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Numerical wave propagation on the hexagonal C-grid. (English) Zbl 1220.76018
Summary: Inertio-gravity mode and Rossby mode dispersion properties are examined for discretizations of the linearized rotating shallow water equations on a regular hexagonal C-grid in planar geometry. It is shown that spurious non-zero Rossby mode frequencies found by previous authors in the $$f$$-plane case can be avoided by an appropriate discretization of the Coriolis terms. Three generalizations of this discretization that conserve energy even for non-constant Coriolis parameter are presented. A quasigeostrophic $$\beta$$-plane analysis is carried out to investigate the Rossby mode dispersion properties of these three schemes. The Rossby mode dispersion relation is found to have two branches. The primary branch modes are good approximations, in terms of both structure and frequency, to corresponding modes of the continuous governing equations, and offer some improvements over a quadrilateral C-grid scheme. The secondary branch modes have vorticity structures approximating those of small-scale modes of the continuous governing equations, suggesting that the hexagonal C-grid might have an advantage in terms of resolving extra Rossby modes; however, the frequencies of the secondary branch Rossby modes are much smaller than those of the corresponding continuous modes, so this potential advantage is not fully realized.

##### MSC:
 76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 76U05 General theory of rotating fluids 86A05 Hydrology, hydrography, oceanography
NICAM
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##### References:
 [1] Arakawa, A.; Lamb, V.R., Computational design of the basic dynamical processes of the UCLA general circulation model, (), 173-265 [2] 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) [3] 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 [4] Dobricic, S., An improved calculation of Coriolis terms on the C grid, Mon. weather rev., 134, 3764-3773, (2006) [5] Dukowicz, J.K., Mesh effects for Rossby waves, J. comput. phys., 119, 188-194, (1995) · Zbl 0828.76052 [6] Fox-Rabinovitz, M.S., Computational dispersion properties of 3D staggered grids for a nonhydrostatic anelastic system, Mon. weather rev., 124, 498-510, (1996) [7] Gavrilov, M.B.; Tošić, I.A., Propagation of the Rossby waves on two dimensional rectangular grids, Meteorol. atmos. phys., 68, 119-125, (1998) [8] Giraldo, F.X., Lagrange – galerkin methods on spherical geodesic grids: the shallow-water equations, J. comput. phys., 160, 336-368, (2000) · Zbl 0977.76045 [9] 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) [10] Le Roux, D.Y.; Staniforth, A.; Lin, C.A., Finite elements for shallow-water equation Ocean models, Mon. weather rev., 126, 1931-1951, (1998) [11] Majewski, D.; Liermann, D.; Prohl, P.; Ritter, B.; Buchhold, M.; Hanisch, T.; Paul, G.; Wergen, W., The operational global icosahedral – hexagonal gridpoint model GME: description and high-resolution tests, Mon. weather rev., 130, 319-338, (2002) [12] 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, Tokyo, August 4-8, 1986. Japan Meteorological Society, 1986, pp. 317-326. [13] Miura, H.; Kimoto, M., A comparison of grid quality of optimized spherical hexagonal – pentagonal geodesic grids, Mon. weather rev., 133, 2817-2833, (2005) [14] Ničković, S.; Gavrilov, M.B.; Tosić, I.A., Geostrophic adjustment on hexagonal grids, Mon. weather rev., 130, 668-683, (2002) [15] Popović, J.M.; Ničković, S.; Gavrilov, M.B., Frequency of quasigeostrophic modes on hexagonal grids, Meteorol. atmos. phys., 58, 41-49, (1996) [16] Randall, D.A., Geostrophic adjustment and the finite-difference shallow-water equations, Mon. weather rev., 122, 1371-1377, (1994) [17] Ringler, T.; Heikes, R.; Randall, D.A., Modeling the atmospheric general circulation using a spherical geodesic grid, Mon. weather rev., 128, 2471-2490, (2000) [18] 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) [19] R. Sadourny, Numerical integration of the primitive equations on a spherical grid with hexagonal cells, in: Proceedings of WMO/IUGG NWP Symposium, Tokyo, Japan, Japan Meteorological Agency, 1969, pp. 45-52. [20] 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 [21] 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 [22] Thuburn, J., A PV-based shallow water model on a hexagonal-icosahedral grid, Mon. weather rev., 125, 2328-2347, (1997) [23] Thuburn, J., Rossby wave propagation on the C-grid, Atmos. sci. lett., 8, 37-42, (2007) [24] Thuburn, J.; Staniforth, A., Conservation and linear Rossby-mode dispersion on the spherical C grid, Mon. weather rev., 132, 641-653, (2004) [25] Torsvik, T.; Thiem, Ø.; Bernsten, J., Stability analysis of geostrophic adjustment on hexagonal grids for regions with variable depth, Mon. weather rev., 133, 3335-3344, (2005) [26] Wajsowicz, R.C., Free planetary waves in finite-difference numerical models, J. phys. oceanogr., 16, 773-789, (1986) [27] Williamson, D., Integration of the barotropic vorticity equation on a spherical geodesic grid, Tellus, 20, 624-653, (1968)
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