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

One way to reduce the computational cost of a spectral model using spherical harmonics (SH) is to use double Fourier series (DFS) instead of SH. The transform method using SH usually requires O (N-3) operations, where N is the truncation wavenumber, and the computational cost significantly increases at high resolution. On the other hand, the method using DFS requires only O (N-2 log N) operations. This paper proposes a new DFS method that improves the numerical stability of the model compared with the conventional DFS methods by adopting the following two improvements: a new expansion method that employs the leastsquares method (or the Galerkin method) to calculate the expansion coefficients in order to minimize the error caused by wavenumber truncation, and new basis functions that satisfy the continuity of both scalar and vector variables at the poles. Partial differential equations such as the Poisson equation and the Helmholtz equation are solved by using the Galerkin method. In the semi-implicit semi-Lagrangian shallow-water model using the new DFS method, the Williamson test cases and the Galewsky test case give stable results without the appearance of high-wavenumber noise near the poles, even without horizontal diffusion and without a zonal Fourier filter. In the Eulerian advection model using the new DFS method, the Williamson test case 1, which simulates a cosine bell advection, also gives stable results without horizontal diffusion but with a zonal Fourier filter. The shallow-water model using the new DFS method is faster than that using SH, especially at high resolutions and gives almost the same results, except that very small oscillations near the truncation wavenumber in the kinetic energy spectrum appear only in the shallow-water model using SH.