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On the stability of heterogeneous shear flows. Part 2

Published online by Cambridge University Press:  28 March 2006

John W. Miles
John W. Miles
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra
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Abstract

Small disturbances relative to a horizontally stratified shear flow are considered on the assumptions that the velocity and density gradients in the undisturbed flow are non-negative and possess analytic continuations into a complex velocity plane. It is shown that the existence of a singular neutral mode (for which the wave speed is equal to the mean speed at some point in the flow) implies the existence of a contiguous, unstable mode in a wave-number (α), Richardson-number (J) plane. Explicit results are obtained for the rate of growth of nearly neutral disturbances relative to Hølmboe's shear flow, in which the velocity and the logarithm of the density are proportional to tanh (y/h). The neutral curve for this configuration, J = J0(α), is shown to be single-valued. Finally, it is shown that a relatively simple generalization of Hølmboe's density profile leads to a configuration having multiple-valued neutral curves, such that increasing J may be destabilizing for some range (s) of α.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 16 , Issue 2 , June 1963 , pp. 209 - 227
Copyright
© 1963 Cambridge University Press

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