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

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, ∇2ψ=exp(-2ϵ2(1-cos(θ))-CGauss(ϵ)∇2ψ=exp(-2ϵ2(1-cos(θ))-CGauss(ϵ). (More precisely, the forcing is a Gaussian minus the “Gauss constraint constant”, CGaussCGauss; this subtraction is necessary because ψψ 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 ϵϵ. 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 ψψ is everywhere nice. The method of matched asymptotic expansions yields simpler, uniformly valid approximations as series of inverse even powers of ϵϵ that converge very rapidly for the large values of ϵ  (ϵ>40)(ϵ>40) appropriate for geophysical vortex computations. The series converges to a nonzero O(exp(-4ϵ2))O(exp(-4ϵ2)) error everywhere except at the south pole where it diverges linearly with order instead of the usual factorial order.