Three ways to solve the Poisson equation on a sphere with Gaussian forcing

Abstract

Motivated by the needs of vortex methods, we describe three different exact or approximate solutions to the Poisson equation on the surface of a sphere when the forcing is a Gaussian of the three-dimensional distance, ∇ 2 ψ = exp ( - 2 ϵ 2 ( 1 - cos ( θ ) ) - C Gauss ( ϵ ) . (More precisely, the forcing is a Gaussian minus the “Gauss constraint constant”, C Gauss ; this subtraction is necessary because ψ is bounded, for any type of forcing, only if the integral of the forcing over the sphere is zero [Y. Kimura, H. Okamoto, Vortex on a sphere, J. Phys. Soc. Jpn. 56 (1987) 4203–4206; D.G. Dritschel, Contour dynamics/surgery on the sphere, J. Comput. Phys. 79 (1988) 477–483]. The Legendre polynomial series is simple and yields the exact value of the Gauss constraint constant, but converges slowly for large ϵ . The analytic solution involves nothing more exotic than the exponential integral, but all four terms are singular at one or the other pole, cancelling in pairs so that ψ is everywhere nice. The method of matched asymptotic expansions yields simpler, uniformly valid approximations as series of inverse even powers of ϵ that converge very rapidly for the large values of ϵ ( ϵ > 40 ) appropriate for geophysical vortex computations. The series converges to a nonzero O ( exp ( - 4 ϵ 2 ) ) error everywhere except at the south pole where it diverges linearly with order instead of the usual factorial order.