Modelling surface waves on shear current with quadratic depth-dependence

TL;DR

This paper models surface wave propagation over a shear current with a quadratic depth profile, deriving a KdV equation in the Boussinesq regime to understand wave-current interactions.

Contribution

It introduces a quadratic depth-dependent current model and derives a KdV equation for surface waves in this setting, extending previous linear models.

Findings

Derivation of a KdV equation for quadratic shear currents

Analysis of wave propagation in quadratic shear profiles

Insights into wave-current interaction dynamics

Abstract

The currents in the ocean have a serious impact on ocean dynamics, since they affect the transport of mass and thus the distribution of salinity, nutrients and pollutants. In many physically important situations the current depends quadratic-ally on the depth. We consider a single layer of fluid and study the propagation of the surface waves in the presence of depth-dependent current with quadratic profile. We select the scale of parameters and quantities, which are typical for the Boussinesq propagation regime (long wave and small amplitude limit) and we also derive the well known KdV model for the surface waves interacting with current.