# Mathematics63

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 these subjects.

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 bathtub!)

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 system.

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 worse.

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 physical oceanography.

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 year!

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.

Eddies

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 States.

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 the computations.

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 ocean circulation.

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 ocean behavior.

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 website, http://www.claymath.org.)

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 coastal erosion.

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 land.

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 science.

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.

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, Oxford, 1990.
Cartwright, David E. Tides: A Scientific History, Cambridge University Press, Cambridge, 1999.