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Formal stability of circular vortices

Published online by Cambridge University Press:  26 April 2006

R. C. Kloosterziel  and
G. F. Carnevale
R. C. Kloosterziel
Affiliation:
Institute for Nonlinear Science, University of California. San Diego. La Jolla. CA 92093, USA Present address: School of Ocean and Earth Science and Technology, University of Hawaii, Honolulu, HI 96822, USA.
G. F. Carnevale
Affiliation:
Scripps Institute of Oceanography, University of California, San Diego, La Jolla, CA 92093, USA
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Abstract

The second variation of a linear combination of energy and angular momentum is used to investigate the formal stability of circular vortices. The analysis proceeds entirely in terms of Lagrangian displacements to overcome problems that otherwise arise when one attempts to use Arnol'd's Eulerian formalism. Specific attention is paid to the simplest possible model of an isolated vortex consisting of a core of constant vorticity surrounded by a ring of oppositely signed vorticity. We prove that the linear stability regime for this vortex coincides with the formal stability regime. The fact that there are formally stable isolated vortices could imply that there are provable nonlinearly stable isolated vortices. The method can be applied to more complicated vortices consisting of many nested rings of piecewise-constant vorticity. The equivalent expressions for continuous vorticity distributions are also derived.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 242 , September 1992 , pp. 249 - 278
Copyright
© 1992 Cambridge University Press

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