Vortical structures on spherical surfaces

A nonlinear elliptic partial differential equation (pde) is obtained as a generalization of the planar Euler equation to the surface of the sphere. A general solution of the pde is found and specific choices corresponding to Stuart vortices are shown to be determined by two parameters λλ and NN which characterizes the solution. For λ=1λ=1 and N=0N=0 or N=−1N=−1, the solution is globally valid everywhere on the sphere but corresponds to stream functions that are simply constants. The solution is however non-trivial for all integral values of N≥1N≥1 and N≤−2N≤−2. In this case, the solution is valid everywhere on the sphere except at the north and south poles where it exhibits point-vortex singularities with equal circulation. The condition for the solutions to satisfy the Gauss constraint is shown to be independent of the value of the parameter NN. Finally, we apply the general methods of Wahlquist and Estabrook to this equation for the determination of (pseudo) potentials. A realization of this algebra would allow the determination of Bäcklund transformations to evolve more general vortex solutions than those presented in this paper.