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1. 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, eventhose who live far inland, the Earth's ocean -- the vast expanse of watercircling the globe and comprising the Atlantic and Pacific Oceans andnumerous smaller seas -- has long been a source of wonder and awe. From theearliest recorded times, men and women have sought to understand thebehavior of the ocean and of the life within it. Our knowledge of theocean is far from complete, but is steadily advancing -- thanks in great partto new developments in mathematics.

   Millennia of trial-and-error experience led to practical and sometimeselegant solutions to problems in ship-building, navigation, fishingstrategy, and the anticipation of oceanic activity ranging from rough seasto the rhythm of tides. During the last few centuries, our understanding ofthe ocean has become increasingly scientific. The observations andaccumulated wisdom of mariners throughout the ages have been augmented bydetailed measurements of water temperature and salinity and by greaterphysical 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 oceanprocesses and requires mathematical theories to understand theirimplications. Researchers use statistics and signal processing to weavetogether the many separate strands of data from sonar buoys, shipboardinstruments, and satellites. Partial differential equations describe the"mechanics" of fluid motion, from the surface waves that rock sea-goingships to the deep currents that sweep around the globe. Numerical analysishas made it possible to obtain increasingly accurate solutions to theseequations; dynamical systems theory and statistics have provided additionalinsights. Today's oceanographers are really mathematicians, in the besttradition of Galileo and Newton. Mathematics, you might say, is the"salty language" of modern oceanography.

2. What's math got to do with it?

   At its most elemental, any ocean process is all aboutchange.Measurable quantities may change as time passes (for example, tidelines on abeach move from low to high twice a day) or may change from location tolocation (for example, pressure on a submarine increases as it dives deeperinto the sea), but most quantities such as temperature and salinity changebased on both position and time. The areas of mathematics which arecritical to the description of changing processes are calculus anddifferential equations. In particular,partial differentialequations (PDEs, for short) are used to describe quantities that changecontinuously in time and space. All areas of oceanography rely heavily onthese subjects.

   Marine geology andmarine geophysics, for example, study thestructure of the Earth as a whole and the changes it has undergone throughtime. Seismic studies for oil exploration, predictions of tsunamis(devastating waves created by deep sea earthquakes), and investigations intothe formation of the largest mountain ranges on our planet (the oceanicridges) are among the interests of marine geologists and geophysicists.Chemical oceanography studies the chemistry of aquatic environments, withspecial attention to interactions between the Earth's crust, the so-calledbiota (micro-organisms, plants, and animals), and the atmosphere. Marinechemists are particularly interested in understanding both the naturalphenomena and the human-generated changes affecting the chemistry of theworld's oceans, rivers, and lakes.

   Similarly,biological oceanography studies how marine lifeformsinteract with each other and with their ocean environment. Marinebiologists chart the populations of biota in estuaries, in coastal zones,and in the open sea, with the ultimate goal of mathematically modeling andpredicting their growth and migration patterns. Both biological andchemical oceanographers are concerned withecosystem modeling:mathematical representations of interactions between the ocean's biologicaland chemical constituents, such as plants and animals and the nutrients theyfeed on. Can you imagine learning about whales, for example, without askingabout what they eat? Ecosystem modeling, especially in coastal contexts, isconcerned with immediate practical issues such as how to predict the amountof biological productivity, the fate of pollutants, and the appearance ofharmful algal blooms.

   Marine geology, chemistry, and biology occur within the context of thedynamical behavior of the ocean, which is the province ofphysicaloceanography. Physical oceanographers study the full spectrum ofcirculation patterns of the ocean, from breaking waves on stormy beaches tothe great currents and eddies that transport mass and energy (mostly in theform of heat) around the globe and that interact with atmospheric dynamicsto drive the weekly weather and the Earth's long-term climate. Physicaloceanographers rely on geophysical fluid dynamics to characterize thebehavior 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 samemanner as the rotation of a merry-go-round "pushes" on a person walking aradial line from its center to its rim. This phenomenon, called theCoriolis effect, must be included in any description of large-scale oceanphenomena. (The Coriolis 'force' isnot strong enough to affectsmall-scale fluid behavior such as water draining from an ordinary householdbathtub!)

