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Stability of a vortex in radial density stratification: role of wave interactions

Published online by Cambridge University Press:  25 May 2011

HARISH N. DIXIT  and
RAMA GOVINDARAJAN
HARISH N. DIXIT
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India
RAMA GOVINDARAJAN*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India
*
‡Email address for correspondence: [email protected]
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Abstract

We study the stability of a vortex in an axisymmetric density distribution. It is shown that a light-cored vortex can be unstable in spite of the ‘stable stratification’ of density. Using a model flow consisting of step jumps in vorticity and density, we show that a wave interaction mediated by shear is the mechanism for the instability. The requirement is for the density gradient to be placed outside the vortex core but within the critical radius of the Kelvin mode. Conversely, a heavy-cored vortex, found in other studies to be unstable in the centrifugal Rayleigh–Taylor sense, is stabilized when the density jump is placed in this region. Asymptotic solutions at small Atwood number At show growth rates scaling as At1/3 close to the critical radius, and At1/2 further away. By considering a family of vorticity and density profiles of progressively increasing smoothness, going from a step to a Gaussian, it is shown that sharp gradients are necessary for the instability of the light-cored vortex, consistent with recent work which found Gaussian profiles to be stable. For sharp gradients, it is argued that wave interaction can be supported due to the presence of quasi-modes. Probably for the first time, a quasi-mode which decays exponentially is shown to interact with a neutral wave to give exponential growth in the combined case. We finally study the nonlinear stages using viscous direct numerical simulations. The initial exponential instability of light-cored vortices is arrested due to a restoring centrifugal buoyancy force, leading to stable non-axisymmetric structures, such as a tripolar state for an azimuthal wavenumber of 2. The study is restricted to two dimensions, and neglects gravity.

JFM classification

Geophysical and Geological Flows: Internal waves Vortex Flows: Vortex instability Geophysical and Geological Flows: Waves in rotating fluids
Type
Papers
Information
Journal of Fluid Mechanics , Volume 679 , 25 July 2011 , pp. 582 - 615
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

†

Present address: Department of Mathematics, University of British Columbia, Vancouver, Canada.

References

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