Hostname: page-component-84b7d79bbc-4hvwz Total loading time: 0 Render date: 2024-07-30T04:15:30.188Z Has data issue: false hasContentIssue false
  • English
  • Français

Finite-amplitude gravity waves in the atmosphere: travelling wave solutions

Published online by Cambridge University Press:  15 August 2017

Mark Schlutow Open the ORCID record for Mark Schlutow [Opens in a new window] ,
R. Klein  and
U. Achatz
Mark Schlutow*
Affiliation:
Institut für Mathematik, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
R. Klein
Affiliation:
Institut für Mathematik, Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany
U. Achatz
Affiliation:
Institut für Atmosphäre und Umwelt, Goethe-Universität Frankfurt, Altenhöferallee 1, 60438 Frankfurt am Main, Germany
*
†Email address for correspondence: mark.schlutow@fu-berlin.de
Get access Rights & Permissions [Opens in a new window]

Abstract

Wentzel–Kramers–Brillouin theory was employed by Grimshaw (Geophys. Fluid Dyn., vol. 6, 1974, pp. 131–148) and Achatz et al. (J. Fluid Mech., vol. 210, 2010, pp. 120–147) to derive modulation equations for non-hydrostatic internal gravity wave packets in the atmosphere. This theory allows for wave packet envelopes with vertical extent comparable to the pressure scale height and for large wave amplitudes with wave-induced mean-flow speeds comparable to the local fluctuation velocities. Two classes of exact travelling wave solutions to these nonlinear modulation equations are derived here. The first class involves horizontally propagating wave packets superimposed over rather general background states. In a co-moving frame of reference, examples from this class have a structure akin to stationary mountain lee waves. Numerical simulations corroborate the existence of nearby travelling wave solutions under the pseudo-incompressible model and reveal better than expected convergence with respect to the asymptotic expansion parameter. Travelling wave solutions of the second class also feature a vertical component of their group velocity but exist under isothermal background stratification only. These waves include an interesting nonlinear wave–mean-flow interaction process: a horizontally periodic wave packet propagates vertically while draining energy from the mean wind aloft. In the process it decelerates the lower-level wind. It is shown that the modulation equations apply equally to hydrostatic waves in the limit of large horizontal wavelengths. Aside from these results of direct physical interest, the new nonlinear travelling wave solutions provide a firm basis for subsequent studies of nonlinear internal wave instability and for the design of subtle test cases for numerical flow solvers.

