Point vortices on the sphere: stability of relative equilibria

2004/02/26 by Frédéric Laurent-Polz, Laurent-Polz, Frédéric, James Montaldi +3
Earth and Planetary Sciences · Engineering · Mathematics · #Aeolian processes and effects #Fluid Dynamics and Turbulent Flows #Fluid Dynamics and Vibration Analysis #math.DS #msc:37J15 #msc:37J20 #msc:37J25 #msc:70H33 #msc:76B47

paper · pdf · doi:10.48550/arxiv.math/0402430

47 pages, 40 figures

arxiv created 2011/06/03 · arxiv updated 2011/06/06

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.

Related

• Point vortices on a rotating sphere
• Determination of stable branches of relative equilibria of the N-vortex problem on the sphere
• Morse index and stability of the planar N-vortex problem
• Stability of Relative Equilibria in the Planar N-Vortex Problem
• Deformation of geometry and bifurcation of vortex rings