Unsteady analytical solutions of the spherical shallow water equations. (English) Zbl 1078.86001
Summary: A new class of unsteady analytical solutions of the spherical shallow water equations (SSWE) is presented. Analytical solutions of the SSWE are fundamental for the validation of barotropic atmospheric models. To date, only steady-state analytical solutions are known from the literature. The unsteady analytical solutions of the SSWE are derived by applying the transformation method to the transition from a fixed cartesian to a rotating coordinate system. Fundamental examples of the new unsteady analytical solutions are presented for specific wind profiles. With the presented unsteady analytical solutions one can provide a measure of the numerical convergence in the case of a temporally evolving system. An application to the atmospheric model PLASMA shows the benefit of unsteady analytical solutions for the quantification of convergence properties.
MSC:
| 86A10 | Meteorology and atmospheric physics |
| 35Q35 | PDEs in connection with fluid mechanics |
| 86A05 | Hydrology, hydrography, oceanography |
| 35C05 | Solutions to PDEs in closed form |
| 65Z05 | Applications to the sciences |
| 76N15 | Gas dynamics (general theory) |
| 76B60 | Atmospheric waves (MSC2010) |
Keywords:
atmospheric flow; validation of barotropic atmospheric models; atmospheric model PLASMA; convergence propertiesReferences:
| [1] | Behrens, J.; Rakowsky, N.; Hiller, W.; Handorf, D.; Läuter, M.; Päpke, J.; Dethloff, K., Amatos: parallel adaptive mesh generator for atmospheric and oceanic simulation, Ocean Model., 10, 171-183 (2005) |
| [2] | L. Bonaventura, T.R. Ringler, 2004. Analysis of discrete shallow water models on geodesic Delauny grids with C-type staggering. Mon. Weather Rev., submitted.; L. Bonaventura, T.R. Ringler, 2004. Analysis of discrete shallow water models on geodesic Delauny grids with C-type staggering. Mon. Weather Rev., submitted. |
| [3] | 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, 1058-1075 (1989) |
| [4] | Côté, J., A Lagrange multiplier approach for the metric terms of semi-Lagrangian models on the sphere, Q.J.R. Meteorol. Soc., 114, 1347-1352 (1988) |
| [5] | Côté, J.; Staniforth, A., An accurate and efficient finite-element global model of the shallow-water equations, Mon. Weather Rev., 118, 2707-2717 (1990) |
| [6] | Dee, D. P.; Da Silva, A. M., Using Hough harmonics to validate and assess nonlinear shallow-water models, Mon. Weather Rev., 114, 2191-2196 (1986) |
| [7] | Dey, C. H., A note on global forecasting with the Kurihara grid, Mon. Weather Rev., 97, 597-601 (1969) |
| [8] | Dutton, J. A., Dynamics of Atmospheric Motion (1995), Dover Publications: Dover Publications New York |
| [9] | Frickenhaus, S.; Hiller, W.; Best, M., FoSSI: Family of simplified solver interfaces for parallel sparse solvers in numerical atmosphere and ocean modeling, Ocean Model., 10, 185-191 (2005) |
| [10] | Galewsky, J.; Scott, R.; Polvani, L. M., An initial-value problem for testing numerical models of the global shallow water equations, Tellus A, 56, 429-440 (2004) |
| [11] | Giraldo, F. X., A spectral element shallow water model on spherical geodesic grids, Int. J. Numer. Meth. Fluids, 35, 869-901 (2001) · Zbl 1030.76045 |
| [12] | Giraldo, F. X.; Warburton, T., A nodal triangle-based spectral element method for the spherical shallow water equations on unstructured grids, J. Comput. Phys., 207, 129-150 (2005) · Zbl 1177.86002 |
| [13] | Haltiner, G. J.; Williams, R., Numerical Prediction and Dynamic Meteorology (1980), Wiley: Wiley New York |
