1932

Abstract

Internal gravity waves play a primary role in geophysical fluids: They contribute significantly to mixing in the ocean, and they redistribute energy and momentum in the middle atmosphere. Until recently, most studies were focused on plane wave solutions. However, these solutions are not a satisfactory description of most geophysical manifestations of internal gravity waves, and it is now recognized that internal wave beams with a confined profile are ubiquitous in the geophysical context. We discuss the reason for the ubiquity of wave beams in stratified fluids, which is related to the fact that they are solutions of the nonlinear governing equations. We focus more specifically on situations with a constant buoyancy frequency. Moreover, in light of recent experimental and analytical studies of internal gravity beams, it is timely to discuss the two main mechanisms of instability for those beams: () the triadic resonant instability generating two secondary wave beams and () the streaming instability corresponding to the spontaneous generation of a mean flow.

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2018-01-05
2024-06-22
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Literature Cited

  1. Alexander M. 2003. Parametrization of physical process: gravity wave fluxes. Encyclopedia of the Atmospheric Sciences J Holton, J Curry, J Pyle 1669–705 London: Academic [Google Scholar]
  2. Alford M, MacKinnon JA, Zhao Z, Pinkel R, Klymak J, Peacock T. 2007. Internal waves across the Pacific. Geophys. Res. Lett. 34:L24601 [Google Scholar]
  3. Alford M, Peacock T, MacKinnon JA, Nash JD, Buijsman MC. et al. 2015. The formation and fate of internal waves in the South China Sea. Nature 521:65–69 [Google Scholar]
  4. Andrews DG, McIntyre ME. 1976. Planetary waves in horizontal and vertical shear: the generalized Eliassen-Palm relation and the mean zonal acceleration. J. Atmos. Sci. 33:2031–48 [Google Scholar]
  5. Andrews DG, McIntyre ME. 1978. An exact theory of nonlinear waves on a Lagrangian-mean flow. J. Fluid Mech. 89:609–46 [Google Scholar]
  6. Beckebanze F, Maas LRM. 2016. Damping of 3D internal wave attractors by lateral walls. Proc. Int. Symp. Strat. Flows, 8th., 29 Aug.–1 Sept. San Diego: Univ. Calif. San Diego [Google Scholar]
  7. Bell TH. 1975. Lee waves in stratified flows with simple harmonic time dependence. J. Fluid Mech. 67:705–22 [Google Scholar]
  8. Benielli D, Sommeria J. 1998. Excitation and breaking of internal gravity waves by parametric instability. J. Fluid Mech. 374:117–44 [Google Scholar]
  9. Bordes G. 2012. Interactions non-linéaires d'ondes et tourbillons en milieu stratifié ou tournant PhD Thesis, ENS Lyon. https://tel.archives-ouvertes.fr/tel-00733175/en [Google Scholar]
  10. Bordes G, Moisy F, Dauxois T, Cortet PP. 2012a. Experimental evidence of a triadic resonance of plane inertial waves in a rotating fluid. Phys. Fluids 24:014105 [Google Scholar]
  11. Bordes G, Venaille A, Joubaud S, Odier P, Dauxois T. 2012b. Experimental observation of a strong mean flow induced by internal gravity waves. Phys. Fluids 24:086602 [Google Scholar]
  12. Bourget B. 2014. Ondes internes, de l'instabilité au mélange. Approche exprimentale. PhD Thesis, ENS Lyon. https://tel.archives-ouvertes.fr/tel-01073663/en [Google Scholar]
  13. Bourget B, Dauxois T, Joubaud S, Odier P. 2013. Experimental study of parametric subharmonic instability for internal plane waves. J. Fluid Mech. 723:1–20 [Google Scholar]
