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A. McDonald

Abstract

A semi-Lagrangian and semi-implicit two time-level integration scheme has been constructed for integrating the primitive meteorological equations. It can be thought of as working in the following way. During the first half-time step the Coriolis terms are integrated implicitly, while the pressure gradient terms are integrated explicitly. During the second half-time step the Coriolis terms are integrated explicitly, while the pressure gradient terms are integrated implicitly. The advection terms are integrated by means of a multiply-upstream semi-Lagrangian scheme and the nonlinear terms are integrated explicitly, once per time step. The scheme is shown to be unconditionally stable when the equations of motion are linearized about an isothermal basic state. It is also very efficient because the implicit integrations can either be solved directly (in the case of the Coriolis terms) or give rise to a Helmholtz equation for which efficient fast solvers exist (in the case of the pressure gradient terms). A real-time run of this scheme in parallel with the Irish Meteorological Service's operational forecast model showed no deterioration in forecast quality while cutting the required CPU by more than 60%.

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A. McDonald

Abstract

The degree of accuracy of various multiply-upstream semi-Lagrangian schemes is examined by analyzing the driven one-dimensional advection equation. The relationship between the order of accuracy of a given scheme and the order of the interpolations used, both of find the position of, and the value of the fields at, the departure point is established. In the process, it is shown how to construct a semi-Lagrangian scheme which is accurate to third-order.

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A. McDonald

Abstract

An unmeteorological oscillation in the forecast fields produced by a two-time-level semi-Lagrangian and semi-implicit model has been traced to the method used for finding the departure point position. Various alternatives to the traditional method for determining it are examined and tested in a forecast model in order to find the one that removes the unmeteorological oscillation most satisfactorily.

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A. McDonald

Abstract

Transparent boundary conditions for the linearized shallow water equations are constructed by incorporating the boundary conditions into equations that describe unidirectional waves. The shallow water equations are then discretized using a semi-Lagrangian approach and the transparency of the boundaries is demonstrated for three scenarios: adjustment waves radiating out of the area, a geostrophically balanced disturbance being advected in through the boundaries, and a geostrophically balanced disturbance being advected out of the area.

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A. McDonald

Abstract

Most operational numerical weather prediction models update their lateral boundaries via the Davies–Kållberg relaxation scheme. In this paper some potential alternatives that take transparency and well-posedness into consideration are derived and tested for the shallow-water equations. The accuracy of the new boundary strategies is displayed by integrating the shallow-water equations in a nested environment using real meteorological data. In these tests they are seen to be superior to the traditional relaxation boundary strategy.

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A. McDonald

Abstract

The stability and accuracy of the multiply-upstream, semi-Lagrangian method of integrating the advective equation in two dimensions is examined for four different interpolation schemes; namely, bilinear, biquadratic, bicubic and biquartic. All are shown to be consistent and unconditionally stable for constant advecting velocity. Their respective amplitude and phase errors are discussed. They are then used to integrate the test problem of a cone being advected about the plane at constant angular velocity. The merits of the schemes relative to each other and relative to a well tried Eulerian scheme am examined with particular regard to accuracy and computation time.

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A. McDonald

Abstract

The one-dimensional advection equation and the one-dimensional advection adjustment equation with rotation are used as test beds for the design of well-posed boundary conditions for the initial-boundary-value problem using semi-Lagrangian discretization. Three options are found to be stable in experimental tests: trajectory truncation, time interpolation, and a well-posed buffer zone. Stability is proved for all three for the one-dimensional advection equation when linear interpolation is used for the interpolation associated with the semi-Lagrangian discretization.

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Nils Gustafsson and A. McDonald

Abstract

A comparison of two semi-Lagrangian limited area models, one spectral, and the other grid point, is described. Forecasts from both models are compared and contrasted, first on a 55-km horizontal mesh and subsequently on a 22-km horizontal mesh. The weaknesses in the respective models exposed by these tests, and the corrections made to overcome them are described. The final models arrived at are shown to be accurate and more efficient than the Eulerian counterpart for the test dataset. It is also found that the spectral model is as accurate as the gridpoint model and is also computationally competitive.. It is concluded that with sufficient thought and effort the gridpoint and spectral models can be made to produce equally good forecasts at comparable computer costs. Finally, a reassessment of the relative merits and drawbacks of the spectral and gridpoint schemes is attempted taking into account the fact that the advection terms are integrated by semi-Lagrangian scheme.

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A. McDonald and Janerik Haugen

Abstract

A two-time-level, three-dimensional semi-Lagrangian, semi-implicit primitive equation gridpoint model that incorporates a sophisticated physics package is presented. It is shown to give accurate 24-h forecasts when integrated over a limited area using a 1.5°×1.5° Arakawa C grid in the horizontal and 16 levels in the vertical for time steps up to 2 h. Also, it is shown to be as accurate as, and approximately twice as efficient as, a three-time-level semi-Lagrangian scheme for time steps up to 2 h but slightly less accurate for a 3-h time step. Finally, it is shown to give accurate forecasts on a 0.5°×0.5° horizontal grid, again using 16 vertical levels, for time steps up to 40 min.

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A. McDonald and Jan Erik Haugen

Abstract

A two time-level, three-dimensional, semi-Lagrangian semi-implicit primitive equation gridpoint model that incorporates a sophisticated physics package and uses hybrid coordinates in the vertical is derived. A simple filter, which is needed to stabilize large time-step forecasts, is introduced. Using it, the model is shown to give accurate 24-h forecasts when integrated over a limited area using a 1.5°×1.5° Arakawa C grid in the horizontal and 16 levels in the vertical for time steps up to 2 h. Also, it is shown to give accurate forecasts on a 0.5°×0.5° horizontal grid, again using 16 vertical levels, for time steps up to 40 min, and to be as accurate as, and approximately twice as efficient as, a three time-level semi-Lagrangian scheme.

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