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The universal aspect ratio of vortices in rotating stratified flows: theory and simulation

Published online by Cambridge University Press:  25 May 2012

Pedram Hassanzadeh ,
Philip S. Marcus  and
Patrice Le Gal
Pedram Hassanzadeh
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
Philip S. Marcus*
Affiliation:
Department of Mechanical Engineering, University of California, Berkeley, CA 94720, USA
Patrice Le Gal
Affiliation:
Institut de Recherche sur les Phénomènes Hors Equilibre, UMR 7342, CNRS - Aix-Marseille Université, 49 rue F. Joliot Curie, 13384 Marseille, CEDEX 13, France
*
†Email address for correspondence: pmarcus@me.berkeley.edu
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Abstract

We derive a relationship for the vortex aspect ratio (vertical half-thickness over horizontal length scale) for steady and slowly evolving vortices in rotating stratified fluids, as a function of the Brunt–Väisälä frequencies within the vortex and in the background fluid outside the vortex , the Coriolis parameter and the Rossby number of the vortex: . This relation is valid for cyclones and anticyclones in either the cyclostrophic or geostrophic regimes; it works with vortices in Boussinesq fluids or ideal gases, and the background density gradient need not be uniform. Our relation for has many consequences for equilibrium vortices in rotating stratified flows. For example, cyclones must have ; weak anticyclones (with ) must have ; and strong anticyclones must have . We verify our relation for with numerical simulations of the three-dimensional Boussinesq equations for a wide variety of vortices, including: vortices that are initially in (dissipationless) equilibrium and then evolve due to an imposed weak viscous dissipation or density radiation; anticyclones created by the geostrophic adjustment of a patch of locally mixed density; cyclones created by fluid suction from a small localized region; vortices created from the remnants of the violent breakups of columnar vortices; and weakly non-axisymmetric vortices. The values of the aspect ratios of our numerically computed vortices validate our relationship for , and generally they differ significantly from the values obtained from the much-cited conjecture that in quasi-geostrophic vortices.

JFM classification

Geophysical and Geological Flows: Rotating flows Geophysical and Geological Flows: Stratified flows Vortex Flows: Vortex Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 706 , 10 September 2012 , pp. 46 - 57
Copyright
Copyright © Cambridge University Press 2012

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