On three-dimensional free surface water flows with constant vorticity

Abstract

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.