An explicit flux‐form semi‐lagrangian shallow‐water model on the sphere

A global shallow-water model based on the flux-form semi-lagrangian scheme is described. the mass-conserving flux-form semi-Lagrangian scheme is a multidimensional semi-Lagrangian extension of the higher order Godunov-type finite-volume schemes (e.g., the piece-wise parabolic method). Unlike the piece-wise parabolic methodology, neither directional splitting nor a Riemann solver is involved. A reverse engineering procedure is introduced to achieve the goal of consistent transport of the absolute vorticity and the mass, and hence, the potential vorticity. Gravity waves are treated explicitly, in a manner that is consistent with the forward-in-time flux-form semi-Lagrangian transport scheme. Due to the finite-volume nature of the flux-form semi-lagrangian scheme and the application of the monotonicity constraint, which can be regarded as a subgrid-scale flux parametrization, essentially noise-free solutions are obtained without additional diffusion. Two selected shallow-water test cases proposed by Williamson et al. (1992) and a stratospheric vortex erosion simulation are presented. Discussions on the accuracy and computational efficiency are given based on the comparisons with a Eulerian spectral model and two advective-form semi-implicit semi-Lagrangian models.

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