Internal gravity waves are propagating disturbances in stably stratified fluids, and can transport momentum and energy over large spatial extents. From a fundamental viewpoint, internal waves are interesting due to the nature of their dispersion relation, and their linear dynamics are reasonably well-understood. From an oceanographic viewpoint, a qualitative and quantitative understanding of significant internal wave generation in the ocean is emerging, while their dissipation mechanisms are being debated. This paper reviews the current knowledge on instabilities in internal gravity waves, primarily focusing on the growth of small-amplitude disturbances. Historically, wave-wave interactions based on weakly nonlinear expansions have driven progress in this field, to investigate spontaneous energy transfer to various temporal and spatial scales. Recent advances in numerical/experimental modeling and field observations have further revealed noticeable differences between various internal wave spatial forms in terms of their instability characteristics; this in turn has motivated theoretical calculations on appropriately chosen internal wave fields in various settings. After a brief introduction, we present a pedagogical discussion on linear internal waves and their different two-dimensional spatial forms. The general ideas concerning triadic resonance in internal waves are then introduced, before proceeding towards instability characteristics of plane waves, wave beams and modes. Results from various theoretical, experimental and numerical studies are summarized to provide an overall picture of the gaps in our understanding. An ocean perspective is then given, both in terms of the relevant outstanding questions and the various additional factors at play. While the applications in this review are focused on the ocean, several ideas are relevant to atmospheric and astrophysical systems too.
Citation: Dheeraj Varma, Manikandan Mathur, Thierry Dauxois. Instabilities in internal gravity waves[J]. Mathematics in Engineering, 2023, 5(1): 1-34. doi: 10.3934/mine.2023016
Internal gravity waves are propagating disturbances in stably stratified fluids, and can transport momentum and energy over large spatial extents. From a fundamental viewpoint, internal waves are interesting due to the nature of their dispersion relation, and their linear dynamics are reasonably well-understood. From an oceanographic viewpoint, a qualitative and quantitative understanding of significant internal wave generation in the ocean is emerging, while their dissipation mechanisms are being debated. This paper reviews the current knowledge on instabilities in internal gravity waves, primarily focusing on the growth of small-amplitude disturbances. Historically, wave-wave interactions based on weakly nonlinear expansions have driven progress in this field, to investigate spontaneous energy transfer to various temporal and spatial scales. Recent advances in numerical/experimental modeling and field observations have further revealed noticeable differences between various internal wave spatial forms in terms of their instability characteristics; this in turn has motivated theoretical calculations on appropriately chosen internal wave fields in various settings. After a brief introduction, we present a pedagogical discussion on linear internal waves and their different two-dimensional spatial forms. The general ideas concerning triadic resonance in internal waves are then introduced, before proceeding towards instability characteristics of plane waves, wave beams and modes. Results from various theoretical, experimental and numerical studies are summarized to provide an overall picture of the gaps in our understanding. An ocean perspective is then given, both in terms of the relevant outstanding questions and the various additional factors at play. While the applications in this review are focused on the ocean, several ideas are relevant to atmospheric and astrophysical systems too.
