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

journal contribution
posted on 10.09.2021, 15:45 by Curtis HooperCurtis Hooper, Karima KhusnutdinovaKarima Khusnutdinova, Roger Grimshaw
We study long surface and internal ring waves propagating in a stratified fluid over a parallel shear current. The far-field modal and amplitude equations for the ring waves are presented in dimensional form. We re-derive the modal equations from the formulation for plane waves tangent to the ring wave, which opens a way to obtaining important characteristics of the ring waves (group speed, wave action conservation law) 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. 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 (i.e. eigenfunctions of the relevant spectral problems) for 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 large 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.



  • Mechanical, Electrical and Manufacturing Engineering
  • Science


  • Mathematical Sciences

Published in

Journal of Fluid Mechanics


Cambridge University Press (CUP)


AM (Accepted Manuscript)

Publisher statement

This article has been published in a revised form in Journal of Fluid Mechanics []. This version is published under a Creative Commons CC-BY-NC-ND. No commercial re-distribution or re-use allowed. Derivative works cannot be distributed. © copyright holder.

Acceptance date









Dr Karima Khusnutdinova. Deposit date: 6 September 2021

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