An energy consistent summation-by-parts finite difference discretization for the non-hydrostatic atmospheric dynamics equations on the cubed-sphere grid

Abstract It is expected that atmospheric models with kilometer-scale horizontal resolution will become accessible for use in climate research in the near future. Such high-resolution models require efficient, accurate and conservative numerical solvers for the non-hydrostatic atmospheric dynamics equations on quasi-uniform spherical meshes. In this article, we address this problem by presenting a novel high-order accurate, energy-conserving spatial approximation for non-hydrostatic dynamics equations on the cubed-sphere mesh. The key novel feature of our work is the application of the Summation-By-Parts Finite-Difference (SBP-FD) method for horizontal approximation. SBP-FD enables high-order accurate and stable approximations on logically rectangular multiblock meshes, such as the cubed-sphere. The horizontal gradient and divergence operators constructed using SBP-FD satisfy the discrete analogue of the Ostrogradsky–Gauss theorem, which is essential for energy conservation. As compared to previous works, we explicitly present expressions for discrete vertical mass and entropy fluxes in height-based vertical coordinates, ensuring the compensation potential and thermodynamics energy tendencies with energy contributions from pressure gradient and gravity forces. The proposed spatial approximation is verified using a suite of widely accepted idealized test cases, the results are in good agreement with reference solutions. The robustness of presented spatial approximation is confirmed with the stability in long-term Held–Suarez experiment.