A comparative study of the performance of various vertical discretization schemes |
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| Authors: | L M Leslie R J Purser |
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| Institution: | (1) Present address: Bureau of Meterology Research Centre, 3001, Melbourne, Australia;(2) University of Madison-Wisconsin, USA;(3) Present address: National Meteorological Center, WWB, W/NMC2, 20233 Washington DC, USA |
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| Abstract: | Summary Most finite-difference numerical weather prediction models employ vertical discretizations that are staggered, and are low-order (usually second-order) approximations for the important terms such as the derivation of the geopotential from the hydrostatic equation, and the calculation of the vertically integrated divergence. In a sigma-coordinate model the latter is used for computing both the surface pressure change and the vertical velocity. All of the above-mentioned variables can diminish the accuracy of the forecast if they are not calculated accurately, and can have an impact on related quantities such as precipitation.In this study various discretization schemes in the vertical are compared both in theory and in practice. Four different vertical grids are tested: one unstaggered and three staggered (including the widely-used  Lorenz grid). The comparison is carried out by assessing the accuracy of the grids using vertical numerics that range from second-order up to sixth-order.The theoretical part of the study examines how faithfully each vertical grid reproduces the vertical modes of the governing equations linearized with a basic state atmosphere. The performance of the grids is evaluated for 2nd, 4th and 6th-order numerical schemes based on Lagrange polynomials, and for a 6th-ordercompact scheme.Our interpretation of the results of the theoretical study is as follows. The most important result is that the order of accuracy employed in the numerics seems to be more significant than the choice of vertical grid. There are differences between the grids at second-order, but these differences effectively vanish as the order of accuracy increases. The sixth-order schemes all produce very accurate results with the grids performing equally well, and with the compact scheme significantly outperforming the Lagrange scheme. A second major result is that for the number of levels typically used in current operational forecast models, second-order schemes (which are used almost universally) all appear to be relatively poor, for other than the lowest modes.The theoretical claims were confirmed in practice using a large number (100) of forecasts with the Australian Bureau of Meteorology Research Centre's operational model. By comparing test model forecasts using the four grids and the different orders of numerics with very high resolution control model forecasts, the results of the theoretical study seem to be corroborated.With 8 Figures |
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