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 O(N² log2 N) operations as opposed to O(N³) 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. 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-3 (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.