Solitary waves with a vortex core in a shallow layer of stratified fluid

Abstract

This paper is concerned with a theoretical model for large amplitude solitary waves with a vortex core in a shallow layer of stratified fluid with a nearly uniform stratification. Previous work has shown that solitary waves can be calculated up to a critical amplitude for which the horizontal velocity, in a frame for which the wave is at rest, approaches zero at the boundary point beneath the wave crest for a wave of elevation (or above the wave crest for a wave of depression). Here we calculate waves with amplitudes slightly greater than the critical amplitude by incorporating a vortex core located near the aforementioned boundary point. The effect of the vortex core is to introduce into the governing equation for the wave amplitude an extra nonlinear term proportional to the 3/2 power of the difference between the wave amplitude and the critical amplitude. We find that as the wave amplitude increases above the critical amplitude, the wave broadens, which is in marked contrast to the case of small amplitude waves. Further, the wave speed is found to depend nonlinearly on the wave amplitude, again in marked contrast to the linear dependence for small amplitude waves. The specific case of linearly stratified fluid is examined in detail.