Oceanic waves represent fundamental challenges in nonlinear science
August 28, 2023 - Dani Rae Wascher
The instability of Stokes waves (steady propagating waves on the surface of an ideal fluid with infinite depth) represents a fundamental challenge in the realm of nonlinear science. A team of researchers recently identified the origin of breaking oceanic waves in a recent publication in the “Proceedings of the National Academy of Sciences."
The research includes two graduates from the UNM Department of Mathematics and Statistics: former students Sergey Dyachenko (now an assistant professor in the Department of Mathematics at the University of Buffalo) and Anastasiya Semenova (now a postdoctoral researcher in the Department of Applied Mathematics at the University of Washington). The team also included Professor Bernard Deconinck from the Department of Applied Mathematics at the University of Washington and UNM Distinguished Professor Pavel Lushnikov.
Steady propagating surface gravity waves, discovered in the 19th century by Stokes, are the key structure of ocean swells easily seen from beaches, airplanes, and ocean liners. Distinguished Professor Lushnikov explains, “A Stokes wave is a surface gravity wave that propagates in the ocean with a constant velocity and is spatially periodic in the direction of propagation.” The dominant instability of these waves depends on their steepness. Distinguished Professor Lushnikov further explains, “We studied the instability of large amplitude gravity waves on the surface of the ocean. The tallest such waves have eluded analysis, and their dynamics remains largely unexplored which motivated our study.”
Since the 1960s, the Benjamin-Feir or modulational instability has dominated the dynamics of small-amplitude waves, resulting in a slow variation of the swell. The team demonstrated that, for steeper waves, another instability caused by disturbances localized at the wave crest significantly surpasses the growth rate of the modulational instability.
“Benjamin-Feir or modulational instability characterizes the growth of disturbances of small amplitude Stokes waves. That instability has the spatial scale greatly exceeding the spatial period of Stokes wave. Disturbances in that case grow on the temporal scales greatly exceeding the temporal period of Stokes wave,” said Lushnikov. “We used mathematical techniques of conformal mappings and developed a new matrix-free approach to address the large-scale eigenvalue problem. These tools allow us to reveal the nature of the instability of Stokes waves.”
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