• Arakawa, A., 1966: Computational design for long-term numerical integration of the equations of fluid motion: Two-dimensional incompressible flow. Part I. J. Comput. Phys.,1, 119–143.

  • ——, and V. Lamb, 1981: A potential enstrophy and energy conserving scheme for the shallow water equations. Mon. Wea. Rev.,109, 18–36.

  • Bleck, R., 1973: Numerical forecasting experiments based on the conservation of potential vorticity on isentropic surfaces. J. Appl. Meteor.,12, 737–752.

  • Egger, J., 1996: Volume conservation in phase space: A fresh look at numerical integration schemes. Mon. Wea. Rev.,124, 1955–1964.

  • Haken, H., 1988: Information and Self-Organization. Springer, 196 pp.

  • Huang, K., 1987: Statistical Mechanics. John Wiley and Sons, 493 pp.

  • Johnson, D. 1997: “General coldness of climate models” and the second law: Implications for modeling the earth system. J. Climate,10, 2826–2846.

  • Mesinger F., and A. Arakawa 1976: Numerical methods used in atmospheric models. 65 pp. [Available from World Meteorological Organization. GARP Publ. Ser., Vol. 17, WMO, Case postale No. 5, CH-1211 Geneva 20, Switzerland.].

  • Peixoto, J., A. Oort, M. Almeida, and A. Tome, 1991: Entropy budget of the atmosphere. J. Geophys. Res.,96 (D6), 10 981–10 988.

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Numerical Generation of Entropies

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  • 1 Meteorologlisches Institut, Universität München, Munich, Germany
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Abstract

The spurious numerical generation and/or destruction of various types of entropies in models is investigated. It is shown that entropy sθ of dry matter tends to be generated if potential temperature is advected by a damping scheme. There is no mean tendency of entropy if the reversible leapfrog scheme is used. Generalized entropies can be assigned to conserved quantities. In particular, the generalized entropy sζ of the vorticity of two-dimensional nondivergent flow is shown to grow in presence of irreversible diffusive processes. This entropy increases numerically if the vorticity equation is integrated with an upstream scheme. There are weak oscillations of sζ if a leapfrog time step is combined with the Arakawa scheme. Similar results are obtained for an entropy sp related to potential vorticity. Information entropy provides a gross measure of the information contained in ensemble forecasts. It is shown that information entropy decreases spuriously if schemes are used that are contracting in phase space. It is argued that the evaluation of entropies provides a useful check of the quality of numerical schemes.

Corresponding author address: Dr. Joseph Egger, Meteorologisches Institut, Universität München, Theresienstraße 37, Arbeitsgruppe fur Theoretische Meteorologie, 80333 München, Germany.

Email: J.Egger@lrz.uni-muenchen.de

Abstract

The spurious numerical generation and/or destruction of various types of entropies in models is investigated. It is shown that entropy sθ of dry matter tends to be generated if potential temperature is advected by a damping scheme. There is no mean tendency of entropy if the reversible leapfrog scheme is used. Generalized entropies can be assigned to conserved quantities. In particular, the generalized entropy sζ of the vorticity of two-dimensional nondivergent flow is shown to grow in presence of irreversible diffusive processes. This entropy increases numerically if the vorticity equation is integrated with an upstream scheme. There are weak oscillations of sζ if a leapfrog time step is combined with the Arakawa scheme. Similar results are obtained for an entropy sp related to potential vorticity. Information entropy provides a gross measure of the information contained in ensemble forecasts. It is shown that information entropy decreases spuriously if schemes are used that are contracting in phase space. It is argued that the evaluation of entropies provides a useful check of the quality of numerical schemes.

Corresponding author address: Dr. Joseph Egger, Meteorologisches Institut, Universität München, Theresienstraße 37, Arbeitsgruppe fur Theoretische Meteorologie, 80333 München, Germany.

Email: J.Egger@lrz.uni-muenchen.de

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