Inspection of hexagonal and triangular C-grid discretizations of the shallow water equations

C-grid discretizations based on a hexagonal or triangular mesh can be investigated with the help of a planar trivariate coordinate system, where the vector components are either defined tangentially (hexagonal C-grid) or perpendicularly (triangular C-grid) to the coordinate lines. Inspecting the Helmholtz decomposition of a vector in case of the linearly dependent trivariate coordinate description reveals insights into the structure and stencil of the discretized divergence and vorticity on such grids. From a vector Laplacian, which is consistent with the Helmholtz decomposition, a general formulation for the inner product operator at grid edges can be derived. Thus, the vector reconstruction of the tangential wind for the Coriolis term in the shallow water equations can be given even for a slightly distorted tesselation as present on icosahedral spherical grids. rmore, a rigorous comparison of the triangular and the hexagonal C-grid linear shallow water equations is performed from a theoretical and also from an experimental viewpoint. It turns out that the additional degree of freedom in the height field in the triangular C-grid case compared to the hexagonal C-grid is responsible for the decoupling of divergence values on upward and downward directed triangles. This problem occurs especially for small Rossby deformation radii, and practically requires additional explicit diffusion. In contrast, the hexagonal C-grid discretization has remarkable similarity to the quadrilateral C-grid case in the wave dispersion properties and eigenvector structure. Numerical experiments performed with that option proved resilent.