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

TL;DR

This paper investigates how nonconstant vorticity, specifically an exponentially decaying shear current, influences the formation of stagnation points beneath solitary water waves using numerical methods.

Contribution

It provides a detailed numerical analysis of stagnation point emergence in flows with exponentially sheared currents, extending understanding beyond constant vorticity cases.

Findings

Flow can have zero, one, or two stagnation points.

Stagnation points emerge from streamlines with sharp corners.

Bifurcation between different stagnation point configurations.

Abstract

While several articles have been written on water waves on flows with constant vorticity, little is known about the extent to which a nonconstant 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 streamfunction. Our findings suggest that the flow beneath the waves can have zero, one or two 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.