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Vorticity inversion and action-at-a-distance instability in stably stratified shear flow

Published online by Cambridge University Press:  14 January 2011

A. RABINOVICH ,
O. M. UMURHAN ,
N. HARNIK ,
F. LOTT  and
E. HEIFETZ
A. RABINOVICH
Affiliation:
Department of Geophysics and Planetary Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
O. M. UMURHAN
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, London E1 4NS, UK Department of Astronomy, City College of San Francisco, San Francisco, CA 94112, USA
N. HARNIK
Affiliation:
Department of Geophysics and Planetary Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
F. LOTT
Affiliation:
Laboratoire de Météorologie Dynamique, École Normale Supérieure, 24 rue Lhomond, 75231 Paris CEDEX 05, France
E. HEIFETZ*
Affiliation:
Department of Geophysics and Planetary Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
*
†Email address for correspondence: [email protected]
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Abstract

The somewhat counter-intuitive effect of how stratification destabilizes shear flows and the rationalization of the Miles–Howard stability criterion are re-examined in what we believe to be the simplest example of action-at-a-distance interaction between ‘buoyancy–vorticity gravity wave kernels’. The set-up consists of an infinite uniform shear Couette flow in which the Rayleigh–Fjørtoft necessary conditions for shear flow instability are not satisfied. When two stably stratified density jumps are added, the flow may however become unstable. At each density jump the perturbation can be decomposed into two coherent gravity waves propagating horizontally in opposite directions. We show, in detail, how the instability results from a phase-locking action-at-a-distance interaction between the four waves (two at each jump) but can as well be reasonably approximated by the interaction between only the two counter-propagating waves (one at each jump). From this perspective the nature of the instability mechanism is similar to that of the barotropic and baroclinic ones. Next we add a small ambient stratification to examine how the critical-level dynamics alters our conclusions. We find that a strong vorticity anomaly is generated at the critical level because of the persistent vertical velocity induction by the interfacial waves at the jumps. This critical-level anomaly acts in turn at a distance to dampen the interfacial waves. When the ambient stratification is increased so that the Richardson number exceeds the value of a quarter, this destructive interaction overwhelms the constructive interaction between the interfacial waves, and consequently the flow becomes stable. This effect is manifested when considering the different action-at-a-distance contributions to the energy flux divergence at the critical level. The interfacial-wave interaction is found to contribute towards divergence, that is, towards instability, whereas the critical-level–interfacial-wave interaction contributes towards an energy flux convergence, that is, towards stability.

JFM classification

Geophysical and Geological Flows: Atmospheric flows Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 670 , 10 March 2011 , pp. 301 - 325
Copyright
Copyright © Cambridge University Press 2011

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