Nonlinear two-dimensional water waves with arbitrary vorticity

TL;DR

This paper develops a comprehensive mathematical framework for modeling two-dimensional water waves with arbitrary vorticity, providing explicit formulas and analyzing vortex interactions in various wave regimes.

Contribution

It introduces new explicit expressions for the Dirichlet-Neumann operator and Green function in the water-wave problem with vorticity, and analyzes vortex interactions in different wave regimes.

Findings

Explicit formulas for Dirichlet-Neumann operator and Green function.

Analysis of point vortex interactions with free surface.

Derivation of coupled equations in Boussinesq and KdV regimes.

Abstract

We consider the two-dimensional water-wave problem with a general non-zero vorticity field in a fluid volume with a flat bed and a free surface. The nonlinear equations of motion for the chosen surface and volume variables are expressed with the aid of the Dirichlet-Neumann operator and the Green function of the Laplace operator in the fluid domain. Moreover, we provide new explicit expressions for both objects. The field of a point vortex and its interaction with the free surface is studied as an example. In the small-amplitude long-wave Boussinesq and KdV regimes, we obtain appropriate systems of coupled equations for the dynamics of the point vortex and the time evolution of the free surface variables.