A Dynamical Core with Double Fourier Series: Comparison with the Spherical Harmonics Method

Abstract

A dynamical core of a general circulation model with the spectral method using double Fourier series (DFS) as basis functions is presented. The model uses the hydrostatic balance approximation and sigma coordinate system in the vertical direction and includes no topography. The model atmosphere is divided into 25 layers with equal sigma depths. Prognostic equations for the vorticity, divergence, temperature, and logarithmic surface pressure are solved by the DFS spectral-transform method with the Fourier filtering at middle and high latitudes. A semi-implicit time-stepping procedure, which deals with the eigendecomposition and inversion of the 3D Helmholtz equation associated with the gravity wave terms, is incorporated for the gravity wave?related terms. The DFS model is tested in terms of the solution of the 3D Helmholtz equation, balanced initial state, developing baroclinic waves, and short- and long-term Held?Suarez?Williamson simulations for T42, T62, T84, and T106 resolutions. It is found that the DFS model is stable and accurate and produces almost the same results as the spherical harmonics method (SHM). The normalized difference (i.e., L2 norm error) measured from the results of highest-resolution SHM-T106 showed a desirable convergence of the DFS solution with the resolution. The convergence property, however, varies with the test case and prognostic variables. The total mass (or global integrated surface pressure) is conserved to a good approximation in the long-term simulations. Computing on the high-performance computer NEC SX-5 (parallel-vector architecture) indicated that DFS is more efficient than the SHM and the efficiency increases with the resolution, for example, by factors of 2.09 and 7.68 for T212 and T1022, respectively.