Shallow-water equations on a spherical multiple-cell grid

The shallow-water equations (SWEs) are discretized on a spherical multiple-cell (SMC) grid and validated with classical tests, including steady zonal flow over flat or hill floor, the Rossby-Haurwitz wave and the unstable zonal jet. The numerical schemes follow the conventional latitude-longitude (lat-lon) grid ones at low latitudes but switch to a fixed-reference direction system to define vector variables at high latitudes for reduction of polar curvature errors. Semi-implicit schemes are used for both the Coriolis terms and the potential energy gradients. A C-grid mass-conserving advection-diffusion scheme is used for the thickness variable and mixed A- and D-grid schemes are applied on the momentum equation. Tests demonstrate that the two reference directions work fine on the SMC grid and the SWEs model is stable as long as numerical noises are suppressed with enough smoothing. This implies that other reduced grids could be used for dynamical models if the vector polar problem is properly solved. The unstructured SMC grid also supports multi-resolutions in mesh refinement style and a refined area is included to show grid refinement effects. For steady smooth flows, the refinement is almost unnoticeable while in unstable jet case, it may stir up instability ripples in the jet flow unless strong average is applied.