| [1] | T. Gerkema, J. T. F. Zimmerman, An introduction to internal waves, Texel: Royal NIOZ, 2008. |
| [2] | B. R. Sutherland, Internal gravity waves, Cambridge University Press, 2010. https://doi.org/10.1017/CBO9780511780318 |
| [3] |
R. M. Robinson, The effects of a vertical barrier on internal waves, Deep-Sea Res., 16 (1969), 421–429. https://doi.org/10.1016/0011-7471(69)90030-8 doi: 10.1016/0011-7471(69)90030-8
|
| [4] | P. Müller, N. Xu, Scattering of oceanic internal gravity waves off random bottom topography, J. Phys. Oceanogr., 22 (1992), 474–488. |
| [5] |
M. J. Mercier, N. B. Garnier, T. Dauxois, Reflection and diffraction of internal waves analyzed with the Hilbert transform, Phys. Fluids, 20 (2008), 086601. https://doi.org/10.1063/1.2963136 doi: 10.1063/1.2963136
|
| [6] |
B. R. Sutherland, K. Yewchuk, Internal wave tunnelling, J. Fluid Mech., 511 (2004), 125–134. https://doi.org/10.1017/S0022112004009863 doi: 10.1017/S0022112004009863
|
| [7] |
M. Mathur, T. Peacock, Internal wave interferometry, Phys. Rev. Lett., 104 (2010), 118501. https://doi.org/10.1103/PhysRevLett.104.118501 doi: 10.1103/PhysRevLett.104.118501
|
| [8] |
C. Garrett, W. Munk, Internal waves in the ocean, Ann. Rev. Fluid Mech., 11 (1979), 339–369. https://doi.org/10.1146/annurev.fl.11.010179.002011 doi: 10.1146/annurev.fl.11.010179.002011
|
| [9] | Y. Z. Miropol'Sky, Dynamics of internal gravity waves in the ocean, Dordrecht: Springer, 2001. https://doi.org/10.1007/978-94-017-1325-2 |
| [10] |
C. Wunsch, Internal tides in the ocean, Rev. Geophys., 13 (1975), 167–182. https://doi.org/10.1029/RG013i001p00167 doi: 10.1029/RG013i001p00167
|
| [11] |
R. T. Pollard, On the generation by winds of inertial waves in the ocean, Deep-Sea Res., 17 (1970), 795–812. https://doi.org/10.1016/0011-7471(70)90042-2 doi: 10.1016/0011-7471(70)90042-2
|
| [12] |
C. Garrett, E. Kunze, Internal tide generation in the deep ocean, Ann. Rev. Fluid Mech., 39 (2007), 57–87. https://doi.org/10.1146/annurev.fluid.39.050905.110227 doi: 10.1146/annurev.fluid.39.050905.110227
|
| [13] |
M. H. Alford, J. A. MacKinnon, H. L. Simmons, J. D. Nash, Near-inertial internal gravity waves in the ocean, Ann. Rev. Mar. Sci., 8 (2016), 95–123. https://doi.org/10.1146/annurev-marine-010814-015746 doi: 10.1146/annurev-marine-010814-015746
|
| [14] | P. G. Baines, Topographic effects in stratified flows, Cambridge University Press, 1998. |
| [15] |
F. Pétrélis, S. L. Smith, W. R. Young, Tidal conversion at a submarine ridge, J. Phys. Oceanogr., 36 (2006), 1053–1071. https://doi.org/10.1175/JPO2879.1 doi: 10.1175/JPO2879.1
|
| [16] |
O. Bühler, M. Holmes-Cerfon, Decay of an internal tide due to random topography in the ocean, J. Fluid Mech., 678 (2011), 271–293. https://doi.org/10.1017/jfm.2011.115 doi: 10.1017/jfm.2011.115
|
| [17] |
M. H. Alford, Redistribution of energy available for ocean mixing by long-range propagation of internal waves, Nature, 423 (2003), 159–162. https://doi.org/10.1038/nature01628 doi: 10.1038/nature01628
|
| [18] |
W. Munk, C. Wunsch, Abyssal recipes Ⅱ: Energetics of tidal and wind mixing, Deep-Sea Res., 45 (1998), 1977–2010. https://doi.org/10.1016/S0967-0637(98)00070-3 doi: 10.1016/S0967-0637(98)00070-3
|
| [19] |
