Editorial Type:
Article
Article Type:
Research Article

On the Group-Velocity Property for Wave-Activity Conservation Laws

J. Vanneste Department of Physics, University of Toronto, Toronto, Ontario, Canada

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T. G. Shepherd Department of Physics, University of Toronto, Toronto, Ontario, Canada

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Print Publication:
01 Mar 1998
DOI:
https://doi.org/10.1175/1520-0469(1998)055<1063:OTGVPF>2.0.CO;2
Page(s):
1063–1068
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Abstract

The density and the flux of wave-activity conservation laws are generally required to satisfy the group-velocity property: under the WKB approximation (i.e., for nearly monochromatic small-amplitude waves in a slowly varying medium), the flux divided by the density equals the group velocity. It is shown that this property is automatically satisfied if, under the WKB approximation, the only source of rapid variations in the density and the flux lies in the wave phase. A particular form of the density, based on a self-adjoint operator, is proposed as a systematic choice for a density verifying this condition.

Corresponding author address: Dr. T. G. Shepherd, Department of Physics, University of Toronto, Toronto, ON M5S 1A7, Canada.

Email: tgs@atmosp.physics.utoronto.ca

Abstract

The density and the flux of wave-activity conservation laws are generally required to satisfy the group-velocity property: under the WKB approximation (i.e., for nearly monochromatic small-amplitude waves in a slowly varying medium), the flux divided by the density equals the group velocity. It is shown that this property is automatically satisfied if, under the WKB approximation, the only source of rapid variations in the density and the flux lies in the wave phase. A particular form of the density, based on a self-adjoint operator, is proposed as a systematic choice for a density verifying this condition.

Corresponding author address: Dr. T. G. Shepherd, Department of Physics, University of Toronto, Toronto, ON M5S 1A7, Canada.

Email: tgs@atmosp.physics.utoronto.ca

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  • Brunet, G., 1994: Empirical normal mode analysis of atmospheric data. J. Atmos. Sci.,51, 932–952.

  • ——, and P. H. Haynes, 1996: Low-latitude reflection of Rossby wave trains. J. Atmos. Sci.,53, 482–496.

  • Durran, D. R., 1995: Pseudomomentum diagnostics for two-dimensional stratified compressible flows. J. Atmos. Sci.,52, 3997–4009.

  • Hayes, M., 1977: A note on group-velocity. Proc. Roy. Soc. London, Ser. A,354, 533–535.

  • Haynes, P. H., 1988: Forced, dissipative generalizations of finite-amplitude wave-activity conservation relations for zonal and nonzonal basic flows. J. Atmos. Sci.,45, 2352–2362.

  • Magnusdottir, G., and P. H. Haynes, 1996: Wave activity diagnostics applied to baroclinic life cycles. J. Atmos. Sci.,53, 2317–2353.

  • McIntyre, M. E., and T. G. Shepherd, 1987: An exact local conservation theorem for finite-amplitude disturbances to non-parallel shear flows, with remarks on Hamiltonian structure and on Arnol’d’s stability theorems. J. Fluid Mech.,181, 527–565.

  • Olver, P. J., 1993: Application of Lie Groups to Differential Equations. Springer-Verlag, 513 pp.

  • Pedlosky, J., 1987: Geophysical Fluid Dynamics. Springer-Verlag, 710 pp.

  • Plumb, R. A., 1985: On the three-dimensional propagation of stationary waves. J. Atmos. Sci.,42, 217–229.

  • Scinocca, J. F., and T. G. Shepherd, 1992: Nonlinear wave-activity conservation laws and Hamiltonian structure for the two-dimensional anelastic equations. J. Atmos. Sci.,49, 5–25.

  • ——, and W. R. Peltier, 1994a: Finite-amplitude wave-activity diagnostics for Long’s stationary solution. J. Atmos. Sci.,51, 613–622.

  • ——, and ——, 1994b: The instability of Long’s stationary solution and the evolution toward severe downslope windstorm flow. Part II: The application of finite-amplitude local wave-activity flow diagnostics. J. Atmos. Sci.,51, 623–653.

  • Shepherd, T. G., 1990: Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics. Advances in Geophysics, Vol. 32, Academic Press, 287–338.

  • Vanneste, J., 1997: On the derivation of fluxes for conservation laws in Hamiltonian systems. I.M.A. J. Appl. Math., in press.

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