Cartwright, David E. Tides: A Scientific History, Cambridge
University Press, Cambridge, 1999.
- Gill, Adrian E. Atmosphere-Ocean Dynamics, Academic Press, San Diego,
- Kundu, Pijush. Fluid Mechanics, Academic Press, San Diego, 1990.
- LeBlond, Paul H., and Mysak, Lawrence A. Waves in the Ocean, Elsevier
Scientific Publishing Company, Amsterdam, 1978.
- Lighthill, James. Waves in Fluids, Cambridge University Press,
- Nansen, Fridtjof (editor). The Norwegian North Polar Expedition
1893-1896: Scientific Results, Vol. V, Greenwood Press, New York, 1969.
- Open University (Oceanography Course Team), Ocean Circulation,
Pergamon Press, Oxford, 1989.
- Pedlosky, Joseph. Geophysical Fluid Dynamics, Springer-Verlag, New
- Pickard, George L. and Emery, William J. Descriptive Physical
Oceanography, Pergamon Press, Oxford, 1982.
- Pond, Stephen, and Pickard, George L. Introductory Dynamical
Oceanography, second edition, Pergamon Press, Oxford, 1983.
- Summerhayes, C.P., and Thorpe, S.A. (editors). Oceanography: An
Illustrated Guide, John Wiley & Sons, New York, 1996.
- Van Dyke, Milton. An Album of Fluid Motion, Parabolic Press,
- Online references
National Oceanic & Atmospheric Administration
National Oceanic & Atmospheric Administration: El Nino
Lefschetz Center for Dynamical Systems, Brown University: Dynamics in the
Ocean and Atmosphere
Oregon State University College of Oceanic & Atmospheric Sciences
California Institute of Technology: A Dynamical Systems Approach to
Transport and Mixing in Geophysical Flows
Clay Mathematics Institute
About the Authors
Barry Cipra is a freelance mathematics writer based in
Northfield, Minnesota. Katherine Socha is a graduate student in mathematics
at the University of Texas at Austin.
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