Improved double Fourier series on a sphere and its application to a semi-implicit semi-Lagrangian shallow-water model

Abstract. One way to reduce the computational cost of a spectral model using spherical harmonics (SH) is use double Fourier series (DFS) instead SH. The transform method SH usually requires O(N3) operations, where N truncation wavenumber, and significantly increases at high resolution. On other hand, DFS only O(N2log N) operations. This paper proposes new that improves numerical stability compared with conventional methods by adopting following two improvements: expansion employs least-squares (or Galerkin method) calculate coefficients in order minimize error caused wavenumber truncation, basis functions satisfy continuity both scalar vector variables poles. Partial differential equations such as Poisson equation Helmholtz are solved method. In semi-implicit semi-Lagrangian shallow-water method, Williamson test cases Galewsky case give stable results without appearance high-wavenumber noise near poles, even horizontal diffusion zonal filter. Eulerian advection 1, which simulates cosine bell advection, also gives but faster than SH, especially resolutions almost same results, except very small oscillations kinetic energy spectrum appear