The Journal of Geometric Mechanics (JGM)

Point vortices on the sphere: Stability of symmetric relative equilibria

Pages: 439 - 486, Volume 3, Issue 4, December 2011      doi:10.3934/jgm.2011.3.439

       Abstract        References        Full Text (2017.1K)       Related Articles       

Frederic Laurent-Polz - Institut Non Linéaire de Nice, 1361 route des Lucioles, 06560 Valbonne, France (email)
James Montaldi - School of Mathematics, University of Manchester, Manchester, M13 9PL, United Kingdom (email)
Mark Roberts - Department of Mathematics, University of Surrey, Guildford GU2 7XH, United Kingdom (email)

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.

Keywords:  Point vortices, Hamiltonian systems, stability, bifurcations, symmetry.
Mathematics Subject Classification:  Primary: 37J15, 37J20, 37J25, 70H33, 76B47.

Received: March 2011;      Revised: May 2011;      Available Online: February 2012.