Long ring waves in a stratified fluid over a shear flow

TL;DR

This paper develops a theoretical framework for understanding long ring waves in a stratified fluid with shear flow, revealing how shear influences wavefront distortion and deriving a 2+1D KdV-type equation for wave amplitudes.

Contribution

It introduces a novel linear modal decomposition for ring waves in shear flows and derives a generalized 2+1D KdV equation for wave amplitude evolution.

Findings

Wavefronts are distorted differently for surface and interfacial waves.

A new singular solution describes wavefront distortion in shear flows.

The theory applies to two-layer fluids with piecewise-constant currents.

Abstract

Oceanic waves registered by satellite observations often have curvilinear fronts and propagate over various currents. In this paper, we study long linear and weakly-nonlinear ring waves in a stratified fluid in the presence of a depth-dependent horizontal shear flow. It is shown that despite the clashing geometries of the waves and the shear flow, there exists a linear modal decomposition (different from the known decomposition in Cartesian geometry), which can be used to describe distortion of the wavefronts of surface and internal waves, and systematically derive a 2+1 - dimensional cylindrical Korteweg-de Vries - type equation for the amplitudes of the waves. The general theory is applied to the case of the waves in a two-layer fluid with a piecewise-constant current, with an emphasis on the effect of the shear flow on the geometry of the wavefronts. The distortion of the wavefronts is described by the singular solution (envelope of the general solution) of the nonlinear first-order differential equation, constituting generalisation of the dispersion relation in this curvilinear geometry. There exists a striking difference in the shape of the wavefronts of surface and interfacial waves propagating over the same shear flow.