Application of double Fourier series to the shallow-water equations on a sphere

The spectral transform method using a double Fourier series as orthogonal basis functions as in Cheong (2000, J. Comput. Phys. 157, 327) is extended to the solution of the shallow-water equations on a sphere. A spectral filter which mimics the implicit diffusion process with the third-order Laplacian operator is applied to the spectral components of predicted variables to prevent the aliasing error or nonlinear instability. For a predicted variable the spectral filter needs only about 76N(2) operations with N being the zonal and meridional wavenumber truncation. The use of the filter even at every time step does not deteriorate the computational efficiency of the double-Fourier-series model, which comes from the availability of FFTs. The filter requires an additional memory for only 6N(2) elements, so the total memory space of O(N-2) is sufficient in the present model. Along with the incorporation of the polar filter, the semi-implicit time-stepping procedure contributes to a significant increase in the time-step size. A standard test set proposed by Williamson et al. (1992, J. Comput. Phys. 102, 211) is used to evaluate the errors associated with the new method for various resolutions. It is shown that as a whole the accuracy of the method is comparable to that of spherical harmonics model (SHM) though the present method provides more accurate time integration for some cases but does not for other cases. A long time-integration far beyond the period specified in the standard test set also illustrates almost the same accuracy as that given by the SHM. The relative efficiency of the method to the SHM appears from the resolution of 256 x 128 transform grids, and it becomes significant for resolutions higher than 512 x 256. The computational efficiency is expected to increase further with an improved FFT algorithm. The test results suggest that the new method could be extended to three-dimensional numerical models used for weather prediction. (C) 2000 Academic Press.