C. Garrett, W. Munk, Space‐time scales of internal waves: A progress report, J. Geophys. Res., 80 (1975), 291–297. https://doi.org/10.1029/JC080i003p00291 doi: 10.1029/JC080i003p00291
|
| [20] |
C. B. Whalen, C. de Lavergne, A. C. N. Garabato, J. M. Klymak, J. A. Mackinnon, K. L. Sheen, Internal wave-driven mixing: governing processes and consequences for climate, Nat. Rev. Earth Environ., 1 (2020), 606–621. https://doi.org/10.1038/s43017-020-0097-z doi: 10.1038/s43017-020-0097-z
|
| [21] |
K. G. Lamb, Internal wave breaking and dissipation mechanisms on the continental slope/shelf, Ann. Rev. Fluid Mech., 46 (2014), 231–254. https://doi.org/10.1146/annurev-fluid-011212-140701 doi: 10.1146/annurev-fluid-011212-140701
|
| [22] | P. G. Drazin, W. H. Reid, Hydrodynamic stability, Cambridge University Press, 2004. https://doi.org/10.1017/CBO9780511616938 |
| [23] |
J. W. Miles, On the stability of heterogeneous shear flows, J. Fluid Mech., 10 (1961), 496–508. https://doi.org/10.1017/S0022112061000305 doi: 10.1017/S0022112061000305
|
| [24] |
L. N. Howard, Note on a paper of John W. Miles, J. Fluid Mech., 10 (1961), 509–512. https://doi.org/10.1017/S0022112061000317 doi: 10.1017/S0022112061000317
|
| [25] |
S. H. Davis, The stability of time-periodic flows, Ann. Rev. Fluid Mech., 8 (1976), 57-74. https://doi.org/10.1146/annurev.fl.08.010176.000421 doi: 10.1146/annurev.fl.08.010176.000421
|
| [26] |
D. Broutman, C. Macaskill, M. E. McIntyre, J. W. Rottman, On Doppler‐spreading models of internal waves, Geophys. Res. Lett., 24 (1997), 2813–2816. https://doi.org/10.1029/97GL52902 doi: 10.1029/97GL52902
|
| [27] |
F. J. Poulin, G. R. Flierl, J. Pedlosky, Parametric instability in oscillatory shear flows, J. Fluid Mech., 481 (2003), 329–353. https://doi.org/10.1017/S0022112003004051 doi: 10.1017/S0022112003004051
|
| [28] |
O. M. Phillips, Wave interactions-the evolution of an idea, J. Fluid Mech., 106 (1981), 215–227. https://doi.org/10.1017/S0022112081001572 doi: 10.1017/S0022112081001572
|
| [29] | A. D. D. Craik, Wave interactions and fluid flows, Cambridge University Press, 1988. https://doi.org/10.1017/CBO9780511569548 |
| [30] | L. J. Sonmor, G. P. Klaassen, Toward a unified theory of gravity wave stability, J. Atmos. Sci., 54 (1997), 2655–2680. |
| [31] |
C. Staquet, J. Sommeria, Internal gravity waves: from instabilities to turbulence, Ann. Rev. Fluid Mech., 34 (2002), 559–593. https://doi.org/10.1146/annurev.fluid.34.090601.130953 doi: 10.1146/annurev.fluid.34.090601.130953
|
| [32] |
T. Dauxois, S. Joubaud, P. Odier, A. Venaille, Instabilities of internal gravity wave beams, Annu. Rev. Fluid Mech., 50 (2018), 131–156. https://doi.org/10.1146/annurev-fluid-122316-044539 doi: 10.1146/annurev-fluid-122316-044539
|
| [33] |
S. A. Thorpe, On the shape of progressive internal waves, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 263 (1968), 563–614. https://doi.org/10.1098/rsta.1968.0033 doi: 10.1098/rsta.1968.0033
|
| [34] |
S. Martin, W. Simmons, C. Wunsch, The excitation of resonant triads by single internal waves, J. Fluid Mech., 53 (1972), 17–44. https://doi.org/10.1017/S0022112072000023 doi: 10.1017/S0022112072000023
|
| [35] | P. K. Kundu, I. R. Cohen, D. R. Dowling, Fluid Mechanics, 6 Eds., Waltham, Ma: Academic Press, 2016. https://doi.org/10.1016/C2012-0-00611-4 |
| [36] | P. H. LeBlond, L. A. Mysak, Waves in the Ocean, Elsevier, 1981. |
