Streamline topologies for integrable vortex motion on a sphere

Abstract

We describe the instantaneous streamline patterns that occur on the surface of a two-dimensional sphere in the presence of point vortices of general strength for cases in which the system is completely integrable. After stating some general results based on the spherical topology, we categorize all possible instantaneous streamline patterns and describe their stagnation point structure for the cases of two and three vortices. It is found that for the case of two vortices, the only non-degenerate topologies that can arise are a figure eight (lemniscate) or a limacon, which are homotopically equivalent. For the case of three vortices, there are 12 topologically distinct primitives, from which an additional 23 patterns can be produced via continuous deformations on the sphere (homotopies). All possible streamline patterns that arise from three vortex motion can be obtained via linear superposition of the primitive topologies and their homotopic equivalents. In this sense, the primitives can be viewed as the ‘building blocks’ for the general flow patterns. The equations of motion and streamline patterns in the stereographic plane are obtained using the projected equations of motion in Hamiltonian form. We describe streamline patterns for three vortex fixed equilibria and relative equilibria as seen in both a fixed and a rotating frame of reference. Non-equilibrium streamline patterns for a collapsing and special periodic solution are studied, although to generally understand and classify the many bifurcations associated with non-equilibrium patterns would require more extensive computation. We conclude with a discussion of the relevance of the three vortex topological classification scheme and the bifurcation of streamline topologies for understanding global atmospheric weather patterns (spherical isobars) and large scale mixing phenomena.