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Nonlinear aspects of an internal gravity wave co-existing with an unstable mode associated with a Helmholtz velocity profile

Published online by Cambridge University Press:  11 April 2006

R. H. J. Grimshaw
R. H. J. Grimshaw
Affiliation:
Mathematics Department, University of Melbourne, Parkville, Victoria 3052, Australia
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Abstract

Recently Lindzen (1974) has proposed a model of a shear-layer instability which allows unstable modes to co-exist with radiating internal gravity waves. The model is an infinite, continuously stratified, Boussinesq fluid, with a simple jump discontinuity in the velocity profile. Linear stability theory shows that the model is stable for wavenumbers k such that k2 < N2/2U2, where N is the Brunt—Väisälä frequency and 2U is the change in velocity across the discontinuity. For N2/2U2 < k2 < N2/U2 an unstable mode may co-exist with an internal gravity wave. This paper examines the weakly nonlinear aspects of this model for wavenumbers k close to the critical wavenumber N/2½U. An equation governing the evolution of the amplitude of the interfacial displacement is derived. It is shown that the interface may support a stable finite amplitude internal gravity wave.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 76 , Issue 1 , 14 July 1976 , pp. 65 - 84
Copyright
© 1976 Cambridge University Press

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