| [37] |
L. Gostiaux, H. Didelle, S. Mercier, T. Dauxois, A novel internal waves generator, Exp. Fluids, 42 (2007), 123–130. https://doi.org/10.1007/s00348-006-0225-7 doi: 10.1007/s00348-006-0225-7
|
| [38] |
M. J. Mercier, D. Martinand, M. Mathur, L. Gostiaux, T. Peacock, T. Dauxois, New wave generation, J. Fluid Mech., 657 (2010), 308–334. https://doi.org/10.1017/S0022112010002454 doi: 10.1017/S0022112010002454
|
| [39] |
D. E. Mowbray, B. S. H Rarity, A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid, J. Fluid Mech., 28 (1967), 1–16. https://doi.org/10.1017/S0022112067001867 doi: 10.1017/S0022112067001867
|
| [40] |
N. H. Thomas, T. N. Stevenson, A similarity solution for viscous internal waves, J. Fluid Mech., 54 (1972), 495–506. https://doi.org/10.1017/S0022112072000837 doi: 10.1017/S0022112072000837
|
| [41] |
D. G. Hurley, The generation of internal waves by vibrating elliptic cylinders. Part 1. Inviscid solution, J. Fluid Mech., 351 (1997), 105–118. https://doi.org/10.1017/S0022112097007027 doi: 10.1017/S0022112097007027
|
| [42] |
D. G. Hurley, G. Keady, The generation of internal waves by vibrating elliptic cylinders. Part 2. Approximate viscous solution, J. Fluid Mech., 351 (1997), 119–138. https://doi.org/10.1017/S0022112097007039 doi: 10.1017/S0022112097007039
|
| [43] |
B. R. Sutherland, S. B. Dalziel, G. O. Hughes, P. F. Linden, Visualization and measurement of internal waves by 'synthetic schlieren'. Part 1. Vertically oscillating cylinder, J. Fluid Mech., 390 (1999), 93–126. https://doi.org/10.1017/S0022112099005017 doi: 10.1017/S0022112099005017
|
| [44] |
P. Echeverri, M. R. Flynn, K. B. Winters, T. Peacock, Low-mode internal tide generation by topography: an experimental and numerical investigation, J. Fluid Mech., 636 (2009), 91–108. https://doi.org/10.1017/S0022112009007654 doi: 10.1017/S0022112009007654
|
| [45] |
L. R. Maas, D. Benielli, J. Sommeria, F. P. A. Lam, Observation of an internal wave attractor in a confined, stably stratified fluid, Nature, 388 (1997), 557–561. https://doi.org/10.1038/41509 doi: 10.1038/41509
|
| [46] |
P. Echeverri, T. Yokossi, N. J. Balmforth, T. Peacock, Tidally generated internal-wave attractors between double ridges, J. Fluid Mech., 669 (2011), 354–374. https://doi.org/10.1017/S0022112010005069 doi: 10.1017/S0022112010005069
|
| [47] |
Y. C. de Verdiere, L. Saint‐Raymond, Attractors for two‐dimensional waves with homogeneous Hamiltonians of degree 0, Commun. Pure Appl. Math., 73 (2020), 421–462. https://doi.org/10.1002/cpa.21845 doi: 10.1002/cpa.21845
|
| [48] |
H. Scolan, E. Ermanyuk, T. Dauxois, Nonlinear fate of internal wave attractors, Phys. Rev. Lett., 110 (2013), 234501. https://doi.org/10.1103/PhysRevLett.110.234501 doi: 10.1103/PhysRevLett.110.234501
|
| [49] |
C. Brouzet, E. Ermanyuk, S. Joubaud, G. Pillet, T. Dauxois, Internal wave attractors: different scenarios of instability, J. Fluid Mech., 811 (2017), 544–568. https://doi.org/10.1017/jfm.2016.759 doi: 10.1017/jfm.2016.759
|
| [50] |
G. Davis, T. Jamin, J. Deleuze, S. Joubaud, T. Dauxois, Succession of resonances to achieve internal wave turbulence, Phys. Rev. Lett., 124 (2020), 204502. https://doi.org/10.1103/PhysRevLett.124.204502 doi: 10.1103/PhysRevLett.124.204502
|
| [51] |