  14. Bourget B, Scolan H, Dauxois T, Le Bars M, Odier P, Joubaud S. 2014. Finite-size effects in parametric subharmonic instability. J. Fluid Mech. 759:739–50 [Google Scholar]
  15. Bretherton FP. 1969. On the mean motion induced by internal gravity waves. J. Fluid Mech. 36:785–803 [Google Scholar]
  16. Brouzet C, Ermanyuk EV, Joubaud S, Sibgatullin IN, Dauxois T. 2016a. Energy cascade in internal-wave attractors. Europhys. Lett. 113:44001 [Google Scholar]
  17. Brouzet C, Sibgatullin IN, Scolan H, Ermanyuk EV, Dauxois T. 2016b. Internal wave attractors examined using laboratory experiments and 3D numerical simulations. J. Fluid Mech. 793:109–31 [Google Scholar]
  18. Bühler O. 2009. Waves and Mean Flows Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  19. Callies J, Ferrari R, Bühler O. 2014. Transition from geostrophic turbulence to inertia–gravity waves in the atmospheric energy spectrum. PNAS 111:17033–38 [Google Scholar]
  20. Chalamalla VK, Sarkar S. 2016. PSI in the case of internal wave beam reflection at a uniform slope. J. Fluid Mech. 789:347–67 [Google Scholar]
  21. Charney JG, Drazin PG. 1961. Propagation of planetary-scale disturbances from the lower into the upper atmosphere. J. Geophys. Res. 66:83–109 [Google Scholar]
  22. Chraibi H, Wunenburger R, Lasseux D, Petit J, Delville JP. 2011. Eddies and interface deformations induced by optical streaming. J. Fluid Mech. 688:195–218 [Google Scholar]
  23. Clark HA, Sutherland BR. 2010. Generation, propagation, and breaking of an internal wave beam. Phys. Fluids 22:076601 [Google Scholar]
  24. Cole ST, Rudnick DL, Hodges BA, Martin JP. 2009. Observations of tidal internal wave beams at Kauai Channel, Hawaii. J. Phys. Oceanogr. 39:421–36 [Google Scholar]
  25. Craik AD. 1988. Wave Interactions and Fluid Flows Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  26. Dauxois T, Young WR. 1999. Near critical reflection of internal waves. J. Fluid Mech. 390:271–95 [Google Scholar]
  27. Davis RE, Acrivos A. 1967. The stability of oscillatory internal waves. J. Fluid Mech. 30:723–36 [Google Scholar]
  28. Eckart C. 1948. Vortices and streams caused by sound waves. Phys. Rev. 73:68–76 [Google Scholar]
  29. Eliassen A, Palm E. 1961. On the transfer of energy in stationary mountain waves. Geofys. Publ. 22:1–23 [Google Scholar]
  30. Fritts DC, Alexander MJ. 2003. Gravity wave dynamics and effects in the middle atmosphere. Rev. Geophys. 41:1003 [Google Scholar]
  31. Garrett C, Kunze E. 2007. Internal tide generation in deep ocean Annu. Rev. Fluid Mech. 39:57–87 [Google Scholar]
  32. Garrett C, Munk W. 1975. Space-time scales of internal waves: a progress report.. J. Geophys. Res. 80:291–97 [Google Scholar]
  33. Gayen B, Sarkar S. 2013. Degradation of an internal wave beam by parametric subharmonic instability in an upper ocean pycnocline. J. Geophys. Res. Oceans 118:4689–98 [Google Scholar]
  34. Gerkema T, Staquet C, Bouruet-Aubertot P. 2006. Decay of semi-diurnal internal-tide beams due to subharmonic resonance. Geophys. Res. Lett. 33:L08604 [Google Scholar]
  35. Gill AE. 1982. Atmosphere–Ocean Dynamics New York: Academic [Google Scholar]
  36. Gostiaux L, Dauxois T. 2007. Laboratory experiments on the generation of internal tidal beams over steep slopes. Phys. Fluids 19:028102 [Google Scholar]
