Stability analyses of divergence and vorticity damping on gnomonic cubed-sphere grids

Abstract

Divergence and vorticity damping, which operate upon horizontal divergence and relative vorticity, are explicit diffusion mechanisms used in dynamical cores to ensure stability. To avoid numerical blow-up from excessively strong diffusion, there are mesh-dependent upper bounds on the coefficients of the diffusion operators. This work considers such stability limits for three gnomonic cubed-sphere meshes: the 1) equidistant, 2) equiangular, and 3) equi-edge mappings. Stability limits are derived from a von Neumann analysis of damping with a simplified pseudo-Laplacian operator, as used in NOAA GFDL's finite-volume dynamical core on the cubed-sphere (FV3), and with the full curvilinear Laplacian. The resulting stability limits depend on the gnomonic mapping through the cubed-sphere cell areas, aspect ratios, and grid nonorthogonality. The analytical stability limits are compared to practical divergence and vorticity damping upper bounds in FV3, using idealised tests and the equiangular and equi-edge grids. For divergence damping, both the magnitude of maximum stable coefficients and the locations of instability agree with linear theory. Due to implicit vorticity diffusion in the FV3 transport scheme, practical limits for vorticity damping are lower than the explicit stability limits and depend on the choice of horizontal transport scheme.

Figures (10)

• Figure 1: Diagrams of the Arakawa C- and D-grids, showing the horizontal wind ($u,v$) components (red) in the top-left cell. The C-grid stores normal wind components on the cell edges, leading to divergence at the cell centre and vorticity at the cell corners. The D-grid stores tangential wind components on the cell edges, leading to vorticity at the cell centre and divergence at the cell corners. The primary grid is shown in solid black, with the offset grid in dashed lines formed by connecting cell centres on the primary grid; an offset D-grid is generated on the C-grid, and an offset C-grid on the D-grid.
• Figure 2: Cell areas, computed numerically with (\ref{['eq:spherical_excess_area']}), for the cubed-sphere (primary) grids at a C192 resolution. The equidistant grid has the largest variation in areas, and the equiangular grid has the narrowest. The smallest cells for the equidistant and equi-edge grids are at the panel corners, whereas the smallest equiangular cells are at the middle of the panel edges.
• Figure 3: Cell aspect ratios of $\chi = \Delta y/ \Delta x$ for the cubed-sphere (primary) grids at a C192 resolution. The largest/smallest aspect ratios are at the middle of the panel edges for all grids. As the equi-edge grid prioritises more uniform cells along the panel edges, it has the narrowest range of aspect ratios. The maximum aspect ratio for the equidistant and equiangular grids is close to $\sqrt{2}$, but not exactly so, due to the even number of cells on the primary grid, i.e. the middle cells are offset from the centre by $\Delta x/2$ or $\Delta y/2$.
• Figure 4: The grid stability function of $\widetilde{\Psi}$ evaluated on the offset grid for the three gnomonic cubed-sphere grids and a C192 resolution. We focus on its minimum value, $\widetilde{\Psi}_{\text{min}}$, which dictates the diffusive stability limit through (\ref{['eq:damp_stab_limit']}). A smaller $\widetilde{\Psi}_{\text{min}}$ for the equiangular grid (the lowest values are shown in white) corresponds to a stricter upper bound on the damping coefficient. $\widetilde{\Psi}_{\text{min}}$ is located at the smallest cells of each grid (Fig. \ref{['fig:cubed_sphere_areas']}); this is the middle of the panel edges for the equiangular grid, and the panel corners for the equidistant and equi-edge grids.
• Figure 5: Amplification factors of $\widetilde{\Gamma}_{2q}$ against normalised wavenumber for the different orders of damping to show the impact of $q$ on scale selectivity. The oscillation-free stability coefficient (\ref{['eq:stab_limit_osc_free']}) and a C192 resolution are used. In contrast to the hyperviscosities, Laplacian diffusion significantly damps a wider range of wavenumbers. With increasing order, the damping becomes increasingly scale selective, with a smaller range of wavenumbers being noticeably diffused.