   The notion that partial differential equations may be used to describe themotion of physical fluids goes back at least to the Swiss mathematicianLeonhard Euler. In 1755, he gave the first physically and mathematicallysuccessful description of the behavior of an idealized fluid. The Eulerequations, as they're called today, are a set of nonlinear PDEs whichexpress Newton's law of "force equals mass times acceleration" for anon-viscous fluid -- the watery equivalent of a frictionless mechanicalsystem.

   In 1821, Claude Navier improved on Euler's equations by including theeffects of viscosity. Oddly enough, the equations he obtained are correct,even though the physical assumptions on which he based his derivation werewrong! In 1845, George Gabriel Stokes rederived the same set of equations,but on a more sound theoretical basis. The result, known as theNavier-Stokes equations, forms the starting point for all modern fluiddynamics studies. Together with the laws of thermodynamics, which weredeveloped in the latter half of the nineteenth century, they are the basisfor modern physical oceanography.

   The study of nonlinear PDEs is a huge field that underlies much of appliedmathematics. With certain notable exceptions, the presence of nonlinearitymakes it virtually impossible to obtain exact solutions to these equations.This is certainly true of the Navier-Stokes equations. Consequently, muchwork is being carried out in computational fluid dynamics, with the goal ofusing computers to approximate numerically the solution of theNavier-Stokes (and Euler) equations. Researchers also attempt to simplifythe equations in order to emphasize key physical features and to reduce thecomputational problem to a manageable size. An ongoing challenge foroceanographers and mathematicians is to understand enough about the physicalmeaning of the Navier-Stokes equations to make sensible simplifications.The goal is to work with simplified versions that still provide usefulapproximate descriptions and predictions.

   What could be so difficult about simplifying the equations of fluiddynamics? There are two major obstacles which any study of ocean behaviormust overcome: the vast range of temporal and spatial scales present in theocean and the tendency of fluid flows to be unstable. Physical oceanographymust contend with turbulent eddies that span mere centimeters and last mereseconds; traveling surface gravity waves with wavelengths of kilometers andperiods of minutes to hours; ocean tides with wavelengths of thousands ofkilometers and periods of half a day; and ocean currents with spatialextents of thousands of kilometers and lifetimes measured in centuries. Thecomputation of ocean circulation on these scales, from a millimeter up tothe size of the Earth, is an enormous problem. Current theory andtechnology cannot approximate behavior over such a wide scope.

   Similarly, the tendency toward instability complicates the prediction offluid behavior. Even in a stable flow, the trajectory of an idealized fluidparticle can be unpredictable. The eventual path of a fluid particle, orsome object carried by the flow, can be highly sensitive to its initialposition. Put two floating objects -- say Tom Hanks and a volleyball -- sideby side in the ocean, wait a few days, and the chance of finding them stilltogether is a Hollywood coincidence. Instability makes matters that muchworse.

   The basic problem is that small disturbances to a flow may, if they have theright structure, draw energy from the flow and grow rapidly until they areso large as to alter the flow in fundamental ways. This kind ofinstability can lead to turbulence; one atmospheric example is gusts of windon a breezy day. The mathematical and physical elements of oceanicinstabilities are similar to those that operate in the atmosphere and makethe prediction of storms so very difficult for meteorologists. In some waysthe surprising fact is that large-scale patterns, such as the Gulf Stream,are so long-lived despite the ocean's tendency toward instability.

3. Aspects of physical oceanography

   Physical oceanography has many subdisciplines, including planetary-scalecirculation 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 incomplicated ways, most oceanographers specialize in one. A comprehensiveaccount of all these areas would fill many, many volumes of an oceanicencyclopedia, but here are a few examples to suggest the tang of modernphysical oceanography.

