A complete set of two-dimensional harmonic vortices on a spherical surface

The solutions of the Euler equations on a spherical surface are constructed, starting from a vector velocity potential (A) over right arrow in the radial direction and with a two-dimensional spherical harmonic variation of order m and well-defined parity under phi bar right arrow -phi. The solutions are well-behaved on the entire surface and continuous at the position of a parallel circle theta = theta(0), where the vorticity is shown to be harmonically distributed. The velocity field is evaluated as the curl of the vector potential: it is shown that the velocity is divergenceless and distributed on the spherical surface. Its polar components at the parallel circle are shown to be continuous, confirming its divergenceless nature, while its azimuthal components are discontinuous at the circle, and their discontinuity is a measure of the vorticity in the radial direction. A closed form for the velocity field lines is also obtained in terms of fixed values of the scalar harmonic function associated with the vector potential. Additionally, the connections of the solutions on a spherical surface with their circular, elliptic and bipolar counterparts on the equatorial plane are implemented via stereographic projections.