Mathematics Awareness Month 2001: Mathematics and the Ocean

April 9, 2001

Each April, the Joint Policy Board for Mathematics invites mathematical scientists to join in the celebration of Mathematics Awareness Month. SIAM, responsible for this year's event,* recommended the theme "Mathematics and the Ocean." The deep and versatile theme, coupled with an insightful advisory committee---Christopher Jones of the Lefschetz Center for Dynamical Systems at Brown University (chair), Roberto Camassa of the Department of Mathematics at the University of North Carolina at Chapel Hill, Arjen Doelman of the Department of Mathematics at the University of Amsterdam, Igor Mezic of the Division of Applied Sciences and Engineering at Harvard University, Roger Samelson of the College of Oceanic and Atmospheric Sciences at Oregon State University, and Carl Wunsch of the Department of Earth, Atmospheric, and Planetary Sciences at the Massachusetts Institute of Technology---resulted in materials that faculty and students should find informative and stimulating.

The poster (shown at right) was mailed to U.S. mathematics departments in late February. The essay, which was written by Barry Cipra and UT Austin grad student Katherine Socha, touches on some of the many ways in which mathematical and computational methods have been brought to bear on ocean phenomena that affect all of our lives. No dry account, the essay will convince readers at almost any level that the mathematical sciences have been an integral part of remarkable recent advances in ocean studies, and that challenging problems remain.

A few extracts from the essay are printed here.

From the essay's opening section, "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. . . .


From Section II, "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. . . .

After outlining the concerns of marine geology and marine geophysics, chemical and biological oceanography, and physical oceanography (to which Section III is devoted), Cipra and Socha explain the central role of nonlinear PDEs in ocean studies:

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.


From Section III, "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. . . . 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.

From the essay's "Internal waves and turbulence" example:

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. . . . Becalmed ships continued to trouble navigators of coastal and polar waters through the centuries. . . . Eventually, mariners recognized that dead water appears where there is a great influx of fresh, cold water forming a layer over the salty sea. . . .

From the Mathematics Awareness Month 2001 poster: NOAA polar-orbiting satellites measure the heat emitted by the ocean surface. This image was created by combining imagery acquired over several days and keeping data from ocean areas not covered by clouds. Johns Hopkins University/Applied Physics Laboratory provided the image;

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. . . . 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.

And from the "Eddies" example:

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. . . .

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. . . .

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. . . .

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. . . .


In "A fluids future," Cipra and Socha describe some current research areas and, for those sufficiently intrigued, directions for future research:

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. . . .

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. . . .

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 Ni´┐Żo 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. . . .

The essay concludes with a range of sources for further information, general and technical, print and Web-based.

*The three JPBM member societies---the American Mathematical Society, the Mathematical Association of America, and SIAM---share the task of developing materials.

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