   Planetary-scale circulation and climate

   During 1982-83, an environmental condition calledEl Niño wasblamed for a multitude of natural disasters: severe damage to the PacificOcean's coral populations; droughts in Indonesia and the Amazon rain foreststhat led to destructive wildfires; and the loss of over 2000 lives in theUnited States due to great storms that caused floods in the Gulf states andtorrential rains and high tides in California. In 1998, the return ofEl Niño led to the death by starvation of thousands of seals and sealions in the California channel islands because the fish on which theynormally feed were driven away by atmospheric and oceanic conditions.

   What isEl Niño? Basically, it's a warming of the upper layers ofthe tropical Pacific Ocean, caused by interaction with the atmosphere.Normally the winds over the Pacific form a circular pattern above theequator: near the sea surface, the trade winds blow west across the Pacific,from South America to Indonesia, where they cycle up through the atmosphereto form the Upper Westerlies, blowing east back across the Pacific. Thesestrong winds drive the ocean to create an upwelling of cooler, nutrient-richwaters along the tropical coast of South America and along the equator.DuringEl Niño years, the trade winds weaken and upwelling isreduced. This causes surface temperature to rise over a vast area of theocean, and these temperature changes greatly affect the local climate.

   Normally, high rainfall occurs north of the equator and in the tropicalsouthwest Pacific area. InEl Niño years, the areas of highrainfall are over the ocean, rather than over Indonesia and Australia. Theweakened trade winds and reduced upwelling reduce the nutrients available tothe phyto- and zoo-plankton that form the foundation of the marine foodchain. This has proved disastrous for Peruvian fisheries, and hasnecessitated a ban on fishing off the coast of Peru during these years. Innormal years, about 20 percent (by weight) of the entire world's fishharvest has been caught there!El Niño effects can lead to morehurricanes in the Pacific, fewer hurricanes in the Gulf of Mexico, anddroughts and floods throughout the world.

   A related effect ofEl Niño is a dramatic increase of surfaceatmospheric pressure over Indonesia and Australia. This atmospheric portionof theEl Niño effects is called the Southern Oscillation. TheEl Niño Southern Oscillation (ENSO) pattern can occur two or threetimes a decade.

   Modelling the ENSO phenomenon has been a great challenge for oceanographers,requiring the use of sophisticated mathematical techniques. Whileresearchers have a pretty good understanding of the physical dynamics thatcauseEl Niño Southern Oscillation, accurate predictions are veryhard to make. Oceanographers and meteorologists find it difficultpredicting even when anEl Niño year will occur -- let alonepredicting the number and intensity of hurricanes that may form during thatyear!

   A central enigma to physical oceanographers is the structure of the"thermocline," the distribution of water temperatures throughout theocean. Due to variations in solar and other incoming thermal energy, theocean is not heated uniformly at the surface. This variable heatingcontributes to the existence of ocean currents, which in turn lead tovariations in water temperatures throughout the full depth of the ocean.The temperature variations in the surface waters can have an enormous andimmediate impact on all life in and out of the ocean, especially throughtheir influence on climate, as observed in the studies ofEl Niño.During the last twenty years, several breakthroughs in physical andmathematical understanding of the thermocline have been achieved, throughthe work of a group of geophysical fluid dynamicists including JosephPedlosky, at the Woods Hole Oceanographic Institution, Peter Rhines, now atthe University of Washington, and William Young, now at the ScrippsInstitution of Oceanography.

   Internal waves and turbulence

   In c. 600 B.C., the despot Periander sent off, by ship, the sons of certainnoble families with orders that the boys be castrated. Though under fullsail, the ship suddenly halted dead in the water. According to thehistorian Pliny, the cause was a kind of mollusk which attached itself tothe ship's hull, preventing its progress and thus rescuing the boys. Plinyprovides other accounts of ships under full power being suddenly held fastin the water, often blaming not a mollusk but a small clinging fish called aRemora. Even one Remora could, it was supposed, halt an entire ship!

