Wavefronts and modal structure of long surface and internal ring waves on a parallel shear current

TL;DR

This paper investigates the wavefronts and modal structures of long surface and internal ring waves in stratified fluids with shear currents, deriving new equations and analyzing their properties in various fluid configurations.

Contribution

It introduces a new spectral problem for modal equations of ring waves and develops a method to analyze their wavefronts and structures in stratified shear flows.

Findings

Modal equations form a new spectral problem.

Wavefronts are characterized for different fluid configurations.

Comparison between homogeneous and stratified fluid wave modes.

Abstract

We study long surface and internal ring waves propagating in a stratified fluid over a parallel shear flow. The far-field modal and amplitude equations for the ring waves are presented in dimensional form. We re-derive them from the formulation for plane waves tangent to the ring wave, which opens a way to obtaining important characteristics of the ring waves (e.g. group speed) and to constructing more general `hybrid solutions' consisting of a part of a ring wave and two tangent plane waves. The modal equations constitute a new spectral problem, and are analysed for a number of examples of surface ring waves in a homogeneous fluid and internal ring waves in a stratified fluid. The detailed analysis is developed for the case of a two-layered fluid with a linear shear current where we study their wavefronts and two-dimensional modal structure. Comparisons are made between the modal functions of the surface waves in a homogeneous and two-layered fluids, as well as the interfacial waves described exactly and in the rigid-lid approximation. We also analyse the wavefronts of surface and interfacial waves for a family of power-law upper-layer currents, which can be used to model wind generated currents, river inflows and exchange flows in straits. A global and local measure of the deformation of wavefronts are introduced and evaluated.