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 blowup 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 Geophysical Fluid Dynamics Laboratory’s (GFDL) 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 idealized tests and the equiangular and equi-edge grids. For divergence damping, both the magnitude of the 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. Significance Statement Many weather and climate models require the removal of energy at the small scales for a stable computational simulation. One way to achieve stability is through explicit diffusion, and two such diffusion mechanisms we examine are the damping of divergent and rotational horizontal motions. However, explicit diffusion needs to be applied carefully, as making it too strong will instead introduce a diffusive instability. This work derives theoretical limits to the strength of damping the divergent and rotational motions. The analysis applies to a type of cubed-sphere model, where the computational grid is obtained by projecting straight lines from the six faces of a reference cube onto great circles on the Earth.