• Ball, F. K., 1956: Energy changes involved in disturbing a dry atmosphere. Quart. J. Roy. Meteor. Soc., 82 , 1529.

  • Bohren, C. F., and B. A. Albrecht, 1998: Atmospheric Thermodynamics. Oxford University Press, 402 pp.

  • Boltzmann, L., 1896: Vorlesungen über Gastheorie. Part 1, Barth, 204 pp.

  • COESA, 1976: U.S. Standard Atmosphere, 1976. U.S. Government Printing Office, 227 pp.

  • Dutton, J. A., 1973: The global thermodynamics of atmospheric motion. Tellus, 25 , 89110.

  • Emden, R., 1926: Thermodynamik der Himmelskörper. Astronomie, B.Encyklopädie der Mathematischen Wissenschaften, VI, 2, K. Schwarzschild and S. Oppenheim, Eds., Teubner, 373–532.

    • Search Google Scholar
    • Export Citation
  • Ertel, H., 1938: Methoden und Probleme der Dynamischen Meteorologie. Springer, 122 pp.

  • Exner, F. M., 1925: Dynamische Meteorologie. Springer, 421 pp.

  • Gibbs, J. W., 1961: Thermodynamics. Vol. 1, The Scientific Papers of J. Willard Gibbs, Dover, 434 pp.

  • Holton, J. R., 1992: An Introduction to Dynamic Meteorology. Academic Press, 507 pp.

  • Maxwell, J. C., 1888: Theory of Heat. 9th ed. Longmans, Green, 333 pp.

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On Maximum Entropy Profiles

W. T. M. VerkleyRoyal Netherlands Meteorological Institute (KNMI), De Bilt, Netherlands

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T. Gerkema Laboratoire des Écoulements Géophysiques et Industriels, Grenoble, France

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Abstract

A column of dry air in hydrostatic equilibrium is considered, bounded by two fixed values of the pressure, and the question is asked, what vertical temperature profile maximizes the total entropy of the column? Using an elementary variational calculation, it is shown how the result depends on what is kept fixed in the maximization process. If one assumes that there is no net heat exchange between the column and its surroundings—implying that the vertical integral of the absolute temperature remains constant—an isothermal profile is obtained in accordance with classical thermodynamics and the kinetic theory of gases. If instead the vertical integral of the potential temperature is kept fixed—as argued by several authors to be appropriate in the case of convective mixing—an isentropic profile results. It is argued that, if one wishes to apply the latter constraint, it should be used as an additional, rather than as an alternative, constraint. The variational problem with both constraints leads to a profile in between the isothermal and the isentropic extremes. This profile has the merit of reproducing very accurately the tropospheric part of the U.S. Standard Atmosphere, 1976.

Corresponding author address: Dr. W. T. M. Verkley, Royal Netherlands Meteorological Institute, P.O. Box 201, 3730 AE De Bilt, Netherlands. Email: verkley@knmi.nl

Abstract

A column of dry air in hydrostatic equilibrium is considered, bounded by two fixed values of the pressure, and the question is asked, what vertical temperature profile maximizes the total entropy of the column? Using an elementary variational calculation, it is shown how the result depends on what is kept fixed in the maximization process. If one assumes that there is no net heat exchange between the column and its surroundings—implying that the vertical integral of the absolute temperature remains constant—an isothermal profile is obtained in accordance with classical thermodynamics and the kinetic theory of gases. If instead the vertical integral of the potential temperature is kept fixed—as argued by several authors to be appropriate in the case of convective mixing—an isentropic profile results. It is argued that, if one wishes to apply the latter constraint, it should be used as an additional, rather than as an alternative, constraint. The variational problem with both constraints leads to a profile in between the isothermal and the isentropic extremes. This profile has the merit of reproducing very accurately the tropospheric part of the U.S. Standard Atmosphere, 1976.

Corresponding author address: Dr. W. T. M. Verkley, Royal Netherlands Meteorological Institute, P.O. Box 201, 3730 AE De Bilt, Netherlands. Email: verkley@knmi.nl

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