Discretization of generalized Coriolis and friction terms on the deformed hexagonal C‐grid

Abstract

This article discusses the generalized Coriolis and friction terms on the hexagonal C‐grid from two perspectives: (a) within the linearized discretized momentum equations on an equilateral grid, and (b) as nonlinear terms on a distorted mesh.The discrete linearized momentum equations are formulated using a trivariate coordinate system. The tendencies of the different forcing terms for each wind component must be linearly dependent. This constraint determines unique discretizations for each term. The linearized vorticity flux term around a zonal mean current requires only the four rhombus potential vorticities (PVs) next to an edge. The vector Laplacian must be formulated with the vorticity on vertices defined as the average of three rhombus vorticities.A modified generalized Coriolis term is defined on the deformed mesh. The baroclinic wave test on the sphere does not reveal any sign of nonlinear Hollingsworth instability, even though it is demonstrated that the vector‐invariant form and the advective form of momentum advection are not equivalent.Physical constraints determine the shape of the stress tensor. These are invariance to the addition of solid‐body rotation and a resulting positive‐definite dissipation rate. An appropriate stress‐tensor formulation does not deliver Laplacian momentum diffusion in the linear case. On the deformed mesh, parts of this stress tensor are obtained by a least‐squares reconstruction of wind gradients. This approach avoids spurious deformations diagnosed for constant flow in the vicinity of pentagon cells.