A variational formulation for steady surface water waves on a Beltrami flow

Abstract

This paper considers steady surface waves `riding' a Beltrami flow (a three-dimensional flow with parallel velocity and vorticity fields). It is demonstrated that the hydrodynamic problem can be formulated as two equations for two scalar functions of the horizontal spatial coordinates, namely the elevation $η$ of the free surface and the potential $Φ$ defining the gradient part (in the sense of the Hodge-Weyl decomposition) of the horizontal component of the tangential fluid velocity there. These equations are written in terms of a nonlocal operator $H(η)$ mapping $Φ$ to the normal fluid velocity at the free surface, and are shown to arise from a variational principle. In the irrotational limit the equations reduce to the Zakharov-Craig-Sulem formulation of the classical three-dimensional steady water-wave problem, while $H(η)$ reduces to the familiar Dirichlet-Neumann operator.