Chaotic advection in the restricted four-vortex problem on a sphere

Shane D. Ross
Engineering Science and Mechanics, Virginia Polytechnic Institute and State University

(with P K Newton)


The chaotic advection of tracer particles in the field of a perturbed latitudinal ring of point vortices on a sphere is considered. We focus on a configuration of three identical point vortices initially evenly spaced on a ring of fixed latitude in the northern hemisphere. The equilateral triangle formed by the vortices is known to be a nonlinearly stable relative equilibrium configuration. When nonlinearly perturbed, the vortex motion induces chaotic particle advection which is analyzed by means of stroboscopic Poincare maps as a function of the dimensionless energy of the system, which can be related to the size of the perturbation from equilibrium. A critical energy is identified which separates the vortex motion into two distinct dynamical regimes. For energies below critical, the vortices undergo periodic partner exchange while retaining their relative orientation. For values above critical, the relative orientation of the vortices changes throughout the periodic cycle. We consider how the streamline topologies bifurcate both as a function of the energy and during the course of their evolution, as well as the role that the evolution of instantaneous streamline structures play in the mixing and transport of particles. Of particular interest is the extent of the mixing region generated by the perturbed ring, both in the northern hemisphere where the vortices reside, and in the southern hemisphere. The geometric extent of the mixing region on the full sphere is considered, and computational evidence of ergodicity in this region is obtained. Global mixing on the sphere does not seem to increase monotonically with energy, but appears to be maximized for values near critical.

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