A. Tabaei, T. R. Akylas, Nonlinear internal gravity wave beams, J. Fluid Mech., 482 (2003), 141–161. https://doi.org/10.1017/S0022112003003902 doi: 10.1017/S0022112003003902
|
| [52] |
B. R. Sutherland, Excitation of superharmonics by internal modes in non-uniformly stratified fluid, J. Fluid Mech., 793 (2016), 335–352. https://doi.org/10.1017/jfm.2016.108 doi: 10.1017/jfm.2016.108
|
| [53] | A. H. Nayfeh, Perturbation methods, John Wiley & Sons, 2008. |
| [54] |
W. F. Simmons, A variational method for weak resonant wave interactions, Proc. R. Soc. Lond. A, 309 (1969), 551–577. https://doi.org/10.1098/rspa.1969.0056 doi: 10.1098/rspa.1969.0056
|
| [55] |
O. M. Phillips, On the dynamics of unsteady gravity waves of finite amplitude Part 1. The elementary interactions, J. Fluid Mech., 9 (1960), 193–217. https://doi.org/10.1017/S0022112060001043 doi: 10.1017/S0022112060001043
|
| [56] |
K. Hasselmann, On the non-linear energy transfer in a gravity-wave spectrum Part 1. General theory, J. Fluid Mech., 12 (1962), 481–500. https://doi.org/10.1017/S0022112062000373 doi: 10.1017/S0022112062000373
|
| [57] |
L. J. Sonmor, G. P. Klaassen, Higher-order resonant instabilities of internal gravity waves, J. Fluid Mech., 324 (1996), 1–23. https://doi.org/10.1017/S0022112096007811 doi: 10.1017/S0022112096007811
|
| [58] |
J. Klostermeyer, Two-and three-dimensional parametric instabilities in finite-amplitude internal gravity waves, Geophys. Astro. Fluid, 61 (1991), 1–25. https://doi.org/10.1080/03091929108229035 doi: 10.1080/03091929108229035
|
| [59] |
S. J. Ghaemsaidi, M. Mathur, Three-dimensional small-scale instabilities of plane internal gravity waves, J. Fluid Mech., 863 (2019), 702–729. https://doi.org/10.1017/jfm.2018.921 doi: 10.1017/jfm.2018.921
|
| [60] |
K. Hasselmann, A criterion for nonlinear wave stability, J. Fluid Mech., 30 (1967), 737–739. https://doi.org/10.1017/S0022112067001739 doi: 10.1017/S0022112067001739
|
| [61] | O. M. Phillips, The dynamics of the upper ocean, 2 Eds., Cambridge University Press, 1977. |
| [62] |
R. P. Mied, The occurrence of parametric instabilities in finite-amplitude internal gravity waves, J. Fluid Mech., 78 (1976), 763–784. https://doi.org/10.1017/S0022112076002735 doi: 10.1017/S0022112076002735
|
| [63] | C. M. Bender, S. A. Orszag, Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory, New York, NY: Springer, 2013. https://doi.org/10.1007/978-1-4757-3069-2 |
| [64] |
J. Klostermeyer, On parametric instabilities of finite-amplitude internal gravity waves, J. Fluid Mech., 119 (1982), 367–377. https://doi.org/10.1017/S0022112082001396 doi: 10.1017/S0022112082001396
|
| [65] |
P. G. Drazin, On the instability of an internal gravity wave, Proc. R. Soc. Lond. A, 356 (1977), 411–432. https://doi.org/10.1098/rspa.1977.0142 doi: 10.1098/rspa.1977.0142
|
| [66] |
R. Thom, Structural stability, catastrophe theory, and applied mathematics, SIAM Rev., 19 (1977), 189–201. https://doi.org/10.1137/1019036 doi: 10.1137/1019036
|
| [67] | E. C. Zeeman, Catastrophe theory, In: Structural stability in physics, Berlin, Heidelberg: Springer, 1979, 12–22. https://doi.org/10.1007/978-3-642-67363-4_3 |
| [68] |
P. N. Lombard, J. J. Riley, Instability and breakdown of internal gravity waves. I. Linear stability analysis, Phys. Fluids, 8 (1996), 3271–3287. https://doi.org/10.1063/1.869117 doi: 10.1063/1.869117
