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

TL;DR

This paper derives and compares stability limits for divergence and vorticity damping on different gnomonic cubed-sphere grids, informing stable parameter choices in atmospheric models.

Contribution

It provides analytical stability bounds for divergence and vorticity damping on three gnomonic cubed-sphere meshes, validated with numerical tests in NOAA's FV3 model.

Findings

Analytical stability limits depend on grid geometry and mapping.

Maximum stable damping coefficients match linear theory for divergence.

Practical vorticity damping limits are lower due to implicit diffusion and transport schemes.

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.