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Table of contents
Chapter 1. Introduction
Chapter 2. Basic equations and discretization
Chapter 3. Semi-Lagrangian formulation
Chapter 4. Computational details
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1.1 Overview
Since the original demonstration of the efficiency advantage of the semi-Lagrangian semi-implicit method over a decade ago by André Robert, this numerical integration scheme is being used in an increasing range of atmospheric models. Most of the applications have been in grid-point models. Shallow-water-equations studies have included three-time-level versions by Robert (1981, 1982) and Staniforth and Temperton (1986), and two-time-level schemes by Temperton and Staniforth (1987), Purser and Leslie (1988), McDonald and Bates (1989), and Côté and Staniforth (1990). There also have been various applications in baroclinic grid-point models. Three-time-level sigma-coordinate versions have been presented by Robert et al. (1985) and Tanguay et al. (1989), and the extension of the three-time-level approach to a non-hydrostatic coordinate has been demonstrated by Tanguay et al. (1990) . Bates and McDonald (1982), McDonald (1986), Leslie and Purser (1991), McDonald and Haugen (1992), and Bates et al. (1993) have developed two-time-level sigma-coordinate schemes, McDonald and Haugen (1993) have presented the two-time-level extension to a hybrid vertical coordinate, and Golding (1992) has applied a split two-time-level semi-Lagrangian scheme in a non-hydrostatic model.
For spectral models, a semi-Lagrangian semi-implicit shallow-water equation model was presented by Ritchie (1988) for a three-time-level version, and adapted by Côté and Staniforth (1988) for a two-time-level scheme. Baroclinic three-time-level spectral model formulations have been demonstrated by Ritchie (1991) for operational numerical weather prediction in a sigma-coordinate model, and recently by Williamson and Olson (1994) for climate simulations with a hybrid coordinate model.
In a broader context, the semi-Lagrangian scheme, as incorporated in spectral numerical weather prediction models, may be considered as an economical variant of the spectral Lagrange-Galerkin method (Süli and Ware, 1991).
Experience at ECMWF (Simmons et al., 1989) suggests that the accuracy of medium-range forecasts has steadily improved with increasing resolution. Consequently, in its four-year plan for the period 1989-1992, ECMWF proposed development of a high-resolution version of its forecast model. A target resolution of a spectral representation with a triangular truncation of 213 waves in the horizontal and 31 levels in the vertical (T213/L31) was set, entailing a doubling of the horizontal resolution and an approximate doubling of the vertical resolution in the troposphere compared to the T106/L19 configuration that was operational at the time (Simmons et al., 1989). In view of the anticipated computer resources, it was clear that major efficiency gains would be necessary in order to attain this objective. These gains have been provided by the introduction of the semi-Lagrangian treatment of advection permitting a substantial increase in the size of the time-step, the use of a reduced Gaussian grid giving a further advantage of about 25%, the introduction of economies in the Legendre transforms, and improvements to the model's basic architecture.
The layout for the remainder of the document is as follows. In Chapter 2 `Basic equations and discretization' we present the reformulation of the Eulerian model in order to transform the vorticity-divergence formulation into a momentum-equation version in preparation for a subsequent semi-Lagrangian vector treatment of the equations of motion. The vertical discretization of the ECMWF hybrid coordinate on a staggered grid is also considered. The semi-Lagrangian treatment is discussed in some detail in Chapter 3 `Semi-Lagrangian formulation' , including the adaptation to accommodate the reduced Gaussian grid. Several important computational details relevant for efficient execution of the high resolution model on a modestly parallel supercomputer are discussed in Chapter 4 `Computational details' .
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08.04.2002 |
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