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Stability of an isolated pancake vortex in continuously stratified-rotating fluids

Published online by Cambridge University Press:  25 July 2016

Eunok Yim ,
Paul Billant  and
Claire Ménesguen
Eunok Yim*
Affiliation:
LadHyX, CNRS, École Polytechnique, F-91128 Palaiseau CEDEX, France
Paul Billant
Affiliation:
LadHyX, CNRS, École Polytechnique, F-91128 Palaiseau CEDEX, France
Claire Ménesguen
Affiliation:
LPO, IFREMER, CNRS, BP 70, 29280 Plouzané, France
*
†Email address for correspondence: eunok@ladhyx.polytechnique.fr
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Abstract

This paper investigates the stability of an axisymmetric pancake vortex with Gaussian angular velocity in radial and vertical directions in a continuously stratified-rotating fluid. The different instabilities are determined as a function of the Rossby number $Ro$, Froude number $F_{h}$, Reynolds number $Re$ and aspect ratio ${\it\alpha}$. Centrifugal instability is not significantly different from the case of a columnar vortex due to its short-wavelength nature: it is dominant when the absolute Rossby number $|Ro|$ is large and is stabilized for small and moderate $|Ro|$ when the generalized Rayleigh discriminant is positive everywhere. The Gent–McWilliams instability, also known as internal instability, is then dominant for the azimuthal wavenumber $m=1$ when the Burger number $Bu={\it\alpha}^{2}Ro^{2}/(4F_{h}^{2})$ is larger than unity. When $Bu\lesssim 0.7Ro+0.1$, the Gent–McWilliams instability changes into a mixed baroclinic–Gent–McWilliams instability. Shear instability for $m=2$ exists when $F_{h}/{\it\alpha}$ is below a threshold depending on $Ro$. This condition is shown to come from confinement effects along the vertical. Shear instability transforms into a mixed baroclinic–shear instability for small $Bu$. The main energy source for both baroclinic–shear and baroclinic–Gent–McWilliams instabilities is the potential energy of the base flow instead of the kinetic energy for shear and Gent–McWilliams instabilities. The growth rates of these four instabilities depend mostly on $F_{h}/{\it\alpha}$ and $Ro$. Baroclinic instability develops when $F_{h}/{\it\alpha}|1+1/Ro|\gtrsim 1.46$ in qualitative agreement with the analytical predictions for a bounded vortex with angular velocity slowly varying along the vertical.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Vortex Flows: Vortex Flows Vortex Flows: Vortex instability
Type
Papers
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Journal of Fluid Mechanics , Volume 801 , 25 August 2016 , pp. 508 - 553
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© 2016 Cambridge University Press 

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