   Becalmed ships continued to trouble navigators of coastal and polar watersthrough the centuries. In fact, Norwegian sailors encountered it sofrequently in their fjords that their word now defines the effect:dödvand, in English "dead water." Eventually, mariners recognized thatdead water appears where there is a great influx of fresh, cold waterforming 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 watersin front of the ship; running the entire crew up and down the ship; workingthe rudder; drawing a heavy rope under the ship, stem to stern; banishingmonks from the ship; and even firing guns into the water or using oars andhandspikes to beat the water.

   The phenomenon of dead water was finally explained scientifically when thetheory of "internal waves" was developed. These are waves that can occurat the boundary between two fluids of different densities. For example, in1762, Benjamin Franklin described how swinging a suspended glass containingoil on water created a "great commotion" at the water -- oil interface,"tho' the surface of the oil was perfectly tranquil." However, two fluidsneed not be as different as oil and water for internal waves to form. In1904, the noted oceanographer V. Walfrid Ekman confirmed mathematically thatthe passage of a sufficiently large ship through a layered region (fresh,lower-density water atop salty, higher-density water) generates great wavesat the interface between the fresh and salt waters. This causes drag on thevessel, as the momentum of the ship is transferred to the waves that itsentry to the two-layer region initiated. The mathematics required to studythis phenomenon comes from what are called eigenvalue problems; that is, themotion may be modeled by a collection of "modes" (for example,corresponding to different frequencies), and the fluid state is computed byadding together the contributions from each mode.

   Eddies

   The ocean is rich with eddies: tiny short-lived swirls near rockycoastlines; fascinating vortex rings which "pinch off" from the GulfStream; and gigantic ocean gyres which span thousands of kilometers and lastfor decades. Their presence has implications for all areas of oceanographicresearch, because eddies (at all space and time scales) are responsible fortransporting and mixing different waters.

   One intriguing area of study for oceanographers is the formation andproperties of eddies that are 50 to 200 kilometers in size and haverotational periods of one to a few months. These are called "mesoscaleeddies," meaning they are of intermediate size and lifespan. Mesoscaleeddies are the oceanic equivalent of hurricanes.

   One example of mesoscale eddies is given by the eddy rings that pinch offfrom the Gulf Stream. The rings that form on the continental side of theGulf Stream typically consist of a core of warm, biologically unproductivewater from the Sargasso Sea surrounded by a ring of colder Gulf Streamwater. Similarly, cold core "Gulf rings" may pinch off from the oppositeside of the Gulf Stream and wander into the warm Sargasso Sea. Physicaloceanographers study the formation and evolution of Gulf rings. Marinebiologists and marine geochemists study these eddies because they exchangeheat, nutrients, and chemical elements such as salt between the Sargasso Seaand the cold, nutrient-rich waters off the Atlantic coast of the UnitedStates.

   The importance of mesoscale eddies was unsuspected until the early 1970s. Atthat time, a massive experiment called the Mid-Ocean Dynamics Experiment(MODE) was conducted in the Atlantic Ocean east of the Gulf Stream. MODEgathered data about the ocean dynamics on space and time scales far smallerthan general circulation scales. Mathematical analysis of the MODE resultsrevealed the astonishing conclusion that water motions at intermediatescales were almost entirely driven and dominated by mesoscale eddies. Thisled to intense experimental, numerical, and mathematical studies of theformation and behavior of these eddies. Researchers have discovered thatmesoscale eddies are often created from instabilities at boundaries betweenocean 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 itstudied was in fact very small: the physical challenges of gathering andanalyzing enough data at a scale which permits recognizing and trackingeddies are enormous. Similarly, numerically simulating the mathematicalmodels in enough detail to analyze eddy behavior requires so many datapoints that only recently have computers grown powerful enough to carry outthe computations.

