β Scribed by John P. Boyd; Cheng Zhou
No coin nor oath required. For personal study only.
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, r 2 w ΒΌ expΓ°Γ2 2 Γ°1 Γ cosΓ°hΓΓ Γ C Gauss Γ°Γ. (More precisely, the forcing is a Gaussian minus the ''Gauss constraint constant", C Gauss ; this subtraction is necessary because w 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 . 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 w is everywhere nice. The method of matched asymptotic expansions yields simpler, uniformly valid approximations as series of inverse even powers of that converge very rapidly for the large values of Γ° > 40Γ appropriate for geophysical vortex computations. The series converges to a nonzero OΓ°expΓ°Γ4 2 ΓΓ error everywhere except at the south pole where it diverges linearly with order instead of the usual factorial order.
π SIMILAR VOLUMES
A general vibrational model of a continuous system with arbitrary linear and cubic operators is considered. Approximate analytical solutions are found using the method of multiple scales. The primary resonances of the external excitation and three-to-one internal resonances between two arbitrary nat