A Global Shallow Water Model on the Spherical-Cubic Grid by using CIP/Multi-Moment FVM

A global numerical model for shallow water flows on the spherical-cubic grid is proposed in this paper. The model is constructed by using the CIP/Multi-Moment finite volume method [1]. Different from the conventional finite volume methods, a CIP/Multi-Moment finite volume method usually uses at least two kinds of moments as the prognostic variables. In present study, the point value (PV) and the volume-integrated average (VIA) are adopted. The volume-integrated average is defined for each element same to the traditional finite volume method. Being an additional moment, the point values are defined on the points located at the element’s boundaries. By using the multi-moment concept, the available degrees of freedom are increased, and high-order numerical schemes can be constructed within very compact stencils. Moreover, different moments can be independently advected by different numerical formulations during the computation. In this paper, for example, we use the Lax-Friedrichs upwind splitting to update the PV moment in terms of a derivative Riemann problem. Integrating the governing equations over each element leads to a finite volume formulation for predicting the VIA moment. The numerical fluxes on the element boundaries are calculated directly from the PV moment. The numerical conservation of VIA of any conservative variable is exactly assured. The time marching is accomplished by the multi-step Runge-Kutta scheme. In order to get around the polar singularity and to obtain uniform grid spacing for a spherical geometry, we make use of a gnomonic spherical-cubic grid [2], which results in twelve boundaries separating six identical patches on the sphere. Highly localized reconstruction in CIP/Multi-Moment finite volume method is well suited for the spherical-cubic grid, especially in dealing with the discontinuity in the coordinates between different patches. The mass conservation is completely achieved over the whole globe. The numerical model has been verified by Williamson’s standard test set [3] for shallow water equation model on sphere. The results reveal that the present model is competitive to most existing ones and is a promising framework for global geophysical fluid dynamics.