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Wentzel–Kramers–Brillouin approximation for atmospheric waves

Published online by Cambridge University Press:  16 July 2015

Oleg A. Godin
Oleg A. Godin*
Affiliation:
Cooperative Institute for Research in Environmental Sciences, University of Colorado at Boulder, Boulder, CO 80309-0216, USA NOAA/Earth System Research Laboratory, Physical Sciences Division, Boulder, CO 80305-3328, USA
*
†Email address for correspondence: [email protected]
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Abstract

Ray and Wentzel–Kramers–Brillouin (WKB) approximations have long been important tools in understanding and modelling propagation of atmospheric waves. However, contradictory claims regarding the applicability and uniqueness of the WKB approximation persist in the literature. Here, we consider linear acoustic–gravity waves (AGWs) in a layered atmosphere with horizontal winds. A self-consistent version of the WKB approximation is systematically derived from first principles and compared to ad hoc approximations proposed earlier. The parameters of the problem are identified that need to be small to ensure the validity of the WKB approximation. Properties of low-order WKB approximations are discussed in some detail. Contrary to the better-studied cases of acoustic waves and internal gravity waves in the Boussinesq approximation, the WKB solution contains the geometric, or Berry, phase. The Berry phase is generally non-negligible for AGWs in a moving atmosphere. In other words, knowledge of the AGW dispersion relation is not sufficient for calculation of the wave phase.

JFM classification

Compressible Flows: Compressible Flows Waves/Free-surface Flows: Elastic waves Geophysical and Geological Flows: Internal waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 777 , 25 August 2015 , pp. 260 - 290
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© 2015 Cambridge University Press 

