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Viscous stability properties of a Lamb–Oseen vortex in a stratified fluid

Published online by Cambridge University Press:  22 February 2010

XAVIER RIEDINGER ,
STÉPHANE LE DIZÈS  and
PATRICE MEUNIER
XAVIER RIEDINGER*
Affiliation:
Institut de Recherche sur les Phénomènes Hors Équilibre, CNRS/Universités Aix-Marseille I&II, 49, rue F. Joliot-Curie, BP 146, F-13384 Marseille cedex 13, France
STÉPHANE LE DIZÈS
Affiliation:
Institut de Recherche sur les Phénomènes Hors Équilibre, CNRS/Universités Aix-Marseille I&II, 49, rue F. Joliot-Curie, BP 146, F-13384 Marseille cedex 13, France
PATRICE MEUNIER
Affiliation:
Institut de Recherche sur les Phénomènes Hors Équilibre, CNRS/Universités Aix-Marseille I&II, 49, rue F. Joliot-Curie, BP 146, F-13384 Marseille cedex 13, France
*
†Email address for correspondence: xavier.riedinger@gmail.com
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Abstract

In this work, we analyse the linear stability of a frozen Lamb–Oseen vortex in a fluid linearly stratified along the vortex axis. The temporal stability properties of three-dimensional normal modes are obtained under the Boussinesq approximation with a Chebychev collocation spectral code for large ranges of Froude numbers and Reynolds numbers (the Schmidt number being fixed to 700). A specific integration technique in the complex plane is used in order to apply the condition of radiation at infinity. For large Reynolds numbers and small Froude numbers, we show that the vortex is unstable with respect to all non-axisymmetrical waves. The most unstable mode is however always a helical radiative mode (m = 1) which resembles either a displacement mode or a ring mode. The displacement mode is found to be unstable for all Reynolds numbers and for moderate Froude numbers (F ~ 1). The radiative ring mode is by contrast unstable only for large Reynolds numbers above 104 and is the most unstable mode for large Froude numbers (F > 2). The destabilization of this mode for large Froude numbers is shown to be associated with a resonance mechanism which is analysed in detail. Analyses of the scaling and of the spatial structure of the different unstable modes are also provided.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 645 , 25 February 2010 , pp. 255 - 278
Copyright
Copyright © Cambridge University Press 2010

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