  37. Gostiaux L, Dauxois T, Didelle H, Sommeria J, Viboud S. 2006. Quantitative laboratory observations of internal wave reflection on ascending slopes. Phys. Fluids 18:056602 [Google Scholar]
  38. Gostiaux L, Didelle H, Mercier S, Dauxois T. 2007. A novel internal waves generator. Exp. Fluids 42:123–30 [Google Scholar]
  39. Grisouard N. 2010. Réflexions et réfractions non-linéaires d'ondes de gravité internes PhD Thesis, Univ. Grenoble http://tel.archives-ouvertes.fr/tel-00540608/en/ [Google Scholar]
  40. Grisouard N, Bühler O. 2012. Forcing of oceanic mean flows by dissipating internal tides. J. Fluid Mech. 708:250–78 [Google Scholar]
  41. Grisouard N, Leclair M, Gostiaux L, Staquet C. 2013. Large scale energy transfer from an internal gravity wave reflecting on a simple slope. Proc. IUTAM 8:119–28 [Google Scholar]
  42. Hasselman K. 1967. A criterion for nonlinear wave instability. J. Fluid Mech. 30:737–39 [Google Scholar]
  43. Hazewinkel J, Winters KB. 2011. PSI on the internal tide on a β plane: flux divergence and near-inertial wave propagation. J. Phys. Oceanogr. 41:1673–82 [Google Scholar]
  44. Hibiya T, Nagasawa M, Niwa Y. 2002. Nonlinear energy transfer within the oceanic internal wave spectrum at mid and high latitudes. J. Geophys. Res. 107:3207 [Google Scholar]
  45. Johnston TMS, Merrifield MA, Holloway PE. 2003. Internal tide scattering at the Line Islands Ridge. J. Geophys. Res. 108:3365 [Google Scholar]
  46. Johnston TMS, Rudnick DL, Carter GS, Todd RE, Cole ST. 2011. Internal tidal beams and mixing near Monterey Bay. J. Geophys. Res. 116:C03017 [Google Scholar]
  47. Joubaud S, Munroe J, Odier P, Dauxois T. 2012. Experimental parametric subharmonic instability in stratified fluids. Phys. Fluids 24:041703 [Google Scholar]
  48. Karimi HH. 2015. Parametric subharmonic instability of internal gravity wave beams PhD Thesis, Mass. Inst. Technol. [Google Scholar]
  49. Karimi HH, Akylas TR. 2014. Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wave trains. J. Fluid Mech. 757:381–402 [Google Scholar]
  50. Karimi HH, Akylas TR. 2017. Near-inertial parametric subharmonic instability of internal gravity wave beams. Phys. Rev. Fluids 2:074801 [Google Scholar]
  51. Kataoka T, Akylas TR. 2013. Stability of internal gravity wave beams to three-dimensional modulations. J. Fluid Mech. 736:67–90 [Google Scholar]
  52. Kataoka T, Akylas TR. 2015. On three-dimensional internal gravity wave beams and induced large-scale mean flows. J. Fluid Mech. 769:621–34 [Google Scholar]
  53. Kataoka T, Akylas TR. 2016. Three-dimensional instability of internal gravity beams. Proc. Int. Symp. Strat. Flows, 8th, 29 Aug.–1 Sept. San Diego: Univ. Calif. San Diego [Google Scholar]
  54. Khatiwala S. 2003. Generation of internal tides in an ocean of finite depth: analytical and numerical calculations. Deep Sea Res. Part I 50:3–21 [Google Scholar]
  55. King B, Zhang HP, Swinney HL. 2009. Tidal flow over three-dimensional topography in a stratified fluid. Phys. Fluids 21:116601 [Google Scholar]
  56. Koudella CR, Staquet C. 2006. Instability mechanisms of a two-dimensional progressive internal gravity wave. J. Fluid Mech. 548:165–96 [Google Scholar]
  57. Lamb KG. 2004. Nonlinear interaction among internal wave beams generated by tidal flow over supercritical topography. Geophys. Res. Lett. 31:L09313 [Google Scholar]
