# American Institute of Mathematical Sciences

July  2022, 21(7): 2415-2431. doi: 10.3934/cpaa.2022053

## On three-dimensional free surface water flows with constant vorticity

 Oskar-Morgenstern Platz 1, 1090 Vienna, Austria

Received  October 2021 Revised  January 2022 Published  July 2022 Early access  March 2022

Fund Project: The author greatfully acknowledges the support of the Austrian Science Fund (FWF) through research grant P 33107-N. Comments and remarks from two anonymous referees, for which the author is thankful, have significantly improved the presentation. Many thanks to Prof. Robin S. Johnson (Newcastle University) for many interesting discussions on topics related to exact solutions concerning geophyical water flows.

We present a survey of recent results on gravity water flows satisfying the three-dimensional water wave problem with constant (non-vanishing) vorticity vector. The main focus is to show that a gravity water flow with constant non-vanishing vorticity has a two-dimensional character in spite of satisfying the three-dimensional water wave equations. More precisely, the flow does not change in one of the two horizontal directions. Passing to a rotating frame, and introducing thus geophysical effects (in the form of Coriolis acceleration) into the governing equations, the two-dimensional character of the flow remains in place. However, the two-dimensionality of the flow manifests now in a horizontal plane. Adding also centripetal terms into the equations further simplifies the flow (under the assumption of constant vorticity vector): the velocity field vanishes, but, however, the pressure function is a quadratic polynomial in the horizontal and vertical variables, and, surprisingly, the surface is non-flat.

Citation: Calin I. Martin. On three-dimensional free surface water flows with constant vorticity. Communications on Pure and Applied Analysis, 2022, 21 (7) : 2415-2431. doi: 10.3934/cpaa.2022053
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