Vortex dynamics in complex domains on a spherical surface

We consider the motion of both point vortices and uniform vortex patches in arbitrary, possibly multiply connected, regions bounded by impenetrable walls on the surface of a sphere. By exploiting knowledge of the functional form of the relevant Green’s function in a pre-image circular domain that is conformally equivalent to a stereographic projection of the fluid domain on the spherical surface, we first generalize Kirchhoff–Routh theory for point vortex motion in the plane to point vortex motion on a spherical shell. Next, we study vortex patch motion and show that there is a contour dynamics formulation for the evolution of uniform vortex patches in any finitely connected domain on a spherical shell bounded by impenetrable walls. We describe a novel numerical scheme whereby this motion can be computed. Some illustrative calculations are shown.