   For many years, unrecognized eddies posed great challenges to studyingoceanic circulation, due primarily to how fluid motion was measured. Theoriginal approach, now called the Eulerian description, relied on anchoredbuoys to gather current data. This provides information about the waterflow at one fixed point of latitude and longitude. However, a second,complementary approach is particularly effective at describing eddies. Itis called the Lagrangian description, in honor of the eighteenth centuryFrench mathematician J.L. Lagrange who studied many problems of fluiddynamics. (Ironically, the Lagrangian description is actually also due toEuler, not Lagrange!) The Lagrangian approach is to use freely driftingfloats which track the movement of a small parcel of water and is quitesimilar to tossing the proverbial 'message in a bottle' into the ocean andwaiting to see where it travels. This method of data collection initiallyhad its own difficulties, because the nature of ocean flows causes floats toget lost as they wander in seemingly random fashion -- aside from thevolleyball, very little of the cargo made it to the island with Tom Hanks.However, technological developments have improved scientists' ability totrack drifting floats, making the Lagrangian description practical. Thetendency of floats to meander is, in fact, an advantage, because it givesresearchers access to more of the ocean and tells them more about theformation and dissipation of eddies and other processes.

   Modern oceanographers use information from both the Eulerian and theLagrangian descriptions in order to gain a complete picture of the ocean'sdynamics. For example, researcher Amy Bower of the Woods Hole OceanographicInstitution uses the Lagrangian approach to study so-called 'meddies,' whichare Mediterranean eddies: their westward flow is considered essential inmaintaining the Mediterranean salt tongue in the Atlantic ocean. She alsostudies a large-scale circulation phenomenon called the Conveyor Belt,seeking good observations in order to check the validity of currentmathematical models. Similarly, every oceanographer who uses satellite datanecessarily is using information from an Eulerian description.

   Mesoscale eddies appear in all areas of the world's seas, acting to stir theoceanic soup. In fact, some physical oceanographers believe that most ofthe ocean's kinetic energy resides in these eddies; however, muchmathematical work remains to understand how eddies interact with the generalocean circulation.

4. A fluids future

   One of the most exciting things about becoming an oceanographer or anapplied mathematician today is how rapidly technological and theoreticaladvances are being made. Spectacular technological improvements have allowedthe gathering and analysis of quantities of data that would have beenunimaginable to early oceanographers. The vastly superior computing poweravailable now (compared to even ten years ago) has enabled researchers tocompute very high resolution numerical results of large-scale ocean models,providing for the first time enough theoretical results to compare with thewealth of actual ocean data. Future computational improvements may yieldbetter resolution of model outcomes, providing accurate predictions of localocean behavior.

   Improvements in laboratory equipment and in data analysis techniques meanthat much work can profitably (and less expensively) be carried out in thelaboratory and yet provide useful insights into the nature of the realocean. For example, scientists observed centuries ago that the Earth'srotation substantially affects ocean currents and circulation. ThisCoriolis effect is now being modeled in various laboratory settings,including labs at Woods Hole. Oceanographer Lawrence Pratt and his studentHeather Deese study the behavior of large-scale currents affected by theCoriolis 'force.' (Pratt, who combines theory and experiment, is generallyinterested in understanding the manifestation of theoretically predictedstructures involving chaotic advection.) Their equipment includes a large,fluid-filled cylinder that has a bottom carefully slanted to provide alaboratory-style Coriolis effect. As the cylinder rotates, dye is injectedinto the water, which provides a visual track of the currents and eddiesformed by the fluid motion. The analysis of lab results may improveunderstanding of the flow of cold water from north to south.

   Similarly, Karl Helfrich, another Woods Hole researcher, uses bothlaboratory experiments and theoretical work to study the physics ofnonlinear waves and the hydraulics of rotating flows. He is particularlyinterested in rotating but restricted flows (as in the strait of Gibraltar,in deep ocean sub-basins, and in regions around islands). The mathematicsin his work includes using statistical techniques to draw inferences fromdata and studying numerical analysis to validate model results.

