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Numerical Dispersion of Gravity Waves

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Schroeder, G., & Schluenzen, K. H. (2009). Numerical Dispersion of Gravity Waves. Monthly Weather Review, 137(12), 4344-4354. doi:10.1175/2009MWR2890.1.

When atmospheric gravity waves are simulated in numerical models, they are not only dispersive for physical but also for numerical reasons. Their wave properties (e. g., damping or propagation speed and direction) can depend on grid spacing as well as on the numerical schemes. In this work numerical dispersion relations for atmospheric gravity waves are theoretically derived as well as experimentally measured using the anelastic Mesoscale Transport and Stream model (METRAS). Both the theoretical solution and the numerical model show a retardation of gravity waves with decreasing grid resolution. Furthermore, the influence of a Shapiro seven-point filter is analyzed. The Shapiro seven-point filter causes damping of the shorter waves. Therefore, shorter waves can better be simulated without the seven-point filter. The influence of different advection schemes is analyzed by prescribing a background wind. A first-order upstream scheme and second- and third-order flux integrated essentially nonoscillatory (FIENO) schemes are used. As expected, the damping is the smaller the higher the order of the scheme. The numerical dispersion has severe consequences, when nonuniform grid spacing is used. Waves moving from the fine grid to the coarse are reflected because of numerical dispersion if they are only poorly resolved on the coarse grid. In tests with different refinement factors and wave lengths the reflection is found to be the larger the greater the refinement factor. The results show that refinement factors larger than 3 should not be used with nonuniform grid spacing or two-way nested grids.