A nonlinear Schr\"odinger equation for water waves on finite depth with constant vorticity

TL;DR

This paper derives a nonlinear Schr"odinger equation for water waves with constant vorticity on finite depth, revealing how vorticity affects wave stability and modulational instability properties.

Contribution

It introduces a new nonlinear Schr"odinger equation incorporating constant vorticity and analyzes its impact on wave stability and modulational instability.

Findings

Vorticity significantly alters modulational instability growth rate and bandwidth.

Coupling between mean flow and vorticity is crucial at third order.

Certain shear currents can stabilize waves independently of depth parameters.

Abstract

A nonlinear Schr\"odinger equation for the envelope of two dimensional surface water waves on finite depth with non zero constant vorticity is derived, and the influence of this constant vorticity on the well known stability properties of weakly nonlinear wave packets is studied. It is demonstrated that vorticity modifies significantly the modulational instability properties of weakly nonlinear plane waves, namely the growth rate and bandwidth. At third order we have shown the importance of the coupling between the mean flow induced by the modulation and the vorticity. Furthermore, it is shown that these plane wave solutions may be linearly stable to modulational instability for an opposite shear current independently of the dimensionless parameter kh, where k and h are the carrier wavenumber and depth respectively.