Double Fourier Series on a Sphere: Applications to Elliptic and Vorticity Equations

Abstract

The solution of elliptic and vorticity equations on a sphere is studied using double Fourier series as orthogonal basis functions. The basis functions incorporate sine series weighted by cosine latitude as meridional basis functions for even zonal wavenumbers other than zero to meet the pole condition. As to the solution of Poisson's equation, it is found that the new method gives improved accuracy compared to the method of Yee (Mon. Weather Rev.109, 501, 1981) due to the absence of constraints imposed on spectral coefficients with the operation number being slightly increased. The new method is applied to the vorticity equation along with the use of Fourier and spherical harmonics filters, and its accuracy is tested for the rotated Rossby–Haurwitz wave. It is shown that the basis functions adopted here provide high accuracy for all tests used. Numerical integration without spherical harmonics filters indicates that they are necessary for stable and accurate time integration. Comparison with the spherical harmonics model reveals that the present method is more accurate by a factor of order 112 for the test case. Further application to the advection equation is carried out. The error measure for the strong advection of the cosine bell with various rotation angles indicates that the present method is capable of producing accurate and stable calculations without the pole problem, suggesting that it could be applied to the numerical weather prediction model, including shallow water equations, without difficulty. Extension to the shallow water equations with accuracy tests as in Williamson et al. (J. Comput. Phys.102, 211, 1992) will be given in the future. Additional time could be saved by introducing the reduced grids near poles in the present method, besides the advantage of applying FFT to both longitudinal and latitudinal directions.