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References

Akmaev, R. A. 2011 Whole atmosphere modeling: Connecting terrestrial and space weather. Rev. Geophys. 49, RG4004.CrossRefGoogle Scholar
Ardhuin, F. & Herbers, T. H. C. 2013 Noise generation in the solid Earth, oceans and atmosphere, from nonlinear interacting surface gravity waves in finite depth. J. Fluid Mech. 716, 316–348.CrossRefGoogle Scholar
Astafyeva, E., Shalimov, S., Olshanskaya, E. & Lognonné, P. 2013 Ionospheric response to earthquakes of different magnitudes: larger quakes perturb the ionosphere stronger and longer. Geophys. Res. Lett. 40, 1675–1681.CrossRefGoogle Scholar
Babich, V. M. 1961 Propagation of Rayleigh waves along the surface of a homogeneous elastic body of arbitrary shape. Dokl. Akad. Nauk SSSR 137, 1263–1266.Google Scholar
Babich, V. M. & Kiselev, A. P. 2004 Nongeometrical phenomena in propagation of elastic surface waves. In Surface Waves in Anisotropic and Laminated Bodies and Defects Detection (ed. Goldstein, R. V. & Maugin, G. A.), pp. 119–129. Kluwer.CrossRefGoogle Scholar
Berry, M. V. 1984 Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392 (1802), 45–57.Google Scholar
Berry, M. V. 1990 Budden & Smith’s ‘additional memory’ and the geometric phase. Proc. R. Soc. Lond. A 431, 531–537.Google Scholar
Berry, M. V. 2010 Geometric phase memories. Nat. Phys. 6, 148–150.CrossRefGoogle Scholar
Bohm, A., Mostafazadeh, A., Koizumi, H., Niu, Q. & Zwanziger, J. 2003 The Geometric Phase in Quantum Systems: Foundations, Mathematical Concepts, and Applications in Molecular and Condensed Matter Physics. Springer.CrossRefGoogle Scholar
Brekhovskikh, L. M. 1960 Waves in Layered Media, pp. 168–171 and 193–198. Academic.Google Scholar
Brekhovskikh, L. M. & Godin, O. A. 1998 Acoustics of Layered Media 1: Plane and Quasi-Plane Waves, 2nd edn. Springer.Google Scholar
Brekhovskikh, L. M. & Godin, O. A. 1999 Acoustics of Layered Media 2: Point Sources and Bounded Beams, 2nd edn. Springer.CrossRefGoogle Scholar
Bretherton, F. P. 1968 Propagation in slowly varying waveguides. Proc. R. Soc. Lond. A 302, 555–567.Google Scholar
Bretherton, F. P. 1969 Lamb waves in a nearly isothermal atmosphere. Q. J. R. Meteorol. Soc. 95, 754–757.CrossRefGoogle Scholar
Broutman, D., Rottman, J. W. & Eckermann, S. D. 2004 Ray methods for internal waves in the atmosphere and ocean. Annu. Rev. Fluid Mech. 36, 233–253.CrossRefGoogle Scholar
Budden, K. G. & Smith, M. S. 1976 Phase memory and additional memory in WKB solutions for wave propagation in stratified media. Proc. R. Soc. Lond. A 350 (1660), 27–46.Google Scholar
Coïsson, P., Lognonné, P., Walwer, D. & Rolland, L. M. 2015 First tsunami gravity wave detection in ionospheric radio occultation data. Earth Space Sci. 2, 125–133.CrossRefGoogle Scholar
Duvall, T. L., Jefferies, S. M., Harvey, J. W. & Pomerantz, M. A. 1993 Time–distance helioseismology. Nature 362, 430–432.CrossRefGoogle Scholar
Einaudi, F. & Hines, C. O. 1970 WKB approximation in application to acoustic-gravity waves. Can. J. Phys. 48, 1458–1471.CrossRefGoogle Scholar
Einaudi, F. & Hines, C. O. 1974 WKB approximation in application to acoustic-gravity waves. In The Upper Atmosphere in Motion (ed. Hines, C. O.), Geophys. Monogr. Ser., vol. 18, pp. 508–530. American Geophysical Union.CrossRefGoogle Scholar
Fedoryuk, M. 1987 Méthodes Asymptotiques pour les Equations Différentielles Ordinaires Linéaires. Mir.Google Scholar
Fouchet, T., Guerlet, S., Strobel, D. F., Simon-Miller, A. A., Bézard, B. & Flasar, F. M. 2008 An equatorial oscillation in Saturn’s middle atmosphere. Nature 453, 200–202.CrossRefGoogle ScholarPubMed
Fritts, D. C. & Alexander, M. J. 2003 Gravity wave dynamics and effects in the middle atmosphere. Rev. Geophys. 41, 1003.CrossRefGoogle Scholar
Frömann, N. & Frömann, P. O. 1965 JWKB Approximation. Contributions to the Theory. North-Holland.Google Scholar
Fuller-Rowell, T. J., Akmaev, R. A., Wu, F., Anghel, A., Maruyama, N., Anderson, D. N., Codrescu, M. V., Iredell, M., Moorthi, S., Juang, H.-M., Hou, Y.-T. & Millward, G. 2008 Impact of terrestrial weather on the upper atmosphere. Geophys. Res. Lett. 35, L09808.CrossRefGoogle Scholar
