Non-orthogonal version of the arbitrary polygonal C-grid and a new diamond grid

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