Nonlinear two-dimensional water waves with arbitrary vorticity

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 smallamplitude 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. & COPY; 2023 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

Automated summary

This paper investigates the two-dimensional water-wave problem in a fluid volume with a general non-zero vorticity field, involving the nonlinear equations of motion for the free surface and volume variables in the fluid domain. The Dirichlet-Neumann operator and the Green function of the Laplace operator are used to provide new explicit expressions for these variables. The interaction between a point vortex and the free surface is studied as an example, obtaining coupled equations for the dynamics of the point vortex and the time evolution of the free surface variables in the small-amplitude long-wave Boussinesq and KdV regimes.