Velocity field within a vortex ring with a large elliptical cross section

The velocity field within a steady toroidal vortex is found for arbitrary mean core radius and section ellipticity. The problem is solved by transforming to coordinates that define invariant sets. The method allows the properties of the coordinate system metric tensor to be exploited in the continuity equation in order to obtain the solution. The vorticity is found to decrease monotonically with distance from the symmetry axis. For a given outer radius and outer perimeter velocity, the circulation of the vortex ring can be either smaller or larger than that of Hill's spherical vortex.