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

• Published in: Geoscientific Model Development Vol. 15; no. 6; pp. 2561 - 2597
• Main Author: Yoshimura, Hiromasa
• Format: Journal Article
• Language: English
• Published: Katlenburg-Lindau Copernicus GmbH 28-03-2022; Copernicus Publications
• Subjects: Accuracy; Advection; Atmospheric models; Basis functions; Coefficients; Computational efficiency; Computer applications; Computing costs; Continuity (mathematics); Differential equations; Diffusion; Energy spectra; Fluid dynamics; Fourier series; Galerkin method; Helmholtz equations; Horizontal diffusion; Kinetic energy; Least squares method; Modelling; Numerical stability; Oscillations; Partial differential equations; Physical simulation; Poisson equation; Poles; Shallow water; Spherical harmonics; Thermal expansion; Variables; Water purification; Wavelengths
• ISSN: 1991-9603; 1991-959X; 1991-962X; 1991-9603; 1991-962X
• DOI: 10.5194/gmd-15-2561-2022

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(N3) 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(N2log 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 least-squares 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.