JFM classification

Geophysical and Geological Flows: Atmospheric flows Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 826 , 10 September 2017 , pp. 1034 - 1065
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Achatz, U. 2007 Gravity-wave breaking: linear and primary nonlinear dynamics. Adv. Space Res. 40, 719–733.CrossRefGoogle Scholar
Achatz, U., Klein, R. & Senf, F. 2010 Gravity waves, scale asymptotics and the pseudo-incompressible equations. J. Fluid Mech. 663, 120–147.CrossRefGoogle Scholar
Achatz, U., Ribstein, B., Senf, F. & Klein, R. 2017 The interaction between synoptic-scale balanced flow and a finite-amplitude mesoscale wave field throughout all atmospheric layers: weak and moderately strong stratification. Q. J. R. Meteorol. Soc. 143 (702), 342–361.CrossRefGoogle Scholar
Alexander, M. J. & Dunkerton, T. J. 1999 A spectral parameterization of mean-flow forcing due to breaking gravity waves. J. Atmos. Sci. 56 (24), 4167–4182.Google Scholar
Baldwin, M. P., Gray, L. J., Dunkerton, T. J., Hamilton, K., Haynes, P. H., Randel, W. J., Holton, J. R., Alexander, M. J., Hirota, I., Horinouchi, T. et al. 2001 The quasi-biennial oscillation. Rev. Geophys. 39 (2), 179–229.Google Scholar
Becker, E. 2012 Dynamical control of the middle atmosphere. Space Sci. Rev. 168 (1–4), 283–314.Google Scholar
Bölöni, G., Ribstein, B., Muraschko, J., Sgoff, C., Wei, J. & Achatz, U. 2016 The interaction between atmospheric gravity waves and large-scale flows: an efficient description beyond the non-acceleration paradigm. J. Atmos. Sci. 73 (12), 4833–4852.CrossRefGoogle Scholar
Bühler, O. 2009 Waves and Mean Flow, 1st edn. Cambridge University Press.Google Scholar
Chu, V. H. & Mei, C. C. 1970 On slowly-varying Stokes waves. J. Fluid Mech. 41, 873–887.Google Scholar
Danilov, V. G., Omelyanov, G. A. & Shelkovich, V. M. 2003 Weak asymptotics method and interaction of nonlinear waves. In Asymptotic Methods for Wave and Quantum Problems (ed. Karasev, M. V.), pp. 33–163. American Mathematical Soceity.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, 2761–2775.Google Scholar
Dosser, H. V. & Sutherland, B. R. 2011 Weakly nonlinear non-Boussinesq internal gravity wavepackets. Physica D 240 (3), 346–356.Google Scholar
Fritts, D. C. 2003 Gravity wave dynamics and effects in the middle atmosphere. Rev. Geophys. 41, 1–64.Google Scholar
Grimshaw, R. 1974 Internal gravity waves in a slowly varying, dissipative medium. Geophys. Fluid Dyn. 6, 131–148.CrossRefGoogle Scholar
Klein, R. 2011 On the regime of validity of sound-proof model equations for atmospheric flows. In ECMWF Workshop on Non-Hydrostatic Modelling, November 2010, http://www.ecmwf.int/publications/library/do/references/list/201010.Google Scholar
Klein, R., Achatz, U., Bresch, D., Knio, O. M. & Smolarkiewicz, P. K. 2010 Regime of validity of soundproof atmospheric flow models. J. Atmos. Sci. 67 (10), 3226–3237.Google Scholar
Lelong, M.-P. & Dunkerton, T. J. 1998 Inertia-gravity wave breaking in three dimensions. Part I: convectively stable waves. J. Atmos. Sci. 55 (1997), 2473–2488.2.0.CO;2>CrossRefGoogle Scholar
Liu, W., Bretherton, F. P., Liu, Z., Smith, L., Lu, H. & Rutland, C. J. 2010 Breaking of progressive internal gravity waves: convective instability and shear instability. J. Phys. Oceanogr. 40, 2243–2263.CrossRefGoogle Scholar
Lombard, P. N. & Riley, J. J. 1996 Instability and breakdown of internal gravity waves. I. Linear stability analysis. Phys. Fluids 8 (12), 3271–3287.Google Scholar
Mclandress, C. 1998 On the importance of gravity waves in the middle atmosphere and their parameterization in general circulation models. J. Atmos. Sol.-Terr. Phys. 60 (14), 1357–1383.Google Scholar
Mied, R. P. 1976 The occurrence of parametric instabilities in finite-amplitude internal gravity waves. J. Fluid Mech. 78, 763–784.Google Scholar
Miura, R. M. & Kruskal, M. D. 1974 Application of a nonlinear WKB method to the Korteweg–DeVries equation. SIAM J. Appl. Maths 26 (2), 376–395.CrossRefGoogle Scholar
Muraschko, J., Fruman, M. D., Achatz, U., Hickel, S. & Toledo, Y. 2015 On the application of Wentzel–Kramer–Brillouin theory for the simulation of the weakly nonlinear dynamics of gravity waves. Q. J. R. Meteorol. Soc. 141 (688), 676–697.Google Scholar
Rieper, F., Achatz, U. & Klein, R. 2013a Range of validity of an extended WKB theory for atmospheric gravity waves: one-dimensional and two-dimensional case. J. Fluid Mech. 729, 330–363.Google Scholar
Rieper, F., Hickel, S. & Achatz, U. 2013b A conservative integration of the pseudo-incompressible equations with implicit turbulence parameterization. Mon. Weath. Rev. 141 (3), 861–886.Google Scholar
Sutherland, B. R. 2001 Finite-amplitude internal wavepacket dispersion and breaking. J. Fluid Mech. 429, 343–380.Google Scholar
Sutherland, B. R. 2006 Weakly nonlinear internal gravity wavepackets. J. Fluid Mech. 569, 249–258.CrossRefGoogle Scholar
Tabaei, A. & Akylas, T. R. 2007 Resonant long–short wave interactions in an unbounded rotating stratified fluid. Stud. Appl. Maths 119 (3), 271–296.Google Scholar
Whitham, G. B. 1965 A general approach to linear and non-linear dispersive waves using a Lagrangian. J. Fluid Mech. 22 (2), 273–283.Google Scholar