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

The chaotic advection of tracer particles in the field of a perturbed latitudinal ring of point vortices on a sphere is considered. We consider a restricted four-vortex problem where three vortices have equal strength, while the fourth has strength zero. The equal-strength vortices are initially spaced evenly 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 perturbed, the vortex motion induces chaotic particle advection analyzed by means of stroboscopic Poincaré 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 plays in the mixing and transport of particles. The geometric extent of the mixing region on the full sphere is considered (measured as a percentage of the surface area of the sphere) and dynamical properties in the region, such as mixing and stretching rates as well as computational evidence of ergodicity, are obtained. Global mixing on the sphere does not seem to increase monotonically with energy, but appears to be maximized for values near critical.