Mathematics and the ocean

Summary: Mathematicians study and model the forces that cause tides and waves.

Approximately 70% of the Earth’s surface is covered with water, most of which is in a constant state of motion. The causes of this motion include the gravitational pull of celestial bodies in space, like the sun and moon; the rotation and shape of the Earth; and the influence of natural phenomena, like wind and earthquakes. Mathematicians have long studied tides and waves, following in the path of ancient scholars and others who sought to understand these phenomena for many spiritual and practical reasons, such as sailing. In the twenty-first century, people still travel both above and below the surface of the oceans for research, commerce, and pleasure, and there are many problems old and new to be explored. Some interesting mathematical investigations related to tides and waves at the start of the twenty-first century include three-dimensional modeling of extreme waves (also called “rogue waves”), such as those observed during the 2004 Indian Ocean tsunami and the Hurricane Katrina storm surges in 2005. Mathematicians, scientists, and engineers have also explored methods and developed technology to harness tide and wave power as an alternative energy source, including methods that actually create waves in addition to using naturally-occurring ones. Some colleges and universities teach courses on tides and waves that involve substantial mathematics. The theme of Mathematics Awareness Month in 2001 was “Mathematics and the Ocean,” underscoring the importance and relationship of ocean phenomena and mathematics, as well as the depth and breadth of the topics studied.

Tides

Water in Earth’s oceans moves in a variety of ways, including many scales of currents, tides, and waves. Mathematicians and scholars from ancient times up through the Renaissance observed, identified, and quantified tidal patterns. The term “tides” generally refers to the overall cyclic rising and falling of ocean levels with respect to land—though tides have been observed in large lakes, the atmosphere, and Earth’s crust, resulting largely from the same forces that produce ocean tides.

The daily tide cycles are caused by the moon’s gravity, which makes the oceans bulge in the direction of the moon. A corresponding rise occurs on the opposite side of the Earth at the same time, because the moon is also pulling on the Earth itself. Most regions on Earth have two high tides and two low tides every day, known as “semidiurnal tides,” which result from the daily rotation of the Earth relative to the moon. Since the angle of the moon’s orbital plane also affects gravitational pull on Earth’s curved surface, some regions have only one cycle of high and low, known as “diurnal tides.” The height of tides varies according to many variables, including coastline shape; water depth (“bathymetry”); latitude; and the position of the sun, which also exerts gravitational force. “Spring” tides, not named for the season, are extremely high and low tides that occur during full and new moons when the sun and moon are in a straight line with the Earth, and their gravitational effects are additive. A proxigean spring tide occurs roughly once every 1.5 years when the moon is at its proxigee (closest distance to Earth) and positioned between the sun and the Earth. Neap tides minimize the difference between high and low tides. They occur during the moon’s quarter phases when the sun’s gravitational pull is acting at right angles to the moon’s pull with respect to the Earth.

A few of the many contributors to the theory and mathematical description of tides include Galileo Galilei, René Descartes, Johannes Kepler, Daniel Bernoulli, Leonhard Euler, Pierre Laplace, George Darwin, and Horace Lamb. Some mathematicians, like Colin Maclaurin and George Airy, won scientific prizes for their research. Work by mathematician William Thomson (Lord Kelvin) on harmonic analysis of tides led to the construction of tide-predicting machines.

Waves

There are many mathematical approaches to the study of waves in the twenty-first century, and some mathematicians center their research around this topic. In contrast to tides, a wave is a more localized disturbance of water in the form of a propagating ridge or swell that occurs on the surface of a body of water. Despite the fact that surface waves appear to be moving when observed, they do not move water particles horizontally along the entire path of the wave. Rather, they combine limited longitudinal or horizontal motions with transverse or vertical motions. Water particles in a wave oscillate in localized, circular patterns as the energy propagates through the liquid, with a radius that decreases as the water depth or distance from the crest of the wave increases. Wind is a primary cause of surface waves, because of frictional drag between air and water particles. Larger waves, like tsunamis, result from underwater Earth movements, such as earthquakes and landslides.

The Navier–Stokes equations, named for Claude-Louis Navier and George Stokes, are partial differential equations that describe fluid motion and are widely used in the study of tides and waves. Solutions to these equations are often found and verified using numerical methods. The Coriolis–Stokes force, named for George Stokes and Gustave Coriolis, mathematically describes force in a rotating fluid, such as the small rotations in surface waves. A few examples of individuals with diverse approaches who have won prizes in this area include Joseph Keller, who has researched many forms and properties of waves, including geometrical diffraction and propagation; Michael Lighthill and Thomas Benjamin, who jointly posed the Benjamin–Lighthill conjecture regarding nonlinear steady water waves, which continues to spur research in both theoretical and applied mathematics; and Sijue Wu, who has researched the well-posedness of the fully two- and three-dimensional nonlinear wave problem in various function spaces, using techniques like harmonic analysis. In other theoretical and applied areas, some techniques from dynamical systems theory, statistical analysis, and data assimilation, which combines data and partial differential equations, have been useful for formulating and solving wave problems.

Bibliography

Cartwright, David. Tides: A Scientific History. Cambridge, England: Cambridge University Press, 2001.

Johnson, R. I. A Modern Introduction to the Mathematical Theory of Water Waves. Cambridge, England: Cambridge University Press, 1997.

Joint Policy Board for Mathematics. “Mathematics Awareness Month April 2001: Mathematics and the Ocean.” http://mathaware.org/mam/01.