3D diffusion in terrain‐following coordinates: testing and stability of horizontally explicit, vertically implicit discretizations. (7th June 2016) - Baldauf, Michael; Brdar, Slavko
Abstract : A numerical discretization of the three‐dimensional (3D) diffusion equation for the scalar case and the vector case (i.e. for the momentum equation) in terrain‐following coordinates on the sphere is described. The discretization uses the horizontally explicit–vertically implicit (HE–VI) approach, which is often applied in atmospheric simulation models. Firstly, a spatially second‐order discretization is proposed, which treats the metric terms of the terrain‐following coordinates in a stable manner even for very steep terrain. A von Neumann stability analysis calculates the maximum stable diffusion Courant number for different implicitness weights as a function of slope angle and grid anisotropy. Secondly, simple analytic solutions of the diffusion equation for both scalar and vector cases are proposed for testing and validation purposes. The implementations in the two atmospheric model systems Icosahedral Non‐hydrostatic (ICON) (global) and Consortium for Small‐scale Modelling (COSMO) (regional) are compared against these exact solutions.