Numerical representation of geostrophic modes on arbitrarily structured C-grids

A 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.

[1]  Francis X. Giraldo,et al.  Lagrange—Galerkin methods on spherical geodesic grids: the shallow water equations , 2000 .

[2]  Srdjan Dobricic.  An improved calculation of Coriolis terms on the C grid , 2006 .

[3]  David A. Randall,et al.  Geostrophic Adjustment and the Finite-Difference Shallow-Water Equations , 1994 .

[4]  G. R. Stuhne,et al.  New icosahedral grid-point discretization of the shallow water equation on the sphere , 1999 .

[5]  David L. Williamson,et al.  Integration of the barotropic vorticity equation on a spherical geodesic grid , 1968 .

[6]  Todd D. Ringler,et al.  Modeling the Atmospheric General Circulation Using a Spherical Geodesic Grid: A New Class of Dynamical Cores , 2000 .

[7]  Akio Arakawa,et al.  Computational Design of the Basic Dynamical Processes of the UCLA General Circulation Model , 1977 .

[8]  J. Gillis,et al.  Methods in Computational Physics , 1964 .

[9]  J. Thuburn.  A PV-Based Shallow-Water Model on a Hexagonal-Icosahedral Grid , 1997 .

[10]  D. Randall,et al.  A Potential Enstrophy and Energy Conserving Numerical Scheme for Solution of the Shallow-Water Equations on a Geodesic Grid , 2002 .

[11]  J. Thuburn.  Numerical wave propagation on the hexagonal C-grid , 2008, J. Comput. Phys..

[12]  Masaki Satoh,et al.  Nonhydrostatic icosahedral atmospheric model (NICAM) for global cloud resolving simulations , 2008, J. Comput. Phys..

[13]  Yoshinobu Masuda,et al.  An Integration Scheme of the Primitive Equation Model with an Icosahedral-Hexagonal Grid System and its Application to the Shallow Water Equations , 1986 .

[14]  Michael Buchhold,et al.  The Operational Global Icosahedral-Hexagonal Gridpoint Model GME: Description and High-Resolution Tests , 2002 .

[15]  F. Mesinger,et al.  A global shallow‐water model using an expanded spherical cube: Gnomonic versus conformal coordinates , 1996 .

[16]  Hiroaki Miura,et al.  A Comparison of Grid Quality of Optimized Spherical Hexagonal–Pentagonal Geodesic Grids , 2005 .

[17]  M. J. P. Cullen,et al.  Integrations of the primitive equations on a sphere using the finite element method , 1974 .

[18]  A. Arakawa,et al.  A Potential Enstrophy and Energy Conserving Scheme for the Shallow Water Equations , 1981 .

[19]  A. Kageyama,et al.  ``Yin-Yang grid'': An overset grid in spherical geometry , 2004, physics/0403123.

[20]  Slobodan Nickovic,et al.  Geostrophic Adjustment on Hexagonal Grids , 2002 .

[21]  Roy A. Walters,et al.  An unstructured grid, three‐dimensional model based on the shallow water equations , 2000 .

[22]  Charles S. Peskin,et al.  On the construction of the Voronoi mesh on a sphere , 1985 .

[23]  Vincenzo Casulli,et al.  Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems , 2002 .

[24]  Yoshio Kurihara,et al.  Numerical Integration of the Primitive Equations on a Spherical Grid , 1965 .

[25]  R. Heikes,et al.  Numerical Integration of the Shallow-Water Equations on a Twisted Icosahedral Grid , 1995 .

[26]  John Thuburn,et al.  Conservation and Linear Rossby-Mode Dispersion on the Spherical C Grid , 2004 .

[27]  Michael S. Fox-Rabinovitz.  Computational Dispersion Properties of 3D Staggered Grids for a Nonhydrostatic Anelastic System , 1996 .

[28]  John Thuburn.  Rossby wave dispersion on the C-grid , 2007 .

[29]  T. Ringler,et al.  Analysis of Discrete Shallow-Water Models on Geodesic Delaunay Grids with C-Type Staggering , 2005 .