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Resonant wave interactions in a stratified shear flow

Published online by Cambridge University Press:  21 April 2006

R. Grimshaw
R. Grimshaw
Affiliation:
School of Mathematics, University of New South Wales, PO Box 1, Kensington, NSW 2033, Australia
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Abstract

Resonant interactions between triads of internal gravity waves propagating in a shear flow are considered for the case when the stratification and the background shear flow vary slowly with respect to typical wavelengths. If ωn, kn(n = 1, 2, 3) are the local frequencies and wavenumbers respectively then the resonance conditions are that ω1 + ω2 + ω3 = 0 and k1 + k2 + k3 = 0. If the medium is only weakly inhomogeneous, then there is a strong resonance and to leading order the resonance conditions are satisfied globally. The equations governing the wave amplitudes are then well known, and have been extensively discussed in the literature. However, if the medium is strongly inhomogeneous, then there is a weak resonance and the resonance conditions can only be satisfied locally on certain space-time resonance surfaces. The equations governing the wave amplitudes in this case are derived, and discussed briefly. Then the results are applied to a study of the hierarchy of wave interactions which can occur near a critical level, with the aim of determining to what extent a critical layer can reflect wave energy.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 190 , May 1988 , pp. 357 - 374
Copyright
© 1988 Cambridge University Press

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