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A mass, energy, vorticity, and potential enstrophy conserving lateral fluid-land boundary scheme for the shallow water equations. (English) Zbl 1194.76208
A numerical scheme for treating fluid-land boundaries in inviscid shallow water flows is derived that conserves the domain-summed mass, energy, vorticity, and potential enstrophy in domains with arbitrarily shaped boundaries. The boundary scheme is derived from a previous scheme that conserved all four domain-summed quantities only in periodic domains without boundaries. The present scheme includes land in the model along with evolution equations for vorticity and extrapolation formulas for the depth at fluid-land boundaries. The authors give proofs of mass, energy, vorticity, and potential enstrophy conservation. Numerical simulations are carried out demonstrating the conservation properties and accuracy of the boundary scheme for inviscid flows and comparing its performance with that of four alternative boundary schemes. The first of these alternatives extrapolates in finite differences the velocity to obtain the vorticity at boundaries; the second enforces the free-slip boundary condition; the third enforces the super-slip condition; and the fourth enforces the no-slip condition. Comparisons of the conservation properties demonstrate that the new scheme is the only one of the five that conserves all four domain-summed quantities, and it is the only one that both prevents a spurious energy cascade to the smallest resolved scales and maintains the correct flow orientation with respect to an external forcing. Comparisons of the accuracy demonstrate that the new scheme generates vorticity fields that have smaller errors than those generated by any of the alternative schemes, and it generates depth and velocity fields that have errors about equal to those in the fields generated by the most accurate alternative scheme.

76M25 Other numerical methods (fluid mechanics) (MSC2010)
76B99 Incompressible inviscid fluids
86A05 Hydrology, hydrography, oceanography
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[1] Arakawa, A., Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. part I, J. comput. phys., 1, 119-143, (1966) · Zbl 0147.44202
[2] Sadourny, R., The dynamics of finite-difference models of the shallow-water equations, J. atmos. sci., 32, 680-689, (1975)
[3] Arakawa, A.; Lamb, V.R., Computational design of the basic dynamical processes of the UCLA general circulation model, (), 173-265
[4] Arakawa, A.; Lamb, V.R., A potential enstrophy and energy conserving scheme for the shallow water equations, Mon. weather rev., 109, 18-36, (1981)
[5] K. Takano, M.G. Wurtele, A fourth order energy and potential enstrophy conserving difference scheme, Tech. Rep. AFGL-TR-82-0205 (NTIS AD-A126626), Air Force Geophysics Laboratory, Hanscom AFB, 1982.
[6] Abramopoulos, F., Generalized energy and potential enstrophy conserving finite difference schemes for the shallow water equations, Mon. weather rev., 116, 650-662, (1988)
[7] Abramopoulos, F., A new fourth-order enstrophy and energy conserving scheme, Mon. weather rev., 119, 128-133, (1991)
[8] Tripoli, G.J., A nonhydrostatic mesoscale model designed to simulate scale interaction, Mon. weather rev., 120, 1342-1359, (1992)
[9] Janjić, Z.I., Pressure gradient force and advection scheme used for forecasting with steep and small scale topography, Contrib. atmos. phys., 50, 186-199, (1977)
[10] Mesinger, F., Horizontal advection schemes of a staggered grid – an enstrophy and energy-conserving model, Mon. weather rev., 109, 467-478, (1981)
[11] Janjić, Z.I., Nonlinear advection schemes and energy cascade on semi-staggered grids, Mon. weather rev., 112, 1234-1245, (1984)
[12] Rančić, M., Fourth-order horizontal advection schemes on the semi-staggered grid, Mon. weather rev., 116, 1274-1288, (1988)
[13] 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, 351-356, (1968)
[14] Salmon, R.; Talley, L.D., Generalizations of arakawa’s Jacobian, J. comput. phys., 83, 247-259, (1989) · Zbl 0672.76002
[15] Perot, B., Conservation properties of unstructured staggered mesh schemes, J. comput. phys., 159, 58-89, (2000) · Zbl 0972.76068
[16] Morton, K.W.; Roe, P.L., Vorticity-preserving lax – wendroff-type schemes for the system wave equation, SIAM J. sci. comput., 23, 170-192, (2001) · Zbl 0994.35011
