The wave stability of solitary waves over a bump for the full Euler equations

TL;DR

This paper numerically investigates the stability of solitary waves over a bump using the full Euler equations, revealing stability for uniform flow perturbations and amplitude-dependent stability for solitary waves, including wave-breaking phenomena.

Contribution

It provides a detailed numerical analysis of wave stability over topography using the full Euler equations, identifying different stability regimes for perturbed flows.

Findings

Steady waves from uniform flow are always stable.

Perturbed solitary waves show stability when initial amplitude is smaller.

Wave-breaking occurs when initial amplitude exceeds steady solution.

Abstract

In this work, we present a numerical study of the wave stability of steady solitary waves over a localised topographic obstacle through the full Euler equations. There are two branches of the solutions: one from the perturbed uniform flow and the other from the perturbed solitary-wave flow. We find that steady waves from the perturbed uniform flow are always stable with respect to perturbations of its amplitude. Regarding the perturbed solitary-wave, when the perturbed initial condition has smaller amplitude than the steady solution we notice a certain type of stability. Yet, when the perturbed initial condition has larger amplitude than the steady solution an onset of wave-breaking seem to occur.