Numerical representation of geostrophic modes on arbitrarily structured C-grids

Abstract

a b s t r a c tA C-grid staggering, in which the mass variable is stored at cell centers and the normal velocity component is stored at cell faces (or edges in two dimensions) is attractive for atmospheric modeling since it enables a relatively accurate representation of fast wave modes. However, the discretization of the Coriolis terms is non-trivial. For constant Coriolis parameter, the linearized shallow water equations support geostrophic modes: stationary solutions in geostrophic balance. A naive discretization of the Coriolis terms can cause geostrophic modes to become non-stationary, causing unphysical behaviour of numerical solutions. Recent work has shown how to discretize the Coriolis terms on a planar regular hexagonal grid to ensure that geostrophic modes are stationary while the Coriolis terms remain energy conserving. In this paper this result is extended to arbitrarily structured C-grids. An explicit formula is given for constructing an appropriate discretization of the Coriolis terms. The general formula is illustrated by showing that it recovers previously known results for the planar regular hexagonal C-grid and the spherical longitude-latitude C-grid. Numerical calculation confirms that the scheme does indeed give stationary geostrophic modes for the hexagonal-pentagonal and triangular geodesic C-grids on the sphere.

Voronoi Tessellations and Their Application to Climate and Global Modeling

“…Centroidal Voronoi tessellations are particularly well-suited for the generation of smoothly varying meshes, thus providing a possible alternative to traditional nesting approaches. With the recent discovery of a class of finite-volume methods that are directly applicable to variable resolution meshes ( (Thuburn et al, 2009;Ringler et al, 2010)), it appears that the creation of variable resolution, global climate system models is now possible.…”

“…C-grid staggering has shown promising results, particularly when applied to variable resolution meshes. See Check: Ringler Dyncore Chapter for a broad discussion of C-grid staggerings and see (Thuburn et al, 2009;Ringler et al, 2010) for an in-depth discussion of C-grid staggering applied to the nonlinear shallow-water equations.…”

“…We apply the methods developed in (Thuburn et al, 2009;Ringler et al, 2010) to test case 5 of the standard shallow-water test cases developed in (Williamson et al, 2001). A flow in geostrophic balance is confronted with a large-scale orographic feature at the start of the simulation, t = 0.…”

DYNAMICO-1.0, an icosahedral hydrostatic dynamical core designed for consistency and versatility

“…A partial generalization has been achieved by Bonaventura and Ringler (2005) but still lacked a discrete conservation of potential vorticity/potential enstrophy and exact discrete geostrophic equilibria, two properties tied together as discussed by Thuburn (2008). A full generalization was obtained later by Thuburn et al (2009) and Ringler et al (2010) assuming a Delaunay-Voronoi pair of primal and dual meshes, or more generally orthogonal primal and dual meshes (see Sect. 2).…”

“…Section 4 is devoted to energetic consistency. The discrete energy budget of DYNAMICO is derived, and the underlying Hamiltonian structure of the TRiSK scheme (Thuburn et al, 2009;Ringler et al, 2010) is identified. In Sect.…”