Solitary waves on flows with an exponentially sheared current and stagnation points

Abstract

Summary While several articles have been written on water waves on flows with constant vorticity, little is known about the extent to which a non-constant vorticity affects the flow structure, such as the appearance of stagnation points. In order to shed light on this topic, we investigate in detail the flow beneath solitary waves propagating on an exponentially decaying sheared current. Our focus is to analyse numerically the emergence of stagnation points. For this purpose, we approximate the velocity field within the fluid bulk through the classical Korteweg-de Vries asymptotic expansion and use the Matlab language to evaluate the resulting stream function. Our findings suggest that the flow beneath the waves can have 0, 1 or 2 stagnation points in the fluid body. We also study the bifurcation between these flows. Our simulations indicate that the stagnation points emerge from a streamline with a sharp corner.