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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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References

Acheson, D. 1976 On over-reflection. J. Fluid Mech. (to appear).
Atlas, D., Metcalf, J. I., Richter, J. H. & Gossard, E. E. 1970 The birth of ‘CAT’ and microscale turbulence. J. Atmos. Sci. 27, 903–913.Google Scholar
Drazin, P. G. 1970 Kelvin—Helmholtz instability of finite amplitude. J. Fluid Mech. 42, 321–335.Google Scholar
Dutton, J. A. & Panofsky, H. A. 1970 Clear air turbulence: a mystery may be unfolding. Science, 167, 937—944.CrossRefGoogle Scholar
Grimshaw, R. 1974 Internal gravity waves in a slowly varying, dissipative medium. Geophys. Fluid Dyn. 6, 131–148.Google Scholar
Hardy, K. R., Reed, R. J. & Mather, G. K. 1973 Observation of Kelvin—Helmholtz billows and their mesoscale environment by radar, instrumental aircraft, and a dense radiosonde network. Quart J. Roy. Met. Soc. 99, 279–293.Google Scholar
Hooke, W. H., Hall, F. F. & Gossard, E. E. 1973 Observed generation of an atmospheric gravity wave by shear instability in the mean flow of the planetary boundary layer. Boundary-Layer Met. 5, 29–42.Google Scholar
Lighthill, M. J. 1960 Studies in magneto-hydrodynamic waves and other anisotropic wave motions. Phil. Trans. A 252, 397–430.Google Scholar
Lindzen, R. S. 1974 Stability of a Helmholtz velocity profile in a continuously stratified infinite Boussinesq fluid — applications to clear air turbulence. J. Atmos. Sci. 31, 1507–1514.Google Scholar
McIntyre, M. & Weissman, M. 1976 Radiating instabilities versus resonant over-reflection with reference to Lindzen's model of a low Richardson number shear layer. (To appear.)
Metcalf, J. I. & Atlas, D. 1973 Microscale ordered motions and atmospheric structure associated with thin echo layers in stably stratified zones. Boundary-Layer Met. 4, 7–36.Google Scholar
Ottersten, H., Hardy, K. R. & Little, C. G. 1973 Radar and sodar probing of waves and turbulence in statically stable clear-air layers. Boundary-Layer Met. 4, 47–89.Google Scholar
Phillips, O. M. 1966 The Dynamics of the Upper Ocean. Cambridge University Press.
Reed, R. J. & Hardy, K. R. 1972 A case study of persistent, intense clear air turbulence in an upper level frontal zone. J. Appl. Met. 11, 541–549.Google Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.