# zbMATH — the first resource for mathematics

Numerical representation of geostrophic modes on arbitrarily structured C-grids. (English) Zbl 1173.86304
Summary: 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.
Reviewer: Reviewer (Berlin)

##### MSC:
 86A10 Meteorology and atmospheric physics 86A05 Hydrology, hydrography, oceanography 76M25 Other numerical methods (fluid mechanics) (MSC2010)
NICAM
Full Text:
##### References:
 [1] Arakawa, A.; Lamb, V.R., Computational design of the basic dynamical processes of the UCLA general circulation model, (), 173-265 [2] Arakawa, A.; Lamb, V.R., A potential enstrophy and energy conserving scheme for the shallow-water equations, Mon. weather rev., 109, 18-36, (1981) [3] Augenbaum, J.M.; Peskin, C.S., On the construction of the Voronoi mesh on a sphere, J. comput. phys., 14, 177-192, (1985) · Zbl 0628.65115 [4] Bonaventura, L.; Ringler, T., Analysis of discrete shallow water models on geodesic Delaunay grids with C-type staggering, Mon. weather rev., 133, 2351-2373, (2005) [5] Casulli, V.; Walters, R.A., An unstructured grid, three-dimensional model based on the shallow water equations, Int. J. numer. methods fluids, 32, 331-348, (2000) · Zbl 0965.76061 [6] Casulli, V.; Zanolli, P., Semi-implicit numerical modeling of nonhydrostatic free-surface flows for environmental problems, Math. comput. modell., 36, 1131-1149, (2002) · Zbl 1027.76034 [7] Cullen, M.J.P., Integration of the primitive equations on a sphere using the finite element method, Quart. J. roy. meteorol. soc., 100, 555-562, (1974) · Zbl 0279.65090 [8] Dobricic, S., An improved calculation of Coriolis terms on the C grid, Mon. weather rev., 134, 3764-3773, (2006) [9] Fox-Rabinovitz, M.S., Computational dispersion properties of 3D staggered grids for a nonhydrostatic anelastic system, Mon. weather rev., 124, 498-510, (1996) [10] Giraldo, F.X., Lagrange – galerkin methods on spherical geodesic grids: the shallow-water equations, J. comput. phys., 160, 336-368, (2000) · Zbl 0977.76045 [11] Heikes, R.; Randall, D.A., Numerical integration of the shallow-water equations on a twisted icosahedral grid. part I: basic design and results of tests, Mon. weather rev., 123, 1862-1887, (1995) [12] A. Kageyama, T. Sato, Yin-Yang grid: an overset grid in spherical geometry, Geochem. Geophys. Geosyst. 5 (2004) Art. No. Q09005. [13] Majewski, D.; Liermann, D.; Prohl, P.; Ritter, B.; Buchhold, M.; Hanisch, T.; Paul, G.; Wergen, W., The operational global icosahedral-hexagonal grid-point model GME: description and high-resolution tests, Mon. weather rev., 130, 319-338, (2002) [14] Y. Masuda, H. Ohnishi, An integration scheme of the primitive equation model with an icosahedral-hexagonal grid system and its application to the shallow-water equations, in: Short- and Medium-Range Numerical Weather Prediction. Collection of Papers Presented at the WMO/IUGG NWP Symposium, Japan Meteorological Society, Tokyo, August 4-8, 1986, pp. 317-326. [15] Miura, H.; Kimoto, M., A comparison of grid quality of optimized spherical hexagonal – pentagonal geodesic grids, Mon. weather rev., 133, 2817-2833, (2005) [16] Ničković, S.; Gavrilov, M.B.; Tosić, I.A., Geostrophic adjustment on hexagonal grids, Mon. weather rev., 130, 668-683, (2002) [17] Rančić, M.; Purser, R.J.; Mesinger, F., A global shallow-water model using an expanded spherical cube: gnomonic versus conformal coordinates, Quart. J. roy. meteorol. soc., 122, 959-982, (1996) [18] Randall, D.A., Geostrophic adjustment and the finite-difference shallow-water equations, Mon. weather rev., 122, 1371-1377, (1994) [19] Ringler, T.; Heikes, R.; Randall, D.A., Modeling the atmospheric general circulation using a spherical geodesic grid, Mon. weather rev., 128, 2471-2490, (2000) [20] Ringler, T.; Randall, D., A potential enstrophy and energy conserving numerical scheme for solution of the shallow-water equations on a spherical geodesic grid, Mon. weather rev., 130, 1397-1410, (2002) [21] R. Sadourny, Numerical integration of the primitive equations on a spherical grid with hexagonal cells, in: Proceedings of WMO/IUGG NWP Symp., Japan Meteorological Agency, Tokyo, Japan, 1969, pp. 45-52. [22] Satoh, M.; Matsuno, T.; Tomita, H.; Nasuno, T.; Iga, S.; Miura, H., Nonhydrostatic icosahedral atmospheric model (NICAM) for global cloud resolving simulations, J. comput. phys., 227, 3486-3514, (2008) · Zbl 1132.86311 [23] Stuhne, G.R.; Peltier, W.R., New icosahedral grid-point discretizations of the shallow water equations on the sphere, J. comput. phys., 148, 23-58, (1999) · Zbl 0930.76067 [24] Thuburn, J., A PV-based shallow water model on a hexagonal-icosahedral grid, Mon. weather rev., 125, 2328-2347, (1997) [25] Thuburn, J., Rossby wave propagation on the C-grid, Atmos. sci. lett., 8, 37-42, (2007) [26] Thuburn, J., Numerical wave propagation on the hexagonal C-grid, J. comput. phys., 227, 5836-5858, (2008) · Zbl 1220.76018 [27] Thuburn, J.; Staniforth, A., Conservation and linear Rossby-mode dispersion on the spherical C-grid, Mon. weather rev., 132, 641-653, (2004) [28] Williamson, D., Integration of the barotropic vorticity equation on a spherical geodesic grid, Tellus, 20, 624-653, (1968)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.