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Shallow water model on a modified icosahedral geodesic grid by using spring dynamics. (English) Zbl 1056.76058
From the summary: We develop a shallow water model on an icosahedral geodesic grid with several grid modifications. Discretizations of differential operators in the equations are based on the finite volume method, so that the global integrations of transported quantities are numerically conserved. Ordinarily, the standard grid is obtained by recursive grid division starting from the lowest order icosahedral grid. From the viewpoint of numerical accuracy of operators, we propose to relocate the variable-defined grid points from the standard positions to the gravitational centers of control volumes. From the other viewpoint of numerical stability, we modify the standard grid configuration by employing the spring dynamics, namely, the standard grid points are connected by appropriate springs, which move grid points until the dynamical system calms down. We find that the latter modification dramatically reduces the grid-noise in the numerical integration of equations. The reason for this is that the geometrical quantities of control volume such as its area and distortion of its shape exhibit the monotonic distribution on the sphere. By a combination of the two modifications, we can integrate the equations both with high accuracy and stability.
Reviewer: Reviewer (Berlin)

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
76U05 General theory of rotating fluids
86A05 Hydrology, hydrography, oceanography
86A10 Meteorology and atmospheric physics
Software:
chammp
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[1] Suda, R., High performance computation of spherical harmonic transform, IPSJ SIG notes, 98-HPC-71, 7, (May 1998)
[2] Suda, R., A fast spherical harmonics transform algorithm, IPSJ SIG notes, 98-HPC-73, 37, (Oct. 1998)
[3] Roache, P.J., Computational fluid dynamics, (1976)
[4] Bryan, K.; Manabe, S.; Pacanowski, R.C., A global Ocean-atmosphere climate model part II. the oceanic circulation, J. phys. oceanogr, 5, 30, (1975)
[5] Takacs, L.L.; Balgovind, R., High latitude filtering in global grid point models, Mon. weather rev, 111, 2005, (1983)
[6] Staniforth, A.; Côté, J., Semi-Lagrangian integration schemes for atmospheric models, Mon. weather rev, 119, 2206, (1991)
[7] Priestley, A., A quasi-conservative version of the semi-Lagrangian advection scheme, Mon. weather rev, 121, 621, (1993)
[8] Sadourny, R.; Arakawa, A.; Mintz, Y., Integration of the nondivergent barotropic vorticity equation with an icosahedral hexagonal grid for the sphere, Mon. weather rev, 96, 351, (1968)
[9] Williamson, D.L., Integration of the barotropic vorticity equation on a spherical geodesic grid, Tellus, 20, 642, (1968)
[10] Phillips, N.A., Numerical integration of the primitive equation on the hemisphere, Mon. weather rev, 87, 333, (1959)
[11] Sadourny, R., Numerical integration of the primitive equations on a spherical grid with hexagonal cells, Proceedings of the WMO/IUGG symposium on numerical weather prediction in Tokyo, tech. rep. of JMA, VII45-VII52, (Nov. 26-Dec. 4, 1969)
[12] Williamson, D.L., Integration of the primitive barotropic model over a spherical geodesic grid, Mon. weather rev, 98, 512, (1969)
[13] Masuda, Y.; Ohnishi, H., An integration scheme of the primitive equation model with an icosahedral-hexagonal grid system and its application to the shallow water equations, Short- and medium-range numerical weather prediction. collection of papers presented at the WMO/IUGG NWP symposium, Tokyo, aug. 4-8 1986, 317, (1986)
[14] Cullen, M.J.P., Integration of the primitive equations on a sphere using the finite element method, Q. J. R. meteorol. soc, 100, 555, (1974) · Zbl 0279.65090
[15] Cullen, M.J.P.; Hall, C.D., Forecasting and general circulation results from finite element models, Q. J. R. meteorol. soc, 105, 571, (1979)
[16] Cooley, J.W.; Turkey, J.W., An algorithm for the machine computation of complex Fourier series, Math. comput, 19, 297, (1965)
[17] Heikes, R.H.; 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, (1995)
[18] Heikes, R.H.; Randall, D.A., Numerical integration of the shallow-water equations on a twisted icosahedral grid. part II: A detailed description of the grid and analysis of numerical accuracy, Mon. weather rev, 123, 1881, (1995)
[19] Williamson, D.L.; Drake, J.B.; Hack, J.J.; Jacob, R.; Swarztrauber, P.N., A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. comput. phys, 102, 211, (1992) · Zbl 0756.76060
[20] Ringler, T.D.; Heikes, R.H.; Randall, D.A., Modeling the atmospheric general circulation using a spherical geodesic grid: A new class of dynamical cores, Mon. weather rev, 128, 2471, (2000)
[21] Stuhne, G.R.; Peltier, W.R., Vortex erosion and amalgamation in a new model of large scale flow on the shere, J. comput. phys, 128, 58, (1996) · Zbl 0861.76045
[22] Stuhne, G.R.; Peltier, W.R., New icosahedral grid-point discretizations of the shallow water equations on the sphere, J. comput. phys, 148, 23, (1999) · Zbl 0930.76067
[23] Thuburn, J., A PV-based shallow-water model on a hexagonal-icosahedral grid, Mon. weather rev, 125, 2328, (1997)
[24] Côté, J., A Lagrange multiplier approach for the metric terms of semi-Lagrangian models on the sphere, Q. J. R. meteorol. soc, 114, 1347, (1988)
[25] Browning, G.L.; Hack, J.J.; Swarztrauber, P.N., A comparison of three numerical methods for solving differential equations on the sphere, Mon. weather rev, 117, 1058, (1995)
[26] K. Ishioka, ispack-0.5 (GFD Dennou Club), available at, http://www.gfd-dennou.org/arch/ispack/, 2000.
[27] Jakob-Chien, R.; Hack, J.J.; Williamson, D.L., Spectral transform solutions to the shallow water test set, J. comput. phys, 119, 164, (1995) · Zbl 0878.76059
[28] Hoskins, B.J., Stability of the rossby – haurwitz wave, Q. J. R. meteorol. soc, 99, 723, (1973)
[29] Thuburn, J.; Li, Y., Numerical simulation of rossby – hauwitz waves, Tellus, 52A, 181, (2000)
[30] Ritchie, H., Application of the semi-Lagrangian method to a spectral model of the shallow-water equations, Mon. weather rev, 116, 1587, (1988)
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