A Vertical Finite-Difference Scheme Based on a Hybrid σ-θ-p Coordinate

Zhengxin Zhu Department of Meteorology, University of Reading, Reading, United Kingdom

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John Thuburn Department of Meteorology, University of Reading, Reading, United Kingdom

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Brian J. Hoskins Department of Meteorology, University of Reading, Reading, United Kingdom

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Peter H. Haynes Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, United Kingdom

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Abstract

A vertical discretization of the primitive equations in a general vertical coordinate is described that enables a primitive equation model to use terrain-following sigma levels near the ground and isentropic levels higher up, with a smooth transition region in between. Therefore, it combines many of the advantages of the computational efficiency of σ coordinates and the predictive and diagnostic potential of θ coordinates, and should be particularly useful for general circulation models to be used for studies of stratosphere-troposphere exchange and middle-atmosphere transport of trace gases. It is shown that the semi-implicit time scheme can be used in a straightforward manner with this discretization. A discussion is given of how to optimize the transition from sigma levels to isentropic levels so as to avoid model levels crossing each other. A numerical problem caused when very shallow, very strong inversions occur in the temperature field is countered by a form of vertical-scale selective dissipation. Baroclinic wave life cycles and full general circulation simulations have been successfully performed with a modified version of the European Centre for Medium-Range Weather Forecasts model.

Abstract

A vertical discretization of the primitive equations in a general vertical coordinate is described that enables a primitive equation model to use terrain-following sigma levels near the ground and isentropic levels higher up, with a smooth transition region in between. Therefore, it combines many of the advantages of the computational efficiency of σ coordinates and the predictive and diagnostic potential of θ coordinates, and should be particularly useful for general circulation models to be used for studies of stratosphere-troposphere exchange and middle-atmosphere transport of trace gases. It is shown that the semi-implicit time scheme can be used in a straightforward manner with this discretization. A discussion is given of how to optimize the transition from sigma levels to isentropic levels so as to avoid model levels crossing each other. A numerical problem caused when very shallow, very strong inversions occur in the temperature field is countered by a form of vertical-scale selective dissipation. Baroclinic wave life cycles and full general circulation simulations have been successfully performed with a modified version of the European Centre for Medium-Range Weather Forecasts model.

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