Fuller-Rowell, T., Wu, F., Akmaev, R., Fang, T.-W. & Araujo-Pradere, E. 2010 A whole atmosphere model simulation of the impact of a sudden stratospheric warming on thermosphere dynamics and electrodynamics. J. Geophys. Res. 115, A00G08.Google Scholar
Garcia, R. F., Doornbos, E., Bruinsma, S. & Hebert, H. 2014 Atmospheric gravity waves due to the Tohoku-Oki tsunami observed in the thermosphere by GOCE. J. Geophys. Res. 119, 4498–4506.CrossRefGoogle Scholar
Garrett, C. J. R. 1968 On the interaction between internal gravity waves and a shear flow. J. Fluid Mech. 34, 711–720.CrossRefGoogle Scholar
Geller, M. A., Alexander, M. J., Love, P. T., Bacmeister, J., Ern, M., Hertzog, A., Manzini, E., Preusse, P., Sato, K., Scaife, A. A. & Zhou, T. 2013 A comparison between gravity wave momentum fluxes in observations and climate models. J. Clim. 26, 6383–6405.CrossRefGoogle Scholar
Gill, A. E. 1982 Atmosphere–Ocean Dynamics. Academic.Google Scholar
Godin, O. A. 1987 A new form of the wave equation for sound in a general layered fluid. In Progress in Underwater Acoustics (ed. Merklinger, H. M.), pp. 337–349. Plenum.CrossRefGoogle Scholar
Godin, O. A. 1997 Reciprocity and energy theorems for waves in a compressible inhomogeneous moving fluid. Wave Motion 25, 143–167.CrossRefGoogle Scholar
Godin, O. A. 2012a Incompressible wave motion of compressible fluids. Phys. Rev. Lett. 108, 194501.CrossRefGoogle ScholarPubMed
Godin, O. A. 2012b Acoustic-gravity waves in atmospheric and oceanic waveguides. J. Acoust. Soc. Am. 132, 657–669.CrossRefGoogle ScholarPubMed
Godin, O. A. 2014a Shear waves in inhomogeneous, compressible fluids in a gravity field. J. Acoust. Soc. Am. 135, 1071–1082.CrossRefGoogle Scholar
Godin, O. A. 2014b Dissipation of acoustic-gravity waves: an asymptotic approach. J. Acoust. Soc. Am. 136, EL411–EL417.CrossRefGoogle ScholarPubMed
Godin, O. A. 2015 Finite-amplitude acoustic-gravity waves: exact solutions. J. Fluid Mech. 767, 52–64.CrossRefGoogle Scholar
Godin, O. A. & Fuks, I. M. 2012 Transmission of acoustic-gravity waves through gas–liquid interfaces. J. Fluid Mech. 709, 313–340.CrossRefGoogle Scholar
Godin, O. A., Zabotin, N. A. & Bullett, T. W. 2015 Acoustic-gravity waves in the atmosphere generated by infragravity waves in the ocean. Earth Planet. Space 67, 47.CrossRefGoogle Scholar
Gossard, E. & Hooke, W. 1975 Waves in the Atmosphere. Elsevier.Google Scholar
Grimshaw, R. 1975 Internal gravity waves: critical layer absorption in a rotating fluid. J. Fluid Mech. 70, 287–304.CrossRefGoogle Scholar
Hargreaves, J. K. & Gadsden, M. 1992 The Solar-Terrestrial Environment. Cambridge University Press.CrossRefGoogle Scholar
Heading, J. 1962 An Introduction to Phase-Integral Methods. Wiley.Google Scholar
Hines, C. O. 1960 Internal atmospheric gravity waves at ionospheric heights. Can. J. Phys. 38, 1441–1481.CrossRefGoogle Scholar
Jia, J. Y., Preusse, P., Ern, M., Chun, H. Y., Gille, J. C., Eckermann, S. D. & Riese, M. 2014 Sea surface temperature as a proxy for convective gravity wave excitation: a study based on global gravity wave observations in the middle atmosphere. Ann. Geophys. 32, 1373–1394.CrossRefGoogle Scholar
Jones, R. M. & Georges, T. M. 1976 Propagation of acoustic-gravity waves in a temperature- and wind-stratified atmosphere. J. Acoust. Soc. Am. 59, 765–779.CrossRefGoogle Scholar
Karal, F. G. & Keller, J. B. 1964 Geometrical theory of elastic surface-wave excitation and propagation. J. Acoust. Soc. Am. 36, 32–40.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Liu, H.-L., Foster, B. T., Hagan, M. E., McInerney, J. M., Maute, A., Qian, L., Richmond, A. D., Roble, R. G., Solomon, S. C., Garcia, R. R., Kinnison, D., Marsh, D. R., Smith, A. K., Richter, J., Sassi, F. & Oberheide, J. 2010 Thermosphere extension of the whole atmosphere community climate model. J. Geophys. Res. 115, A12302.Google Scholar
Makela, J. J., Lognonné, P., Hébert, H., Gehrels, T., Rolland, L., Allgeyer, S., Kherani, A., Occhipinti, G., Astafyeva, E., Coïsson, P., Loevenbruck, A., Clévédé, E., Kelley, M. C. & Lamouroux, J. 2011 Imaging and modeling the ionospheric airglow response over Hawaii to the tsunami generated by the Tohoku earthquake of 11 March 2011. Geophys. Res. Lett. 38, L13305.CrossRefGoogle Scholar
