Mathematics: A Critical Tool for Understanding the Oceans

By Adrian Constantin and George Haller

This post is adapted from the introduction to a special issue on Mathematical Aspects of Physical Oceanography, volume 31, no. 3 of Oceanography, the official magazine of the Oceanographic Society, and is published with the permission of The Oceanography Society.

Our knowledge and understanding of ocean dynamics is far from complete, but is expanding thanks in great part to new developments in mathematics. Some of the most important oceanographic discoveries have been made as a result of an integrated, multidisciplinary approach. The deepest understanding and the most interesting results almost always evolve from the interplay between theory and observation. A substantial body of theory to aid in the interpretation of observations has been developed, yet the ocean offers continually new data to challenge existing ideas—modern fieldwork is much more than cataloguing oceanic features, being designed as much to test theoretical hypotheses as it is to detect new phenomena.

The mathematical subject areas that are essential to the description of the changing spatiotemporal processes in the ocean are partial differential equations and dynamical systems. All subfields of physical oceanography rely heavily on these subjects, with analytical and computational aspects often intertwined and mutually reinforcing each other—their combined effect being stronger than the sum of each separate part. With a few notable exceptions, nonlinearity makes it impossible to obtain exact solutions to the governing equations for ocean flows. Consequently, numerical simulations play a prominent role in modern physical oceanography. However, the available technological means cannot cope with the vast range of temporal and spatial scales present in the ocean, and the prediction of fluid flow behavior becomes unrealistic if small disturbances draw energy from the main flow and subsequently grow rapidly until they become large enough to alter fundamentally the overall flow. An ongoing challenge is to reduce the computational problem to a manageable size, a task contingent upon making sensible simplifications that still provide accurate descriptions and predictions. This procedure often not only permits an in-depth study of a known phenomenon but sometimes also uncovers new processes that may not have been apparent or were overlooked. For successful derivation of adequate simplified models, it is necessary to understand the main ongoing mechanisms very well. This allows identification of physical regimes in which certain factors can be neglected, so that the dynamics is captured by a relatively simpler model. Such a model is amenable to in-depth theoretical studies that often reveal unexpected features and close the gap between real-world observations and idealized theoretical flow patterns.

From January 22 to March 23, 2018, the program “Mathematical Aspects of Physical Oceanography” took place at the Erwin Schrödinger Institute for Mathematics and Physics (Vienna, Austria). The presentations at that event inspired a special edition of Oceanography, the official (and open-access) magazine of The Oceanography Society, on “Mathematical Aspects of Physical Oceanography.”

In this issue, you’ll find papers that discuss emerging theoretical methodologies and computational approaches, and describe high-precision experimental results. The methods are diverse, and reflect the critical role of deep mathematical tools in advancing our knowledge of the oceans, a critical need especially as we recognize that there are systemic changes occurring that are likely to affect global climate and more.

We invite you to read this special issue, and share with your colleagues and students. We hope that increasing awareness will also lead to increased collaboration between oceanographers and mathematical scientists and, ultimately, to accelerating our understanding of our planet.

About the authors
Adrian Constantin is Professor of Mathematics at the University of Vienna, Austria.

George Haller is Professor of Mechanics, at the Swiss Federal Institute of
Technology Zürich, Switzerland.