[17] Ringler, T.D.; Randall, D.A., A potential enstrophy and energy conserving numerical scheme for solution of the shallow-water equations on a geodesic grid, Mon. weather rev., 130, 1397-1410, (2002)
[18] Ringler, T.D.; Randall, D.A., The ZM grid: an alternative to the Z grid, Mon. weather rev., 130, 1411-1422, (2002)
[19] Bonaventura, L.; Ringler, T., Analysis of discrete shallow-water models on geodesic delauney grids with C-type staggering, Mon. weather rev., 133, 2351-2373, (2005)
[20] Salmon, R., Poisson-bracket approach to the construction of energy- and potential-enstrophy-conserving algorithms for the shallow-water equations, J. atmos. sci., 61, 2016-2036, (2004)
[21] Salmon, R., A general method for conserving quantities related to potential vorticity in numerical models, Nonlinearity, 18, R1-R16, (2005) · Zbl 1213.76143
[22] Salmon, R., A general method for conserving energy and potential enstrophy in shallow-water models, J. atmos. sci., 64, 515-531, (2007)
[23] Mesinger, F.; Janjić, Z.I.; Ničković, S.; Gavrilov, D.; Deaven, D.G., The step-mountain coordinate: model description and performance for cases of alpine Lee cyclogenesis and for a case of an Appalachian redevelopment, Mon. weather rev., 116, 1493-1518, (1988)
[24] Adcroft, A.; Hill, C.; Marshall, J., Representation of topography by shaved cells in a height coordinate Ocean model, Mon. weather rev., 125, 2293-2315, (1997)
[25] Hurlburt, H.E.; Thompson, J.D., A numerical study of loop current intrusions and eddy shedding, J. phys. oceanogr., 10, 1611-1651, (1980)
[26] Jiang, S.; Jin, F.-F.; Ghil, M., Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model, J. phys. oceanogr., 25, 764-786, (1995)
[27] Dijkstra, H.A.; de Ruijter, W.R.M., On the physics of the agulhas current: steady retroflection regimes, J. phys. oceanogr., 31, 2971-2985, (2001)
[28] Sura, P.; Fraedrich, K.; Lunkeit, F., Regime transition in a stochastically forced double-gyre model, J. phys. oceanogr., 31, 411-426, (2001)
[29] Simonnet, E.; Ghil, M.; Ide, K.; Temam, R.; Wang, S., Low-frequency variability in shallow-water models of the wind-driven Ocean circulation. part I: steady-state solution, J. phys. oceanogr., 33, 712-728, (2003)
[30] Marchal, O.; Nycander, J., Nonuniform upwelling in a shallow-water model of the antarctic bottom water in the Brazil basin, J. phys. oceanogr., 34, 2492-2513, (2004)
[31] Poulin, F.J.; Flierl, G.R., The influence of topography on the stability of jets, J. phys. oceanogr., 35, 811-825, (2005)
[32] Matsuura, T.; Fujita, M., Two different aperiodic phases of wind-driven Ocean circulation in a double-gyre, two-layer shallow-water model, J. phys. oceanogr., 36, 1265-1286, (2006)
[33] Evans, J.L.; Holland, G.J.; Elsberry, R.L., Interactions between a barotropic vortex and an idealized subtropical ridge. part I: vortex motion, J. atmos. sci., 48, 301-314, (1991)
[34] Hart, R.E.; Evans, J.L., Simulations of dual-vortex interaction within environmental shear, J. atmos. sci., 56, 3605-3621, (1999)
[35] Mundt, M.D.; Vallis, G.K.; Wang, J., Balanced models and dynamics for the large- and mesoscale circulation, J. phys. oceanogr., 27, 1133-1152, (1997)
[36] G.S. Ketefian, Development and testing of a 2D potential enstrophy conserving ocean model and a 3D potential enstrophy conserving, nonhydrostatic, compressible atmospheric model, Ph.D. dissertation, Stanford University, 2006. <http://www.stanford.edu/group/efmh/gsk/ketefian_thesis.pdf>.
[37] Carnevale, G.F.; Cavallini, F.; Crisciani, F., Dynamic boundary conditions revisited, J. phys. oceanogr., 31, 2489-2497, (2001)
[38] Primeau, F.W., Multiple equilibria of a double-gyre Ocean model with super-slip boundary conditions, J. phys. oceanogr., 28, 2130-2147, (1998)
[39] Wang, J.; Vallis, G.K., Emergence of fofonoff states in inviscid and viscous Ocean circulation models, J. mar. res., 52, 83-127, (1994)
[40] Greatbatch, R.J.; Nadiga, B.T., Four-gyre circulation in a barotropic model with double-gyre wind forcing, J. phys. oceanogr., 30, 1461-1471, (2000)
[41] Hollingsworth, A.; Kallberg, P.; Renner, V.; Burridge, D.M., An internal symmetric computational instability, Quart. J. R. meteor. soc., 109, 417-428, (1983)
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