The effect of a background shear current on large amplitude internal solitary waves

Abstract

Large amplitude internal waves interacting with a linear shear current in a system of two layers of different densities are studied using a set of nonlinear evolution equations derived under the long wave approximation without the smallness assumption on the wave amplitude. For the case of uniform vorticity, solitary wave solutions are obtained under the Boussinesq assumption for a small density jump, and the explicit relationship between the wave speed and the wave amplitude is found. It is shown that a linear shear current modifies not only the wave speed, but also the wave profile drastically. For the case of negative vorticity, when compared with the irrotational case, a solitary wave of depression traveling in the positive x direction is found to be smaller, wider, and slower, while the opposite is true when traveling in the negative x direction. In particular, when the amplitude of the solitary wave propagating in the negative x direction is greater than the critical value, a stationary recirculating eddy appears at the wave crest.