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

TL;DR

This paper develops a variational framework for steady surface water waves on a Beltrami flow, reducing the complex hydrodynamic problem to two scalar equations involving a nonlocal operator, generalizing classical water wave formulations.

Contribution

It introduces a novel variational formulation for steady water waves on Beltrami flows, extending classical models to include rotational effects with a new nonlocal operator.

Findings

Formulation reduces to classical water wave equations in the irrotational limit.

Derivation of a variational principle for the Beltrami flow case.

Identification of a nonlocal operator generalizing the Dirichlet-Neumann operator.

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 $\eta$ of the free surface and the potential $\Phi$ 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 $H(\eta)$ mapping $\Phi$ 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 $H(\eta)$ reduces to the familiar Dirichlet-Neumann operator.