Point vortex motion on a sphere with solid boundaries

We consider the motion of a point vortex on the surface of a sphere with solid boundaries. This problem is of interest in oceanography, where coherent vortex structures can persist for long times, and move over distances large enough so that the curvature of the Earth becomes important (see Gill [1982], Chaos [1994]). In this context, the boundary is a first step in modeling the presence of coastlines and shores using inviscid theory. Using the equations of motion for the vortex projected onto the stereographic plane, we construct the appropriate Green’s function using classical image method ideas, as long as the domain has certain symmetry properties. After the solution is obtained in the stereographic plane, it is projected back down to the sphere, yielding the sought after solution to the problem. We demonstrate the utility of the method by solving for the vortex trajectories and streamlines for several canonical examples, including a spherical cap, longitudinal wedge, half longitudinal wedge, channel, and rectangle. The results are compared with the corresponding ones in the physical plane in order to highlight the effect of the spherical geometry.