Point vortices on the sphere:  Stability of symmetric relative equilibria

Frederic Laurent-Polz; James Montaldi; Mark Roberts

doi:10.3934/jgm.2011.3.439

Article Contents

2011, Volume 3, Issue 4: 439-486. Doi: 10.3934/jgm.2011.3.439

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Point vortices on the sphere: Stability of symmetric relative equilibria

1.
Institut Non Linéaire de Nice, 1361 route des Lucioles, 06560 Valbonne
2.
School of Mathematics, University of Manchester, Manchester, M13 9PL
3.
Department of Mathematics, University of Surrey, Guildford GU2 7XH

Received: March 2011

Revised: May 2011

Published: December 2011

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Abstract

We describe the linear and nonlinear stability and instability of certain symmetric configurations of point vortices on the sphere forming relative equilibria. These configurations consist of one or two rings, and a ring with one or two polar vortices. Such configurations have dihedral symmetry, and the symmetry is used to block diagonalize the relevant matrices, to distinguish the subspaces on which their eigenvalues need to be calculated, and also to describe the bifurcations that occur as eigenvalues pass through zero.

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Mathematics Subject Classification: Primary: 37J15, 37J20, 37J25, 70H33, 76B47.

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