Abstract
Abstract. Quasi-uniform grids of the sphere have become popular recently since they avoid parallel scaling bottlenecks associated with the poles of latitude–longitude grids. However quasi-uniform grids of the sphere are often non-orthogonal. A version of the C-grid for arbitrary non-orthogonal grids is presented which gives some of the mimetic properties of the orthogonal C-grid. Exact energy conservation is sacrificed for improved accuracy and the resulting scheme numerically conserves energy and potential enstrophy well. The non-orthogonal nature means that the scheme can be used on a cubed sphere. The advantage of the cubed sphere is that it does not admit the computational modes of the hexagonal or triangular C-grids. On various shallow-water test cases, the non-orthogonal scheme on a cubed sphere has accuracy less than or equal to the orthogonal scheme on an orthogonal hexagonal icosahedron. A new diamond grid is presented consisting of quasi-uniform quadrilaterals which is more nearly orthogonal than the equal-angle cubed sphere but with otherwise similar properties. It performs better than the cubed sphere in every way and should be used instead in codes which allow a flexible grid structure.
Highlights
Quasi-uniform grids of the sphere have become popular recently since they avoid parallel scaling bottlenecks associated with the poles of latitude–longitude grids
The potential vorticity (PV) at the edge, qe, is interpreted as the PV at the primal and dual edges. It is interpolated from surrounding qv values from an upwind-biased stencil using CLUST which was developed for mapping PV from vertices to edges of the polygonal C-grid (Weller, 2012)
Unlike the scheme of Thuburn et al (2014), the new scheme does not rely on the dual grid being centroidal
Summary
Quasi-uniform grids of the sphere have become popular recently since they avoid parallel scaling bottlenecks associated with the poles of latitude–longitude grids. The predominant groups of quasi-uniform grid are hexagonal icosahedral, triangular icosahedral and cubed sphere (Weller et al, 2009). The hexagonal C-grid has become popular since Thuburn (2008), Thuburn et al (2009) and Ringler et al (2010) worked out how to calculate the Coriolis term so as to get steady geostrophic modes. A more efficient discretisation would have the correct ratio of DOFs and would not need to control spurious behaviour in the excess DOFs. The correct ratio of DOFs can be achieved by using grids of quadrilaterals, such as the cubed-sphere grid.
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