The astonishing development of computer technology since the mid-20th century has been accompanied by a corresponding proliferation in the numerical methods that have been developed to improve the simulation of atmospheric flows. This article reviews some of the numerical developments concern the ongoing improvements of weather forecasting and climate simulation models. Early computers were single-processor machines with severely limited memory capacity and computational speed, requiring simplified representations of the atmospheric equations and low resolution. As the hardware evolved and memory and speed increased, it became feasible to accommodate more complete representations of the dynamic and physical atmospheric processes. These more faithful representations of the so-called primitive equations included dynamic modes that are not necessarily of meteorological significance, which in turn led to additional computational challenges. Understanding which problems required attention and how they should be addressed was not a straightforward and unique process, and it resulted in the variety of approaches that are summarized in this article. At about the turn of the century, the most dramatic developments in hardware were the inauguration of the era of massively parallel computers, together with the vast increase in the amount of rapidly accessible memory that the new architectures provided. These advances and opportunities have demanded a thorough reassessment of the numerical methods that are most successfully adapted to this new computational environment. This article combines a survey of the important historical landmarks together with a somewhat speculative review of methods that, at the time of writing, seem to hold out the promise of further advancing the art and science of atmospheric numerical modeling.

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A model using shallow-water equations with an Arakawa-type scheme for momentum terms is tested on a quasi-uniform geometry on the sphere, derived by a spherical expansion of the inscribed cube based on the gnomonic projection. Thereby, a quasi-homogeneous distribution of grid points is achieved, and a global finitedifference model is designed which does not require Fourier filtering or suffer from the burden of redundant computational points at high latitudes. Difficulties resulting from the directional discontinuity of the coordinate lines crossing the edges of the expanded cube are almost completely eliminated by using the Arakawa B-grid, so that only scalar points are placed along the edges. An alternative approach is developed based on numerical orthogonalization of the grid whereby, inter alia, the directional discontinuity at the edges is avoided at the cost of some accumulation of points in the vicinity of the vertices of the cube. In the customary Rossby-Haurwitz wave-4 tests, both approaches are shown to converge to a visually indistinguishable solution as the resolution is increased. However, with the orthogonalized, conformal grid, convergence towards the asymptotic solution was substantially faster.

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ABSTRACT:The application of quasi-uniform grids in global models of the atmosphere is an attempt to increase the computational efficiency by a more cost-effective exploitation of the computing infrastructure. This paper describes the development of a global version of NCEP's regional, step-coordinate, Eta model on two quasi-uniform grids: cubic and octagonal. The governing equations are expressed in a general curvilinear form, so that the cubic and the octagonal versions of the model share the same code in spite of different mapping of the computational domain.The dynamical core of the derived global Eta model is successfully tested in the benchmark test of Held and Suarez. The model with the step-wise formulation of the terrain and full physics is integrated in a series of tests with real data, and the results are compared both with the analysis and the results of the regional Eta model.

A non-hydrostatic version of the regional Eta model used operationally at the National Meteorological Center (NMC) has been developed by implementing the ideas of Juang (1992) and Laprise (1992), who independently recommended a hydrostatically based, vertical coordinate for a fully compressible set of equations. The grid-point model dynamics is based on perturbation equations in the rpvertical coordinate; the base state may be taken from the operational NMC model, and thus updated with time.Thc non-hydrostatic model uses a stepwise treatment of terrain present in the operational version, eliminating the pressure-gradient-term error associated with sigma-coordinate models over steep topography. The compressible equations are written in a form that allows conservation of energy in a horizontally closed domain with appropriate advective schemes.A two-dimensional version of the model, without parametrizations of physical processes, has been used successfully to simulate ascending warm bubbles and collapsing cold bubbles at high resolutions. A two-step scheme for the advection equation with minimized dissipation and dispersion errors. Mon. Weather Rev., 113, 1050-1065 A vertical finite-difference scheme for hydrostatic and nonhydrostatic equations. Mon. Weather Rev., 112, 1398-1402 Some considerations on vertical differencing. J. Meteorol. SOC. Japan., 56,98-111 A simulation of the development of successive cells along a cold outflow boundary. J . Atmos. Sci., 39, 1466-1483 A mesoscale numerical model using the nonhydrostatic pressurebased a-coordinate equations: Model experiments with dry mountain flows. Mon. Weather Rev.,

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