Nonlinear water waves in shallow water in the presence of constant vorticity: A Whitham approach

TL;DR

This paper develops a new nonlinear wave model in shallow water with constant vorticity, incorporating dispersive effects via a Whitham approach, and analyzes how vorticity influences wave behavior, including solitary waves and wave breaking.

Contribution

It introduces a novel single PDE model combining hyperbolic and dispersive effects for shallow water waves with vorticity, extending existing models with a Whitham-type equation.

Findings

Vorticity significantly affects solitary wave properties.

The model predicts changes in wave breaking times due to vorticity.

Dispersive effects are accurately captured in the new PDE.

Abstract

Two-dimensional nonlinear gravity waves travelling in shallow water on a vertically sheared current of constant vorticity are considered. Using Euler equations, in the shallow water approximation, hyperbolic equations for the surface elevation and the horizontal velocity are derived. Using Riemann invariants of these equations, that are obtained analytically, a closed-form nonlinear evolution equation for the surface elevation is derived. A dispersive term is added to this equation using the exact linear dispersion relation. With this new single first-order partial differential equation, vorticity effects on undular bores are studied. Within the framework of weakly nonlinear waves, a KdV-type equation and a Whitham equation with constant vorticity are derived from this new model and the effect of vorticity on solitary waves and periodic waves is considered. Futhermore, within the framework of the new model and the Whitham equation a study of the effect of vorticity on the breaking time of dispersive waves and hyperbolic waves as well is carried out.