|
| [69] |
A. D. McEwan, R. M. Robinson, Parametric instability of internal gravity waves, J. Fluid Mech., 67 (1975), 667–687. https://doi.org/10.1017/S0022112075000547 doi: 10.1017/S0022112075000547
|
| [70] |
A. Lifschitz, E. Hameiri, Local stability conditions in fluid dynamics, Physics of Fluids A: Fluid Dynamics, 3 (1991), 2644–2651. https://doi.org/10.1063/1.858153 doi: 10.1063/1.858153
|
| [71] |
S. Leblanc, Local stability of Gerstner's waves, J. Fluid Mech., 506 (2004), 245–254. https://doi.org/10.1017/S0022112004008444 doi: 10.1017/S0022112004008444
|
| [72] |
A. Constantin, P. Germain, Instability of some equatorially trapped waves, J. Geophys. Res.: Oceans, 118 (2013), 2802–2810. https://doi.org/10.1002/jgrc.20219 doi: 10.1002/jgrc.20219
|
| [73] |
D. Ionescu-Kruse, On the short-wavelength stabilities of some geophysical flows, Phil. Trans. R. Soc. A, 376 (2018), 20170090. https://doi.org/10.1098/rsta.2017.0090 doi: 10.1098/rsta.2017.0090
|
| [74] |
P. N. Lombard, J. J. Riley, On the breakdown into turbulence of propagating internal waves, Dynam. Atmos. Oceans, 23 (1996), 345–355. https://doi.org/10.1016/0377-0265(95)00431-9 doi: 10.1016/0377-0265(95)00431-9
|
| [75] |
C. R. Koudella, C. Staquet, Instability mechanisms of a two-dimensional progressive internal gravity wave, J. Fluid Mech., 548 (2006), 165–196. https://doi.org/10.1017/S0022112005007524 doi: 10.1017/S0022112005007524
|
| [76] |
Y. Onuki, S. Joubaud, T. Dauxois, Simulating turbulent mixing caused by local instability of internal gravity waves, J. Fluid Mech., 915 (2021), A77. https://doi.org/10.1017/jfm.2021.119 doi: 10.1017/jfm.2021.119
|
| [77] |
B. Bourget, T. Dauxois, S. Joubaud, P. Odier, Experimental study of parametric subharmonic instability for internal plane waves, J. Fluid Mech., 723 (2013), 1–20. https://doi.org/10.1017/jfm.2013.78 doi: 10.1017/jfm.2013.78
|
| [78] |
J. Klostermeyer, Parametric instabilities of internal gravity waves in Boussinesq fluids with large Reynolds numbers, Geophys. Astro. Fluid, 26 (1983), 85–105. https://doi.org/10.1080/03091928308221764 doi: 10.1080/03091928308221764
|
| [79] |
D. Cacchione, C. Wunsch, Experimental study of internal waves over a slope, J. Fluid Mech., 66 (1974), 223–239. https://doi.org/10.1017/S0022112074000164 doi: 10.1017/S0022112074000164
|
| [80] |
S. A. Thorpe, A. P. Haines, On the reflection of a train of finite-amplitude internal waves from a uniform slope, J. Fluid Mech., 178 (1987), 279–302. https://doi.org/10.1017/S0022112087001228 doi: 10.1017/S0022112087001228
|
| [81] |
M. Leclair, K. Raja, C. Staquet, Nonlinear reflection of a two-dimensional finite-width internal gravity wave on a slope, J. Fluid Mech., 887 (2020), A31. https://doi.org/10.1017/jfm.2019.1077 doi: 10.1017/jfm.2019.1077
|
| [82] |
T. Dauxois, W. R. Young, Near-critical reflection of internal waves, J. Fluid Mech., 390 (1999), 271–295. https://doi.org/10.1017/S0022112099005108 doi: 10.1017/S0022112099005108
|
| [83] |
R. Bianchini, A. L. Dalibard, L. Saint-Raymond, Near-critical reflection of internal waves, Anal. PDE, 14 (2021), 205–249. https://doi.org/10.2140/apde.2021.14.205 doi: 10.2140/apde.2021.14.205
|
| [84] |