   Despite recent technological advances, there are still many long-standingtheoretical problems for applied mathematicians and oceanographers to studyanalytically. For example, in May 2000, the Clay Mathematics Institute ofCambridge, Massachusetts announced seven "Millennium Prize Problems."These are old and important mathematical problems, each of which now has aone-million dollar prize for its solution. Among them is the challenge todevelop a mathematical theory that will determine if smooth, physicallyreasonable solutions to the Navier-Stokes equations actually exist. (Aprecise statement of the problem can be found at the Clay Institute'swebsite, www.claymath.org.)

   Mathematicians also work directly on developing models of oceanographyproblems. One effect of anEl Niño year, for example, is anacceleration of beach erosion, which is particularly troubling to coastalcommunities. Beach and coastline erosion occur as a result of complexinteractions between the shore, the incoming waves, and the passingcurrents. A mathematical description of these interactions must includecertain features: a PDE to describe the changing sea surface (obtained bysimplifying the Navier-Stokes equations); a "transport equation" todescribe the sediment-laden bottom layer of sea water overlying the beach;"forcing terms" to describe the effects of wind stress and of incomingwaves; and "initial conditions" to describe the starting state of thebeach-ocean system. Two researchers in this field are mathematicians JerryBona of the University of Texas and Juan Restreppo of the University ofArizona. Their mathematical analyses of this type of "coupled problem"may provide further insight into physical mechanisms that could reducecoastal erosion.

   An exciting new development in mathematical oceanography has grown out ofjoint work between applied mathematicians and physical oceanographers:dynamical systems theory can describe the mixing properties and "Lagrangiantransport" created by certain ocean phenomena. For example, Chris Jones ofBrown University's Division of Applied Mathematics uses dynamical systems tostudy the transport of fluid parcels by the Gulf Stream and its associatededdies. Similar geometric techniques are being applied by Roger Samelson ofOregon State University, Chad Couliette of the California Institute ofTechnology, and Stephen Wiggins of the University of Bristol. Thesescientists study localized phenomena such as the transport of fluid in andout of bays (like Monterey Bay in California) and the transport of fluid by"meandering jets" (like the Kuroshio, which is the Pacific Oceanequivalent of the Gulf Stream).

   Similarly, mathematical control theory (also called inverse methods or dataassimilation) is having a huge impact on the understanding of oceancirculation. Mathematical control theory is the result of studying how bestto "drive" a system to achieve some predetermined goal; for example,mechanical engineers may use control theory to move a robotic arm to aparticular position with a prescribed error tolerance in a pre-specifiedamount of time. The physical oceanography analogue of this happens whenresearchers attempt to determine the forces such as winds or heat exchangeswhich drove the ocean from a previous (observed) state to its current state.Oceanographic data assimilation is a rapidly expanding area of study. Forexample, a research group at Oregon State University led by John Allen andRobert Miller use these techniques in combining high-frequency radar maps ofcoastal surface currents and numerical results of circulation models toestimate the structure and evolution of the state of the coastal oceans.Similar work has been carried out by Andrew Bennett of Oregon StateUniversity to model tropical atmosphere-ocean interactions. Substantialapplications of these inverse methods techniques are also being developed byCarl Wunsch of MIT and collaborators from the Scripps Institution ofOceanography at UC San Diego and NASA's Jet Propulsion Laboratory.

   Another important, ongoing modeling problem is to improve the descriptionand representation of small-scale processes and their impact on large-scalefeatures, such as climate change or the frequency ofEl Niño years. Related to this is the study of turbulence, often called the mostdifficult problem faced by researchers in modern fluid mechanics. Turbulenceand physical instabilities are the primary causes of inaccuratemeteorological and oceanographical forecasting. Turbulence in the atmospherecauses airplane pilots to flash the 'FASTEN SEATBELTS' sign, warningpassengers of an unpredictable ride. Turbulence in the oceans can create anequally unpredictable environment for all creatures -- in the sea and on theland.

