Solitary wave-mean flow interaction in strongly nonlineardispersive shallow water waves

TL;DR

This paper analytically studies the interaction between solitary waves and mean flows in strongly nonlinear shallow water waves using Whitham modulation theory, providing accurate predictions validated by numerical solutions.

Contribution

It derives exact modulation equations for SGN equations and uses Riemann invariants to analyze wave interactions, improving understanding of large amplitude wave dynamics.

Findings

Analytical description of solitary wave interactions with mean flows.

Second-order accuracy of DSW fitting method for solitary waves.

Predictions of wave edge amplitude and speed match numerical results.

Abstract

The interaction of a solitary wave and a slowly varying mean background or flow for the Serre-Green-Naghdi (SGN) equations is studied using Whitham modulation theory. The exact form of the three SGN-Whitham modulation equations -- two for the mean horizontal velocity and depth decoupled from one for the solitary wave amplitude field -- are obtained exactly in the solitary wave limit. Although the three equations are not diagonalizable, the restriction of the full system to simple waves for the mean equations is diagonalized in terms of Riemann invariants. The Riemann invariants are used to analytically describe the head-on and overtaking interactions of a solitary wave with a rarefaction wave and dispersive shock wave (DSW), leading to scenarios of solitary wave trapping or transmission by the mean flow. The analytical results for overtaking interactions prove that a simpler, approximate approach based on the DSW fitting method is accurate to the second order in solitary wave amplitude, beyond the first order accurate Korteweg-de Vries approximation. The analytical results also accurately predict the SGN DSW's solitary wave edge amplitude and speed. The analytical results are favourably compared with corresponding numerical solutions of the full SGN equations. Because the SGN equations model the bi-directional propagation of strongly nonlinear, long gravity waves over a flat bottom, the analysis presented here describes large amplitude solitary wave-mean flow interactions in shallow water waves.