Application of Double-Fourier-series Spectral Method to a Large Size Problem: Two-dimensional Simulations of the Shear Instability on the Sphere

Abstract

Double-Fourier-Series (DFS) spectral method is applied to a large-size problem of barotropic instability of double-shear flow on the sphere. The computing source is the NEC SX-5 parallel vector processors, with the maximum vector length of 512. It is demonstrated that the DFS spectral model is robust and stable even for such a large-sized intensively nonlinear problem, and can simulate well the multiple scale phenomenon without losing accuracy. In addition to the efficiency on serial computing, represented with OðN 2 log 2 NÞ operations as opposed to OðN 3 Þ for the spherical harmonics spectral method, with N the truncation, the DFS spectral model also preserves the efficiency on parallel computing on vector architecture, due to its nature of two dimensional transform. The parallel performance increased slightly with the resolution, and nearly 33.5 percent (26.8 GFLOPS) of the theoretical peak performance (80 GFLOPS) was achieved in the highest-resolution experiment.Corresponding author: Hyeong-Bin Cheong, Dept. of Environmental Atmospheric Sciences, Pukyong National University, 599-1 Daeyeon 3-cong, Namgu, Pusan 608-737, Korea. E-mail: hbcheong@pknu.ac.kr ( 2004, Meteorological Society of Japan The zonal-mean absolute vorticity, of which initial condition is characterized as two peaks in both hemispheres, evolves with time into a nearly constant value over the hemisphere. On the other hand, the meridional gradient of the absolute vorticity increases around the equator. The kinetic energy per unit mass is calculated for each total wavenumber, where a disturbance field of a single total wavenumber is separated by an 8th-order spherical harmonics filter. Kinetic energy spectrum shows two distinct subranges, each with a constant slope. The subrange, other than the viscous subrange, shows a slightly increasing slope with time and approaches l À3 (l is the total wavenumber) in the matured stage, when a single large vortex is formed. As the resolution increases, the subrange other than the viscous subrange extends to the higher wavenumber domain, due to low viscosity. Numerical convergence of the solution with a fixed viscosity is discussed in terms of time averaged zonal-mean statistics of the zonal-flow.

Nonhydrostatic icosahedral atmospheric model (NICAM) for global cloud resolving simulations

“…Another problem occurs for the computation on a massively parallel computer: the spectral transform method requires extensive data movement between computer nodes. Although the double Fourier transformation method is proposed by [9][10][11][12] as an alternative, this method still requires global communication between computer nodes.…”

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

“…Figure 1 shows Grids [0], [1], and [−1] when 4 and the grid interval ∆ 4 ⁄ . Grid [0] has been widely used in DFS models, for example, in Merilees (1973b), Orszag (1974), Cheong (2000aCheong ( , 2000b, and Yoshimura and Matsumura (2005). Grid [1] was used, for example, in the DFS expansion in Yee (1981).…”

“…2 (hereafter the new DFS model). We also ran the same test cases in the semi-implicit semi-Lagrangian shallow water model using the DFS 10 method of Yoshimura and Matsumura (2005) with the basis functions of Cheong (2000aCheong ( , 2000b (hereafter the old DFS model), and in the model using the SH method (hereafter the SH model) for comparison. The new DFS model was run for each of Grid 128 and the number of latitudinal grid points 64.…”

“…5 permits discontinuity at the poles and nonisotropic waves, which may lead to a prohibitive timestep restriction and numerical instability, and these problems can be avoided by applying the spherical harmonic filter. 5 Cheong (2000aCheong ( , 2000b proposed expanding as…”

“…3 for odd , and does not guarantee the continuity of ∇ at the poles. The shallow water model and the vorticity equation model using a semi-implicit Eulerian scheme ran stably using high-order horizontal diffusion with O operations to smooth out the high-wavenumber components (Cheong, 2000b;Cheong et al, 2002;Kwon et al, 2004). The semi-implicit Eulerian hydrostatic atmospheric model also ran stably with high-order horizontal diffusion (Cheong, 2006;Koo and Hong, 2013;Park 15 et al, 2013).…”