  58. Leclair M, Grisouard N, Gostiaux L, Staquet C, Auclair F. 2011. Reflexion of a plane wave onto a slope and wave-induced mean flow. Proc. Int. Symp. Strat. Flows, 7th, 22–26 Aug. Rome: Sapienza Univ. Roma [Google Scholar]
  59. Lelong M-P, Riley J. 1991. Internal wave-vortical mode interactions in strongly stratified flows. J. Fluid Mech. 232:1–19 [Google Scholar]
  60. Lerisson G. 2017. Étude de la stabilité globale et locale d'une onde de gravité interne en milieu linéairement stratifié PhD Thesis, École Polytech. [Google Scholar]
  61. Liang Y, Zareei A, Alam MR. 2017. Inherently unstable internal gravity waves due to resonant harmonic generation. J. Fluid Mech. 811:400–20 [Google Scholar]
  62. Lien R-C, Gregg MC. 2001. Observations of turbulence in a tidal beam and across a coastal ridge. J. Geophys. Res. 106:4575–91 [Google Scholar]
  63. Lighthill J. 1978a. Acoustic streaming. J. Sound Vib. 61:391–418 [Google Scholar]
  64. Lighthill J. 1978b. Waves in Fluids Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  65. MacKinnon JA, Alford MH, Sun O, Pinkel R, Zhao Z, Klymak J. 2013. Parametric subharmonic instability of the internal tide at 29°N. J. Phys. Oceanogr. 43:17–28 [Google Scholar]
  66. MacKinnon JA, Winters KB. 2005. Subtropical catastrophe: significant loss of low-mode tidal energy at 28.9°. Geophys. Res. Lett. 32:L15605 [Google Scholar]
  67. Maugé R, Gerkema T. 2008. Generation of weakly nonlinear nonhydrostatic internal tides over large topography: a multi-modal approach. Nonlinear Process. Geophys. 15:233–44 [Google Scholar]
  68. Maurer P, Joubaud S, Odier P. 2016. Generation and stability of inertia-gravity waves. J. Fluid Mech 808:539–61 [Google Scholar]
  69. McEwan AD. 1971. Degeneration of resonantly-excited standing internal gravity waves. J. Fluid Mech. 50:431–48 [Google Scholar]
  70. McEwan AD. 1973. Interactions between internal gravity wave and their traumatic effect on a continuous stratification. Bound.-Layer Meteorol. 5:159–75 [Google Scholar]
  71. McEwan AD, Plumb RA. 1977. Off-resonant amplification of finite internal wave packets. Dyn. Atmos. Oceans 2:83–105 [Google Scholar]
  72. Mercier MJ, Garnier N, Dauxois T. 2008. Reflection and diffraction of internal waves analyzed with the Hilbert transform. Phys. Fluids 20:086601 [Google Scholar]
  73. Mercier MJ, Martinand D, Mathur M, Gostiaux L, Peacock T, Dauxois T. 2010. New wave generation. J. Fluid Mech. 657:308–34 [Google Scholar]
  74. Mied RP. 1976. The occurrence of parametric instabilities in finite-amplitude internal gravity waves. J. Fluid Mech. 78:763–84 [Google Scholar]
  75. Moudjed B, Botton V, Henry D, Ben Hadid H, Garandet JP. 2014. Scaling and dimensional analysis of acoustic streaming jets. Phys. Fluids 26:093602 [Google Scholar]
  76. Mowbray DE, Rarity BS. 1967. A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified fluid. J. Fluid Mech. 28:1–16 [Google Scholar]
  77. Nazarenko S. 2011. Wave Turbulence Berlin: Springer-Verlag [Google Scholar]
  78. Nyborg WL. 1965. Acoustic streaming. Properties of Polymers and Nonlinear Acoustics WP Mason 265–331 Phys. Acoust. Ser. 2B San Diego: Academic [Google Scholar]
  79. Pairaud I, Staquet C, Sommeria J, Mahdizadeh M. 2010. Generation of harmonics and sub-harmonics from an internal tide in a uniformly stratified fluid: numerical and laboratory experiments. IUTAM Symposium on Turbulence in the Atmosphere and Oceans D Dritschel 51–62 IUTAM Bookser. 28 Berlin: Springer-Verlag [Google Scholar]
