Exponential asymptotics and the generation of free-surface flows by submerged line vortices

TL;DR

This paper develops analytical methods using exponential asymptotics to study free-surface water waves generated by submerged vortices at low Froude numbers, revealing wave formation and trapping phenomena.

Contribution

It introduces a boundary-integral approach combined with exponential asymptotics to analyze nonlinear wave generation by submerged vortices, including trapped waves between vortices.

Findings

Single vortex induces waves via Stokes lines.

Multiple vortices can trap waves between them.

Trapped waves occur at specific Froude numbers.

Abstract

There has been significant recent interest in the study of water waves coupled with non-zero vorticity. We derive analytical approximations for the exponentially-small free-surface waves generated in two-dimensions by one or several submerged point vortices when driven at low Froude numbers. The vortices are fixed in place, and a boundary-integral formulation in the arclength along the surface allows the study of nonlinear waves and strong point vortices. We demonstrate that for a single point vortex, techniques in exponential asymptotics prescribe the formation of waves in connection with the presence of Stokes lines originating from the vortex. When multiple point vortices are placed within the fluid, trapped waves may occur, which are confined to lie between the vortices. We also demonstrate that for the two-vortex problem, the phenomenon of trapped waves occurs for a countably infinite set of values of the Froude number. This work will form a basis for other asymptotic investigations of wave-structure interactions where vorticity plays a key role in the formation of surface waves.