Global shallow water models based on multi-moment constrained finite volume method and three quasi-uniform spherical grids

This is a review article to present several accurate and computationally efficient global models for shallow water equations recently developed under a general numerical framework, the multi-moment constrained finite volume (MCV) method. The multi-moment constrained finite volume method defines the unknowns (prognostic variables) as the point values at the solution points located over each mesh element. The time evolution equations to update these unknowns are derived through the constraint conditions on different moments, e.g. the point value (PV) and the volume-integrated average (VIA). Rigorous numerical conservation is guaranteed by the constraint on the VIA moment through a finite volume formulation of flux form. The resulted numerical schemes are very simple, efficient and easy to implement for both structured and unstructured grids. We have implemented the MCV method to three major spherical grids, Yin-Yang overset grid, cubed-sphere grid and geodesic icosahedral grid, which have overall quasi-uniform grid spacings and are highly popular in the community of global modeling for atmospheric and oceanic dynamics. In this paper, we present the global shallow water models based on these three spherical grids and the third-order MCV scheme. We evaluate and compare the models by widely used benchmark tests, which show the third-order convergence rate for all models, and the numerical results are competitive to other exiting models. Using MCV method as a numerical formulation is well-balanced between solution quality and computational simplicity, the proposed models provide accurate and practical bases for developing dynamic core of general circulation models on different spherical grids.