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

Abstract 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.