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An Implicit Formulation for Horizontal Diffusion in Gridpoint Models

Georges JakimowUniversité du Québec, Département de physique, Montréal, Québec, Canada

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Evhen YakimiwUniversité du Québec, Département de physique, Montréal, Québec, Canada and Recherche en prévision numérique, Atmospheric Environment Service, Dorval, Québec, Canada

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André RobertUniversité du Québec, Département de physique, Montréal, Québec, Canada

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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.

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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