On The Spherical Yin-Yang Grid And Application Of A Novel Multi-moment Finite Volume Method

Currently the latitude-longitude (LAT/LON) spherical grid is widely adoptedin the most atmospheric general circulation model (GCM) since it is simple todesign and well orthogonal for advection computation. However, there are twonumerical problems in the poles: the coordinate singularity and the grid conver-gence near the poles. In fact, the coordinate singularity is not a really physicalsingularity, but just a computational singularity. The problem of the grid con-vergence is more serious. In the high-resolution case these problems become notonly so difficult, but the new essential physical problems appear: the physicalfluid dynamics problems represented in the equatorial and polar grid scale aredifferent.The Yin-Yang grid, which is a new overlapping grid system in sphericalgeometry with two perpendicularly-oriented latitude-longitude grid components(called Yin and Yang respectively) and effectively avoids the coordinate singular-ity and the grid convergence near the poles, is suggested. After the transferringdata between the Yin and Yang components are successfully solved, the semi-Lagrangian advection scheme is succesfully implemented on the Yin-Yang grid inthe paper. Numerical experiments show that the numerical accuracy in the origi-nal LAT/LON semi-Lagrangian scheme is effectively maintained in the Yin-Yanggrid.However, the Yin-Yang grid brings the extra numerical problem in the over-lapping zone. For the sake of the precision and conservation in the Yin-Yanggrid, a novel unified numerical formulation for high order, conservative compu-tational fluid dynamics method—multi-moment finite volume method, is firstlyintroduced into the Yin-Yang spherical grid. Different from the traditional fi-nite volume method, the high order interpolation can be spatially constructedover single grid cell. The one-dimensional, two-dimensional shallow water nu-merical tests indicate that the multi-moment finite volume method is high order,compact, conservative numerical algorithm. As a result, the combination of themulti-moment finite volume method and the semi-Lagrangian scheme makes the overlapping zone between Yin and Yang components minimal. Under the condi-tions of keeping the Yin/Yang component grid conservative, the present shallowwater model tests in the Yin-Yang spherical grid suggest that it has competi-tive numerical accuracy and computational efficiency even in the presence of fastwave, and that the conservation in total mass is adequate for at least middle termweather predictions or short term climate simulations. In spite of the rigorousconservativeness across the Yin-Yang boundary that requires further investiga-tion, the numerical techniques presented in this paper constitute a promising andpractical numerical framework to develop atmospheric and oceanic GCMs in thespherical geometry.