Analytical solutions for distributed multipolar vortex equilibria on a sphere

Analytical solutions of the steady Euler equations corresponding to stationary multipolar vortices on a sphere are derived. The solutions represent localized regions of distributed vorticity consisting of uniform vortex patches with a finite set of superposed point vortices. The mathematical method combines stereographic projection with conformal mapping theory to generalize a class of exact solutions for planar multipolar vortices developed by Crowdy [Phys. Fluids 11, 2556 (1999)] to the physically more important scenario of multipolar vortices on a spherical surface. The solutions are believed to be the first examples of analytical solutions of the Euler equations on a sphere involving patches of distributed vorticity with nontrivial shape.