  80. Peacock T, Echeverri P, Balmforth NJ. 2008. An experimental investigation of internal tide generation by two-dimensional topography. J. Phys. Oceanogr. 38:235–42 [Google Scholar]
  81. Phillips OM. 1966. The Dynamics of the Upper Ocean New York: Cambridge Univ. Press [Google Scholar]
  82. Plumb RA. 1977. The interaction of two internal waves with the mean flow: implications for the theory of the quasi-biennial oscillation. J. Atmos. Sci. 34:1847–58 [Google Scholar]
  83. Plumb R, McEwan A. 1978. The instability of a forced standing wave in a viscous stratified fluid: a laboratory analogue of the quasi-biennial oscillation. J. Atmos. Sci. 35:1827–39 [Google Scholar]
  84. Rainville L, Pinkel R. 2006. Propagation of low-mode internal waves through the ocean. J. Phys. Oceanogr. 36:1220–36 [Google Scholar]
  85. Riley N. 2001. Steady streaming. Annu. Rev. Fluid Mech. 33:43–65 [Google Scholar]
  86. Sarkar S, Scotti A. 2016. Turbulence during generation of internal tides in the deep ocean and their subsequent propagation and reflection. Annu. Rev. Fluid Mech. 49:195–220 [Google Scholar]
  87. Scolan H, Ermanyuk E, Dauxois T. 2013. Nonlinear fate of internal waves attractors. Phys. Rev. Lett. 110:234501 [Google Scholar]
  88. Semin B, Facchini G, Pétrélis F, Fauve S. 2016. Generation of a mean flow by an internal wave. Phys. Fluids 28:096601 [Google Scholar]
  89. Squires TM, Quake SR. 2013. Microfluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77:977–1026 [Google Scholar]
  90. Staquet C, Sommeria J. 2002. Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34:559–93 [Google Scholar]
  91. Sun O, Pinkel R. 2013. Subharmonic energy transfer from the semi-diurnal internal tide to near-diurnal motions over Kaena Ridge, Hawaii. J. Phys. Oceanogr. 43:766–89 [Google Scholar]
  92. Sutherland BR. 2010. Internal Gravity Waves Cambridge, UK: Cambridge Univ. Press [Google Scholar]
  93. Sutherland BR. 2013. The wave instability pathway to turbulence. J. Fluid Mech. 724:1–4 [Google Scholar]
  94. Tabaei A, Akylas TR. 2003. Nonlinear internal gravity wave beams. J. Fluid Mech. 482:141–61 [Google Scholar]
  95. Tabaei A, Akylas TR, Lamb KG. 2005. Nonlinear effects in reflecting and colliding internal wave beams. J. Fluid Mech. 526:217–43 [Google Scholar]
  96. Thomas NH, Stevenson TN. 1972. A similarity solution for viscous internal waves. J. Fluid Mech. 54:495–506 [Google Scholar]
  97. van den Bremer TS, Sutherland BR. 2014. The mean flow and long waves induced by two-dimensional internal gravity wavepackets. Phys. Fluids 26:106601 [Google Scholar]
  98. Voisin B. 2003. Limit states of internal wave beams. J. Fluid Mech. 496:243–93 [Google Scholar]
  99. Westervelt PJ. 1953. The theory of steady rotational flow generated by a sound field. J. Acoust. Soc. Am. 25:60–67 [Google Scholar]
  100. Wunsch C, Ferrari R. 2004. Vertical mixing, energy, and the general circulation of the oceans. Annu. Rev. Fluid Mech. 36:281–314 [Google Scholar]
  101. Xie JH, Vanneste J. 2014. Boundary streaming with Navier boundary condition. Phys. Rev. E 89:063010 [Google Scholar]
  102. Young WR, Tsang Y-K, Balmforth NJ. 2008. Near-inertial parametric subharmonic instability. J. Fluid Mech. 607:25–49 [Google Scholar]
  103. Zhou Q, Diamessis PJ. 2013. Reflection of an internal gravity wave beam off a horizontal free-slip surface. Phys. Fluids 25:036601 [Google Scholar]
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