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

• Authors: John P. Boyd;Cheng Zhou
• Venue: Journal of Computational Physics
• Year: 2009

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@j=exp(-2@e^2(1-cos(@q))-C^G^a^u^s^s(@e). (More precisely, the forcing is a Gaussian minus the ''Gauss constraint constant'', C^G^a^u^s^s; this subtraction is necessary because @j 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 @e. 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 @j is everywhere nice. The method of matched asymptotic expansions yields simpler, uniformly valid approximations as series of inverse even powers of @e that converge very rapidly for the large values of @e(@e40) appropriate for geophysical vortex computations. The series converges to a nonzero O(exp(-4@e^2)) error everywhere except at the south pole where it diverges linearly with order instead of the usual factorial order.