Vortical structures on spherical surfaces

Abstract

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 $\lambda$ and $N$ which characterizes the solution. For $\lambda =1$ and $N=0$ or $N=-1$, 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 $N\ge 1$ and $N\le -2$. 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 $N$. 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.