5. Dreaming of the deeps

   Few other aspects of the world around us have evoked such lyricism andstoicism, rapture and despair, science and superstition, awe and fear as theocean. The beauty of the seas has resonated in human consciousness throughthe centuries, drawing out of our imagination art, music, poetry, andscience.

   The ocean embodies the most fundamental force of nature on planet Earth. Itshapes coastlines, links with the atmosphere to create the climate, andprovides a home to countless creatures. Undoubtedly, mankind will remainfascinated with the siren song of the seas, using every resourceavailable -- including mathematics -- to live with the ocean, to understandit, and to heed its call.

    

6. For further reading
    
   
   1. Bascom, Willard.Waves and Beaches: The Dynamics of the OceanSurface, Anchor Press/Doubleday, Garden City, New York, 1980.
   2. Earle, Sylvia A.Sea Change: A Message of the Oceans, G. P. Putnam'sSons, New York, 1995.
   3. Stommel, H.A View of the Sea, Princeton University Press, Princeton,New Jersey, 1987.
   4. Summerhayes, C.P., and Thorpe, S.A. (editors).Oceanography: AnIllustrated Guide, John Wiley & Sons, New York, 1996.
   
    
    
   
7. Technical references
    
   
   1. Acheson, D.J. Elementary Fluid Dynamics, Oxford University Press,Oxford, 1990.
   2. Cartwright, David E.Tides: A Scientific History, CambridgeUniversity Press, Cambridge, 1999.
   3. Gill, Adrian E.Atmosphere-Ocean Dynamics, Academic Press, San Diego,1982.
   4. Kundu, Pijush.Fluid Mechanics, Academic Press, San Diego, 1990.
   5. LeBlond, Paul H., and Mysak, Lawrence A.Waves in the Ocean, ElsevierScientific Publishing Company, Amsterdam, 1978.
   6. Lighthill, James.Waves in Fluids, Cambridge University Press,Cambridge, 1978.
   7. Nansen, Fridtjof (editor).The Norwegian North Polar Expedition1893-1896: Scientific Results, Vol. V, Greenwood Press, New York, 1969.
   8. Open University (Oceanography Course Team),Ocean Circulation,Pergamon Press, Oxford, 1989.
   9. Pedlosky, Joseph.Geophysical Fluid Dynamics, Springer-Verlag, NewYork, 1979.
   10. Pickard, George L. and Emery, William J.Descriptive PhysicalOceanography, Pergamon Press, Oxford, 1982.
   11. Pond, Stephen, and Pickard, George L.Introductory DynamicalOceanography, second edition, Pergamon Press, Oxford, 1983.
   12. Summerhayes, C.P., and Thorpe, S.A. (editors).Oceanography: AnIllustrated Guide, John Wiley & Sons, New York, 1996.
   13. Van Dyke, Milton.An Album of Fluid Motion, Parabolic Press,Stanford, 1982.
   
    
    
   
8. Online references
    
   
   1. National Oceanic & Atmospheric Administration
      http://www.noaa.gov/

      National Oceanic & Atmospheric Administration: El Nino

      http://www.elnino.noaa.gov/

      Lefschetz Center for Dynamical Systems, Brown University: Dynamics in theOcean and Atmosphere

      http://www.dam.brown.edu/lcds/oceanresearch.html

      Oregon State University College of Oceanic & Atmospheric Sciences

      http://www.oce.orst.edu/

      California Institute of Technology: A Dynamical Systems Approach toTransport and Mixing in Geophysical Flows

      http://transport.caltech.edu/

      Clay Mathematics Institute

      http://www.claymath.org/

About the Authors
Barry Cipra is a freelance mathematics writer based inNorthfield, Minnesota. Katherine Socha is a graduate student in mathematicsat the University of Texas at Austin.