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Gravity waves, scale asymptotics and the pseudo-incompressible equations

Published online by Cambridge University Press:  27 September 2010

ULRICH ACHATZ*
Affiliation:
Institut für Atmosphäre und Umwelt, Goethe-Universität Frankfurt, Altenhöferallee 1, 60438 Frankfurt am Main, Germany
R. KLEIN
Affiliation:
Institut für Mathematik, Freie Universität Berlin, Arnimallee 6, 14195 Berlin-Dahlem, Germany
F. SENF
Affiliation:
Leibniz-Institut für Atmosphärenphysik an der Universität Rostock, Schlossstraße 6, 18225 Kühlungsborn, Germany
*
Email address for correspondence: achatz@iau.uni-frankfurt.de

Abstract

Multiple-scale asymptotics is used to analyse the Euler equations for the dynamical situation of a gravity wave (GW) near breaking level. A simple saturation argument in combination with linear theory is used to obtain the relevant dynamical scales. As a small expansion parameter, the ratio of the inverse of the vertical wavenumber and potential temperature and pressure scale heights is used, which we allow to be of the same order of magnitude here. It is shown that the resulting equation hierarchy is consistent with that obtained from the pseudo-incompressible equations, both for non-hydrostatic and hydrostatic GWs, while this is not the case for the anelastic equations unless the additional assumption of sufficiently weak stratification is adopted. To describe vertical propagation of wavepackets over several atmospheric-scale heights, Wentzel–Kramers–Brillouin (WKB) theory is used to show that the pseudo-incompressible flow divergence generates the same amplitude equation that also obtains from the full Euler equations. This gives a mathematical justification for the use of the pseudo-incompressible equations in the study of GW breaking in the atmosphere for arbitrary background stratification. The WKB theory interestingly even holds at wave amplitudes close to static instability. In the mean-flow equations, we obtain in addition to the classic wave-induced momentum-flux divergences a wave-induced correction of hydrostatic balance in the vertical momentum equation, which cannot be obtained from Boussinesq or anelastic dynamics.

Type
Papers
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Achatz, U. 2007 Gravity-wave breaking: linear and primary nonlinear dynamics. Adv. Space Res. 40, 719733.Google Scholar
Bannon, P. R. 2001 On the anelastic approximation for a compressible atmosphere. J. Atmos. Sci. 53, 36183628.Google Scholar
Batchelor, G. K. 1953 The conditions for dynamical similarity of motions of a frictionless perfect-gas atmosphere. Q. J. R. Meteorol. Soc. 79, 224235.Google Scholar
Bretherton, F. P. 1966 The propagation of groups of internal gravity waves in a shear flow. Q. J. R. Meteorol. Soc. 92, 466480.Google Scholar
Davies, T., Staniforth, A., Wood, N. & Thuburn, J. 2003 Validity of anelastic and other equation sets as inferred from normal-mode analysis. Q. J. R. Meteorol. Soc. 129, 27612775.Google Scholar
Dunkerton, T. J. 1997 Shear instability of internal inertia-gravity waves. J. Atmos. Sci. 54, 16281641.Google Scholar
Durran, D. R. 1989 Improving the anelastic approximation. J. Atmos. Sci. 46, 14531461.Google Scholar
Durran, D. R. 2008 A physically motivated approach for filtering acoustic waves from the equations governing compressible stratified flow. J. Fluid Mech. 601, 365379.Google Scholar
Durran, D. R. & Arakawa, A. 2007 Generalizing the Boussinesq approximation to stratified compressible flow. C. R. Mec. 355, 655664.Google Scholar
Fritts, D. C. & Alexander, M. J. 2003 Gravity wave dynamics and effects in the middle atmosphere. Rev. Geophys. 41 (1), 1003.Google Scholar
Fritts, D. C., Vadas, S. L., Wan, K. & Werne, J. A. 2006 Mean and variable forcing of the middle atmosphere by gravity waves. J. Atmos. Sol.-Terr. Phys. 68, 247265.Google Scholar
Grimshaw, R. 1975 Internal gravity waves: critical layer absorption in a rotating fluid. J. Fluid Mech. 70, 287304.Google Scholar
Klein, R. 2000 Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods. Z. Angew. Math. Mech. 80, 765777.Google Scholar
Klein, R. 2009 Asymptotics, structure, and integration of sound-proof atmospheric flow equations. Theor. Comput. Fluid Dyn. 23, 161195.Google Scholar
Klein, R., Achatz, U., Bresch, D., Knio, O. M. & Smolarkiewicz, P. K. 2010 Regime of validity of sound-proof atmospheric flow models. J. Atmos. Sci. (in press).Google Scholar
Lindzen, R. S. 1981 Turbulence and stress owing to gravity wave and tidal breakdown. J. Geophys. Res. 86, 97079714.Google Scholar
Lipps, F. 1990 On the anelastic approximation for deep convection. J. Atmos. Sci. 47, 17941798.Google Scholar
Lipps, F. & Hemler, R. 1982 A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci. 29, 21922210.Google Scholar
Müller, P. 1976 On the diffusion of momentum and mass by internal gravity waves. J. Fluid Mech. 77, 789823.Google Scholar
Nance, L. B. 1994 On the inclusion of compressibility effects in the scorer parameter. J. Atmos. Sci. 54, 362367.Google Scholar
Nance, L. B. & Durran, R. D. 1994 A comparison of three anelastic systems and the pseudoincompressible system. J. Atmos. Sci. 51, 35493565.Google Scholar
Ogura, Y. & Phillips, N. A. 1962 A scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci. 19, 173179.Google Scholar
Prusa, J. M., Smolarkiewicz, P. K. & Wyszogrodzki, A. A. 2008 EULAG: a computational model for multiscale flows. Comput. Fluids 37, 11931207.Google Scholar
Shaw, T. A. & Shepherd, T. G. 2008 Wave-activity conservation laws for the three-dimensional anelastic and Boussinesq equations with a horizontally homogeneous background flow. J. Fluid Mech. 594, 493506.Google Scholar
Shaw, T. A. & Shepherd, T. G. 2009 A theoretical framework for energy and momentum consistency in subgrid-scale parameterization for climate models. J. Atmos. Sci. 66, 30953114.Google Scholar