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Notes: 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.

Notes: Summary With the increasingly widespread adoption of massively parallel processing (MPP) computers for applications in computational fluid dynamics it becomes appropriate to reconsider the geometrical configuration of the computational grid that best suits the problem. In the case of global numerical weather prediction we have recently advocated a conformal spherical-cubic geometry. Among its merits, this grid lends itself naturally to simple domain-decomposition and obviates the need for polar filtering. Here we extend the same principles, but with an emphasis on the problem of regional forecasting. In this case we observe that it is possible to cover the global domain with a conformal grid geometry based on the mapping to the sphere of a back-to-back pair of octagonal regions. In the most symmetrical case, each octagon maps to a hemisphere. By compounding this mapping with a nonhomogeneous conformal mapping of the sphere to itself, one can also arrange to have quasi-uniform enhanced resolution of the resulting grid inside any chosen circle on the sphere, at the expense of relatively coarse resolution degrading gradually with distance outside the circle of interest. With appropriate grid dimensions, the new ‘conformal octagon’ decomposes naturally into several identical square subdomains for efficient distribution over the nodes of an MPP computer.

Notes: Summary We propose and objective method whereby the density of Shannon's “information” associated with the retrieval of a profile of atmospheric variables from satellite-derived infrared radiance measurements may be estimated. The technique is a natural extension of one we previously proposed to estimate the effective data density in a profile. We test the method in a comparison of simulated satellite instruments to show that the method does indeed provide an objective summary of the spatial distribution of each instrument's information content. We propose that further extensions of the method be developed to include other more traditional data sources in a fully three-dimensional scheme. We also note that analogous and compatible methods may be used to diagnose the information content of meteorological analysis and forecast fields relative to the information contained in the covariance, at the appropriate season, of the corresponding climate fields.