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André Robert

Abstract

Atmospheric models based on the Euler equations exist and are used occasionally to carry out numerical experiments. Such a model is used here to simulate the motion of warm bubbles in a dry isentropic atmosphere. For the time integration, this model uses a scheme that is semi-Lagrangian and semi-implicit. It is the stability and the efficiency of this integration scheme that is examined in the context of an isentropic vertical stratification. Some results are presented and are compared with the results generated by similar experiments reported in the scientific literature.

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André Robert

Abstract

A brief examination of some recent applications of the spectral method will clearly establish and display the rapid progress accomplished by this technique over the past several years. A discussion of the feasibility of extending this method to cover weather forcasting problems will follow.

Some results of numerical integrations of atmospheric models by the spectral method are presented. A scheme developed for the prediction of precipitation is then described, followed by a very simple numerical experiment.

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Eyhen Yakimiw
and
André Robert

Abstract

A fully implicit scheme proposed by a group located at the Courant Institute is examined and analyzed for its stability and accuracy. The scheme seems to be easily adaptable to numerical integration in large-scale atmospheric models and could be more efficient than the semi-implicit or the split-explicit algorithms currently used in these models.

Linear analysis shows us that the fractional step method, applied in the scheme in order to simplify the calculations, introduces truncation errors that affect the Rossby wave propagation. These errors we unimportant when short time steps are used but they become quite substantial for large steps. This result suggests that the proposed technique may not be as attractive as originally anticipated.

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Monique Tanguay
and
André Robert

Abstract

A modification is introduced in a semi-implicit version of a grid point model of the shallow water equations. The new model is simpler, runs one-third, and after 5 days of integration, the forecasts differ by less than 1 m.

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ANDRÉ J. ROBERT

Abstract

This project used a series of 500-mb. charts prepared originally for the study of planetary waves. These charts, covering both hemispheres, provided the initial conditions for a spectral barotropic model. In this model, the calculations proceeded from functions equivalent to spherical harmonics with the stream field represented by 153 degrees of freedom. A set of five integrations carried to 72 hr. produced reasonably good forecasts that did not appear to be affected seriously by the deficiencies of the observational network.

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Chantal Rivest
,
Andrew Staniforth
, and
André Robert

Abstract

Semi-Lagrangian semi-implicit techniques are now well established and used by an increasing number of meteorological centers. However, it is demonstrated by both analysis and numerical integration that there is a serious problem incorporating orographic forcing into semi-Lagrangian models, since spurious resonance can develop in mountainous regions for Courant numbers larger than unity. A solution, consisting of two classes of schemes, is proposed, analyzed, and then evaluated using a global shallow-water model. Simply off-centering the semi-implicit scheme eliminates the spurious resonances. Although this can be achieved with a first-order scheme, it is at the expense of decreased accuracy, and therefore a second-order scheme is recommended.

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ANDRÉ ROBERT
,
JOHN HENDERSON
, and
COLIN TURNBULL

Abstract

A semi-implicit time integration algorithm developed earlier for a barotropic model resulted in an appreciable economy of computing time. An extension of this method to baroclinic models is formulated, including a description of the various steps in the calculations. In the proposed scheme, the temperature is separated into a basic part dependent only on the vertical coordinate and a corresponding perturbation part. All terms involving the perturbation temperature are calculated from current values of the variables, while a centered finite-difference time average is applied to the horizontal pressure gradient, the divergence, and the vertical motion in the remaining terms. This method gives computationally stable integrations with relatively large time steps.

The model used to test the semi-implicit scheme does not include topography, precipitation, diabatic heating, and other important physical processes. Five-day hemispheric integrations from real data with time steps of 60 and 30 min show differences of the order of 3 m. These errors are insignificant when compared to other sources of error normally present in most numerical models. Presently, this model produces relatively good short-range predictions, and this is a strong factor in favor of inserting the major physical processes as soon as possible.

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Monique Tanguay
,
André Robert
, and
René Laprise

Abstract

The semi-implicit algorithm, originally developed by Robert for an economical integration of the primitive equations in large-scale models of the atmospheric, is here generalized in order to integrate the fully compressible, nonhydrostatic equations. We show that there is little computational overhead associated with the integration of the full, and hence presumably more correct, set of equations that do not invoke the hydrostatic assumption to exclude the high frequency, vertically propagating acoustic modes.

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Georges Jakimow
,
Evhen Yakimiw
, and
André Robert

Abstract

Explicit formulations for horizontal diffusion in atmospheric models are only conditionally stable. As a consequence, the coefficient of diffusion cannot be increased beyond a critical value that depends on the grid length and the time step. When a semi-Lagrangian and semi-implicit integration scheme is used with large time steps, the upper limit imposed on the coefficient of diffusion seems to be unreasonably low for some particular applications. Global integration on a regular spherical grid is one example of an application where an implicit formulation appears to be desirable.

An implicit formulation that is unconditionally stable is proposed for gridpoint models. It is tested in a tridimensional hemispheric model. The model is integrated to 5 days with various values of the coefficient of diffusion. All forecasts are verified, and the value of the coefficient that gives the best score is retained. It is found that this optimum value is larger than the values commonly used in most models. These values are used strictly to test the proposed implicit formulation.

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MICHAEL KWIZAK
and
ANDRÉ J. ROBERT

Abstract

A semi-implicit time integration scheme tested earlier with a spectral model is now adapted to a grid point model of the primitive equations. Predictions prepared by the implicit method compare quite favorably with the forecasts produced by an explicit technique. The implicit model runs about four times faster; and after 5 days of integration, the forecasts differ by less than 20 m.

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