Nonlinear Wave-Currents interactions in shallow water

TL;DR

This paper analyzes the extended Green-Naghdi equations to understand how vorticity influences long wave propagation in shallow water, revealing new solitary wave behaviors and wave-current interaction insights.

Contribution

It introduces a detailed analysis of the extended Green-Naghdi equations in vorticity-rich environments, highlighting novel solitary wave behaviors and wave-current interaction phenomena.

Findings

Solitary waves can have a peak at their crest with an angle depending on vorticity.

Presence of vorticity significantly alters wave behavior and interactions.

Numerical validations support the theoretical insights.

Abstract

We study here the propagation of long waves in the presence of vorticity. In the irrotational framework, the Green-Naghdi equations (also called Serre or fully nonlinear Boussinesq equations) are the standard model for the propagation of such waves. These equations couple the surface elevation to the vertically averaged horizontal velocity and are therefore independent of the vertical variable. In the presence of vorticity, the dependence on the vertical variable cannot be removed from the vorticity equation but it was however shown in [?] that the motion of the waves could be described using an extended Green-Naghdi system. In this paper we propose an analysis of these equations, and show that they can be used to get some new insight into wave-current interactions. We show in particular that solitary waves may have a drastically different behavior in the presence of vorticity and show the existence of solitary waves of maximal amplitude with a peak at their crest, whose angle depends on the vorticity. We also show some simple numerical validations. Finally, we give some examples of wave-current interactions with a non trivial vorticity field and topography effects.