Singular continuation of point vortex relative equilibria on the plane and sphere

ISSN： 0951-7715

Source： Nonlinearity, Vol.26, Iss.3, 2013-03, pp. : 777-804

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Previous Menu Next

Abstract

Singular continuations of relative equilibrium configurations of point vortices on the plane and sphere are proved to exist. These continuations are singular in the sense that two or more vortex positions coincide for certain values of the continuation parameter, and the properties of these continuations are relevant to the existence of positive-dimensional solution sets to the relative equilibrium equations. Several different scaling regimes are shown to exist, yielding clustered vortex configurations with different asymptotic properties. On the plane, the vortex strengths vary with the continuation parameter while the rotation rate is fixed; on the sphere, the situation is reversed. The configurations so generated on the sphere are among the few known relative equilibria that do not rely on symmetry for their construction.