• Allen, J. S., , J. A. Barth, , and P. A. Newberger, 1990: On intermediate models for barotropic continental shelf and slope flow fields. Part I: Formulation and comparison of exact solutions. J. Phys. Oceanogr, 20 , 10171042.

    • Search Google Scholar
    • Export Citation
  • Aris, R., 1962: Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Dover, 286 pp.

  • Craig, G. C., 1993: A scaling for the three-dimensional semigeostrophic approximation. J. Atmos. Sci, 50 , 33503355.

  • Cullen, M. J. P., , and R. J. Douglas, 2003: Large-amplitude nonlinear stability results for atmospheric circulations. Quart. J. Roy. Meteor. Soc, 129 , 19691988.

    • Search Google Scholar
    • Export Citation
  • Eliassen, A., 1949: The quasi-static equations of motion with pressure as independent variable. Geofys. Publ, 17 (3) 144.

  • Holton, J. R., 1992: An Introduction to Dynamic Meteorology. International Geophysics Series, Vol. 48, Academic Press, 511 pp.

  • Hoskins, B. J., 1975: The geostrophic momentum approximation and the semi-geostrophic equations. J. Atmos. Sci, 32 , 233242.

  • Hoskins, B. J., , and F. P. Bretherton, 1972: Atmospheric frontogenesis models: Mathematical formulation and solution. J. Atmos. Sci, 29 , 1137.

    • Search Google Scholar
    • Export Citation
  • Kahlig, P., 1974: Der Effekt der Führungsbewegung in hydrodynamischen Gleichungen (in German). Arch. Meteor. Geophys. Biokl. Ser. A, 23 , 87100.

    • Search Google Scholar
    • Export Citation
  • McIntyre, M. E., , and I. Roulstone, 2002: Are there higher-accuracy analogues of semigestrophic theory? Geometric Methods and Models, Vol. 2, Large-Scale Atmosphere–Ocean Dynamics, J. Norbury and I. Roulstone, Eds., Cambridge University Press, 301–364.

    • Search Google Scholar
    • Export Citation
  • Pichler, H., 1997: Dynamik der Atmosphäre (in German). Spektrum Akademischer Verlag, 572 pp.

  • Purser, R. J., 1999: Legendre-transformable semigeostrophic theories. J. Atmos. Sci, 56 , 25222535.

  • Ren, S., 2000: Finite-amplitude wave-activity invariants and nonlinear stability theorems for shallow water semigeostrophic dynamics. J. Atmos. Sci, 57 , 33883397.

    • Search Google Scholar
    • Export Citation
  • Salmon, R., 1988: Semigeostrophic theory as a Dirac-bracket projection. J. Fluid Mech, 196 , 345358.

  • Simmonds, J. G., 1994: A Brief on Tensor Analysis. Springer, 112 pp.

  • Spiegel, M. R., 1959: Schaum's Outline of Theory and Problems of Vector Analysis and an Introduction to Tensor Analysis. McGraw-Hill, 225 pp.

    • Search Google Scholar
    • Export Citation
  • White, A. A., 2002: A view of the equations of meteorological dynamics and various approximations. Analytical Methods and Numerical Models, Vol. 1, Large-Scale Atmosphere-Ocean Dynamics, J. Norbury and I. Roulstone, Eds., Cambridge University Press, 1–100.

    • Search Google Scholar
    • Export Citation
  • Zdunkowski, W., , and A. Bott, 2003: Dynamics of the Atmosphere: A Course in Theoretical Meteorology. Cambridge University Press, 719 pp.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 156 156 3
PDF Downloads 36 36 3

A Vector Derivation of the Semigeostrophic Potential Vorticity Equation

View More View Less
  • 1 Institute for Meteorology and Geophysics, University of Innsbruck, Innsbruck, Austria
© Get Permissions Rent on DeepDyve
Restricted access

Abstract

The semigeostrophic potential vorticity equation is derived in a vector-based notation for the shallow-water and primitive equations, as models for atmospheric flow. The derivation proceeds from knowledge of the functional form of potential vorticity and starts directly from a vector form of the governing equations. The method makes use of highly involved vector identities and provides for a clearer picture of the nature of the final result, as compared to a component-based derivation. It is, however, limited by the need to know a priori the appropriate functional vector form of the dynamically relevant quantity, such as potential vorticity.

Corresponding author address: Dr. Martin Ehrendorfer, Institute for Meteorology and Geophysics, University of Innsbruck, Innrain 52, A-6020 Innsbruck, Austria. Email: martin.ehrendorfer@uibk.ac.at

Abstract

The semigeostrophic potential vorticity equation is derived in a vector-based notation for the shallow-water and primitive equations, as models for atmospheric flow. The derivation proceeds from knowledge of the functional form of potential vorticity and starts directly from a vector form of the governing equations. The method makes use of highly involved vector identities and provides for a clearer picture of the nature of the final result, as compared to a component-based derivation. It is, however, limited by the need to know a priori the appropriate functional vector form of the dynamically relevant quantity, such as potential vorticity.

Corresponding author address: Dr. Martin Ehrendorfer, Institute for Meteorology and Geophysics, University of Innsbruck, Innrain 52, A-6020 Innsbruck, Austria. Email: martin.ehrendorfer@uibk.ac.at

Save