| [14] | Heinze, T.; Hense, A., The shallow water equations on the sphere and their Lagrange-Galerkin solution, Meteorol. Atmos. Phys., 81, 129-137 (2002) |
| [15] | Hoskins, B. J., Stability of the Rossby-Haurwitz wave, Q.J.R. Meteorol. Soc., 99, 723-745 (1973) |
| [16] | C. Jablonowski, Adaptive grids in weather and climate modeling. Ph.D. thesis, The University of Michigan, USA, 2004.; C. Jablonowski, Adaptive grids in weather and climate modeling. Ph.D. thesis, The University of Michigan, USA, 2004. |
| [17] | Jakob-Chien, R.; Hack, J. J.; Williamson, D. L., Spectral transform solutions to the shallow water test set, J. Comput. Phys., 119, 164-187 (1995) · Zbl 0878.76059 |
| [18] | Läuter, M., An adaptive Lagrange-Galerkin method for the shallow water equations on the sphere, Proc. Appl. Math. Mech., 3, 48-51 (2003) · Zbl 1354.76108 |
| [19] | M. Läuter, Großräumige Zirkulationsstrukturen in einem nichtlinearen adaptiven Atmosphärenmodell. Ph.D. thesis, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany, 2004.; M. Läuter, Großräumige Zirkulationsstrukturen in einem nichtlinearen adaptiven Atmosphärenmodell. Ph.D. thesis, Universität Potsdam, Am Neuen Palais 10, 14469 Potsdam, Germany, 2004. |
| [20] | Lin, S.-J.; Rood, R. B., An explicit flux-form semi-Lagrangian shallow water model on the sphere, Q.J.R. Meteorol. Soc., 123, 2477-2498 (1997) |
| [21] | McDonald, A.; Bates, J. R., Semi-Lagrangian integration of a gridpoint shallow water model on the sphere, Mon. Weather Rev., 117, 130-137 (1989) |
| [22] | Merilees, P. E.; Ducharme, P.; Jacques, G., Experiments with a polar filter and a one-dimensional semi-implicit algorithm, Atmosphere, 15, 19-33 (1977) |
| [23] | Pedlosky, J., Geophysical Fluid Dynamics (1987), Springer: Springer New York · Zbl 0713.76005 |
| [24] | Phillips, N. A., Numerical integration of the primitive equations on the hemisphere, Mon. Weather Rev., 87, 333-345 (1959) |
| [25] | Piani, C.; Norton, W. A., Solid-body rotation in the northern hemisphere summer stratosphere, Geophys. Res. Lett., 29, 2117-2120 (2002) |
| [26] | Ringler, T. D.; Randall, D. A., A potential entropy and energy conserving numerical scheme for solution of the shallow-water equations on a geodesic grid, Mon. Weather Rev., 130, 5, 1397-1410 (2002) |
| [27] | 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 |
| [28] | Swarztrauber, P. N.; Williamson, D. L.; Drake, J. B., The cartesian method for solving partial differential equations in spherical geometry, Dyn. Atmos. Oceans, 27, 679-706 (1997) |
| [29] | Takacs, L. L., Effects of using a posteriori methods for the conservation of integral invariants, Mon. Weather Rev., 116, 525-545 (1988) |
| [30] | Tolstykh, M. A., Vorticity-divergence semi-Lagrangian shallow-water model of the sphere based on compact finite differences, J. Comput. Phys., 179, 180-200 (2002) · Zbl 1060.76086 |
| [31] | Tomita, H.; Tsugawa, M.; Satoh, M.; Goto, K., Shallow water model on a modified icosahedral geodesic grid by using spring dynamics, J. Comput. Phys., 174, 579-613 (2001) · Zbl 1056.76058 |
| [32] | Umscheid, L.; Sankar-Rao, M., Further tests of a grid system for global numerical prediction, Mon. Weather Rev., 99, 686-690 (1971) |
| [33] | Williamson, D. L.; Browning, G. L., Comparison of grids and difference approximations for numerical weather prediction over a sphere, J. Appl. Meteorol., 12, 264-274 (1973) |
| [34] | Williamson, D. L.; Drake, J. B.; Hack, J. J.; Jakob, R.; Swarztrauber, P. N., A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys., 102, 211-224 (1992) · Zbl 0756.76060 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