Maruyama, T., Tsugawa, T., Kato, H., Ishii, M. & Nishioka, M. 2012 Rayleigh wave signature in ionograms induced by strong earthquakes. J. Geophys. Res. 117, A08306.Google Scholar
Maslov, V. P. & Fedoriuk, M. V. 1981 Semi-Classical Approximation in Quantum Mechanics. Reidel.CrossRefGoogle Scholar
Matoza, R. S., Vergoz, J., Le Pichon, A., Ceranna, L., Green, D. N., Evers, L. G., Ripepe, M., Campus, P., Liszka, L., Kvaerna, T., Kjartansson, E. & Hoskuldsson, A. 2011 Long-range acoustic observations of the Eyjafjallajökull eruption, Iceland, April–May 2010. Geophys. Res. Lett. 38, L06308.CrossRefGoogle Scholar
Miropol’sky, Yu. Z. 2001 Dynamics of Internal Gravity Waves in the Ocean. Kluwer.CrossRefGoogle Scholar
Nazarenko, S., Kevlahan, N. R. & Dubrulle, B. 1999 WKB theory for rapid distortion of inhomogeneous turbulence. J. Fluid Mech. 390, 325–348.CrossRefGoogle Scholar
Nishida, K., Kobayashi, N. & Fukao, Y. 2013 Background Lamb waves in the Earth’s atmosphere. Geophys. J. Intl 196, 312–316.CrossRefGoogle Scholar
Occhipinti, G., Rolland, L., Lognonné, P. & Watada, S. 2013 From Sumatra 2004 to Tohoku-Oki 2011: the systematic GPS detection of the ionospheric signature induced by tsunamigenic earthquakes. J. Geophys. Res. 118, 3626–3636.CrossRefGoogle Scholar
Olver, F. W. J. 1974 Asymptotics and Special Functions. Academic.Google Scholar
Ostashev, V. E. 1987 Equation for acoustic and gravity-waves in a stratified moving medium. Sov. Phys. Acoust. 33, 95–96.Google Scholar
Ostashev, V. E. 1997 Acoustics in Moving Inhomogeneous Media. E&FN Spon.Google Scholar
Petrukhin, N. S., Pelinovsky, E. N. & Batsyna, E. K. 2011 Reflectionless propagation of acoustic waves through the Earth’s atmosphere. JETP Lett. 93, 564–567.CrossRefGoogle Scholar
Petrukhin, N. S., Pelinovsky, E. N. & Batsyna, E. K. 2012a Reflectionless propagation of acoustic waves in the solar atmosphere. Astron. Lett. 38, 388–393.CrossRefGoogle Scholar
Petrukhin, N. S., Pelinovsky, E. N. & Batsyna, E. K. 2012b Reflectionless acoustic gravity waves in the Earth’s atmosphere. Geomagn. Aeron. 52, 814–819.CrossRefGoogle Scholar
Petrukhin, N. S., Pelinovsky, E. N. & Talipova, T. G. 2012c Nonreflected vertical propagation of acoustic waves in a strongly inhomogeneous atmosphere. Izv. Atmos. Ocean. Phys. 48, 169–173.CrossRefGoogle Scholar
Pierce, A. D. 1965 Propagation of acoustic-gravity waves in a temperature- and wind-stratified atmosphere. J. Acoust. Soc. Am. 37, 218–227.CrossRefGoogle Scholar
Pitteway, M. L. V. & Hines, C. O. 1965 The reflection and ducting of atmospheric acoustic-gravity waves. Can. J. Phys. 43, 2222–2243.CrossRefGoogle Scholar
Podesta, J. J. 2005 Compressible fluid model for the seismic waves generated by a sunquake. Solar Phys. 232, 1–23.CrossRefGoogle Scholar
Schirber, S., Manzini, E., Krismer, T. & Giorgetta, M. 2014 The quasi-biennial oscillation in a warmer climate: sensitivity to different gravity wave parameterizations. Clim. Dyn. 44, 1–12.Google Scholar
Shapere, A. & Wilczek, F. (Ed.) 1989 Geometric Phases in Physics, World Scientific.Google Scholar
Tatarskiy, V. I. 1979 On the theory of sound propagation in a stratified atmosphere. Izv. Atmos. Ocean. Phys. 15, 795–801.Google Scholar
Tromp, J. & Dahlen, F. A. 1992 The Berry phase of a slowly varying waveguide. Proc. R. Soc. Lond. A 437 (1900), 329–342.Google Scholar
Ursin, B. 1983 Review of elastic and electromagnetic wave propagation in horizontally layered media. Geophysics 48, 1063–1081.CrossRefGoogle Scholar
Vadas, S. L. & Liu, H. 2009 Generation of large-scale gravity waves and neutral winds in the thermosphere from the dissipation of convectively generated gravity waves. J. Geophys. Res. 114, A10310.Google Scholar
Vadas, S. L. & Nicolls, M. J. 2012 The phases and amplitudes of gravity waves propagating and dissipating in the thermosphere: theory. J. Geophys. Res. 117, A05322.Google Scholar
Watada, S. 2009 Radiation of acoustic and gravity waves and propagation of boundary waves in the stratified fluid from a time-varying bottom boundary. J. Fluid Mech. 627, 361–377.CrossRefGoogle Scholar
Zabotin, N. A., Godin, O. A., Sava, P. C. & Zabotina, L. Y. 2014 Tracing three-dimensional acoustic wavefronts in inhomogeneous, moving media. J. Comput. Acoust. 22, 1450002.CrossRefGoogle Scholar