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On the non-separable baroclinic parallel flow instability problem

Published online by Cambridge University Press:  29 March 2006

Michael E. McIntyre
Michael E. McIntyre
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge
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Abstract

Perturbation series are developed and mathematically justified, using a straightforward perturbation formalism (that is more widely applicable than those given in standard textbooks), for the case of the two-dimensional inviscid Orr-Sommerfeld-like eigenvalue problem describing quasi-geostrophic wave instabilities of parallel flows in rotating stratified fluids.

The results are first used to examine the instability properties of the perturbed Eady problem, in which the zonal velocity profile has the form u = z + μu1(y, z) where, formally, μ [Lt ] 1. The connexion between baroclinic instability theories with and without short wave cutoffs is clarified. In particular, it is established rigorously that there is instability at short wavelengths in all cases for which such instability would be expected from the ‘critical layer’ argument of Bretherton. (Therefore the apparently conflicting results obtained earlier by Pedlosky are in error.)

For the class of profiles of form u = z + μu1(y) it is then shown from an examination of the O(μ) eigenfunction correction why, under certain conditions, growing baroclinic waves will always produce a counter-gradient horizontal eddy flux of zonal momentum tending to reinforce the horizontal shear of such profiles. Finally, by computing a sufficient number of the higher corrections, this first-order result is shown to remain true, and its relationship to the actual rate of change of the mean flow is also displayed, for a particular jet-like form of profile with finite horizontal shear. The latter detailed results may help to explain at least one interesting feature of the mean flow found in a recent numerical solution for the wave régime in a heated rotating annulus.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 40 , Issue 2 , 3 February 1970 , pp. 273 - 306
Copyright
© 1970 Cambridge University Press

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