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Explicit wave action conservation for water waves on vertically sheared flows

Toledo, Y; Shrira, VI; Quinn, BE

Explicit wave action conservation for water waves on vertically sheared flows Thumbnail


Authors

Y Toledo

BE Quinn



Abstract

This paper addresses a major shortcoming of the current generation of wave models, namely their inability to describe wave propagation upon ambient currents with vertical shear. The wave action conservation equation (WAE) for linear waves propagating in horizontally inhomogeneous vertically-sheared currents is derived following Voronovich (1976). The resulting WAE specifies conservation of a certain depth-averaged quantity, the wave action, a product of the wave amplitude squared, eigenfunctions and functions of the eigenvalues of the boundary value problem for water waves upon a vertically sheared current. The formulation of the WAE is made explicit using known asymptotic solutions of the boundary value problem which exploit the smallness of the current magnitude compared to the wave phase velocity and/or its vertical shear and curvature; the adopted approximations are shown to be sufficient for most of the conceivable applications. In the limit of vanishing current shear, the new formulation reduces to that of Bretherton and Garrett (1968) without shear and the invariant is calculated with the current magnitude taken at the free surface. It is shown that in realistic oceanic conditions, the neglect of the vertical structure of the currents in wave modelling which is currently universal might lead to significant errors in wave amplitude. The new WAE which takes into account the vertical shear can be better coupled to modern circulation models which resolve the three-dimensional structure of the uppermost layer of the ocean.

Journal Article Type Article
Acceptance Date Mar 2, 2017
Online Publication Date Mar 4, 2017
Publication Date 2017-04
Publicly Available Date Mar 28, 2024
Journal Ocean Modelling
Print ISSN 1463-5003
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 112
Pages 33-47
DOI https://doi.org/10.1016/j.ocemod.2017.03.003
Keywords Wave action equation; Vertically sheared currents; Wave rays
Publisher URL http://www.sciencedirect.com/science/article/pii/S1463500317300288

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