E. Horne, J. Schmitt, N. Pustelnik, S. Joubaud, P. Odier, Variational mode decomposition for estimating critical reflected internal wave in stratified fluid, Exp. Fluids, 62 (2021), 110. https://doi.org/10.1007/s00348-021-03206-7 doi: 10.1007/s00348-021-03206-7
|
| [85] |
L. Gostiaux, T. Dauxois, H. Didelle, J. Sommeria, S. Viboud, Quantitative laboratory observations of internal wave reflection on ascending slopes, Phys. Fluids, 18 (2006), 056602. https://doi.org/10.1063/1.2197528 doi: 10.1063/1.2197528
|
| [86] |
N. Grisouard, M. Leclair, L. Gostiaux, C. Staquet, Large scale energy transfer from an internal gravity wave reflecting on a simple slope, Procedia IUTAM, 8 (2013), 119–128. https://doi.org/10.1016/j.piutam.2013.04.016 doi: 10.1016/j.piutam.2013.04.016
|
| [87] |
B. Bourget, H. Scolan, T. Dauxois, M Le Bars, P. Odier, S. Joubaud, Finite-size effects in parametric subharmonic instability, J. Fluid Mech., 759 (2104), 739–750. https://doi.org/10.1017/jfm.2014.550 doi: 10.1017/jfm.2014.550
|
| [88] |
H. H. Karimi, T. R. Akylas, Parametric subharmonic instability of internal waves: locally confined beams versus monochromatic wavetrains, J. Fluid Mech., 757 (2014), 381–402. https://doi.org/10.1017/jfm.2014.509 doi: 10.1017/jfm.2014.509
|
| [89] |
H. A. Clark, B. R. Sutherland, Generation, propagation, and breaking of an internal wave beam, Phys. Fluids, 22 (2010), 076601. https://doi.org/10.1063/1.3455432 doi: 10.1063/1.3455432
|
| [90] |
T. Kataoka, T. R. Akylas, Stability of internal gravity wave beams to three-dimensional modulations, J. Fluid Mech., 736 (2013), 67–90. https://doi.org/10.1017/jfm.2013.527 doi: 10.1017/jfm.2013.527
|
| [91] |
B. Fan, T. R. Akylas, Finite-amplitude instabilities of thin internal wave beams: experiments and theory, J. Fluid Mech., 904 (2020), A13. https://doi.org/10.1017/jfm.2020.682 doi: 10.1017/jfm.2020.682
|
| [92] |
B. Fan, T. R. Akylas, Instabilities of finite-width internal wave beams: from Floquet analysis to PSI, J. Fluid Mech., 913 (2021), A5. https://doi.org/10.1017/jfm.2020.1172 doi: 10.1017/jfm.2020.1172
|
| [93] |
A. Javam, J. Imberger, S. W. Armfield, Numerical study of internal wave–wave interactions in a stratified fluid, J. Fluid Mech., 415 (2000), 65–87. https://doi.org/10.1017/S0022112000008594 doi: 10.1017/S0022112000008594
|
| [94] |
A. Tabaei, T. R. Akylas, K. G. Lamb, Nonlinear effects in reflecting and colliding internal wave beams, J. Fluid Mech., 526 (2005), 217–243. https://doi.org/10.1017/S0022112004002769 doi: 10.1017/S0022112004002769
|
| [95] |
C. H. Jiang, P. S. Marcus, Selection rules for the nonlinear interaction of internal gravity waves, Phys. Rev. Lett., 102 (2009), 124502. https://doi.org/10.1103/PhysRevLett.102.124502 doi: 10.1103/PhysRevLett.102.124502
|
| [96] |
T. R. Akylas, H. H. Karimi, Oblique collisions of internal wave beams and associated resonances, J. Fluid Mech., 711 (2012), 337–363. https://doi.org/10.1017/jfm.2012.395 doi: 10.1017/jfm.2012.395
|
| [97] |
A. Javam, J. Imberger, S. W. Armfield, Numerical study of internal wave reflection from sloping boundaries, J. Fluid Mech., 396 (1999), 183–201. https://doi.org/10.1017/S0022112099005996 doi: 10.1017/S0022112099005996
|
| [98] |
T. Peacock, A. Tabaei, Visualization of nonlinear effects in reflecting internal wave beams, Phys. Fluids, 17 (2005), 061702. https://doi.org/10.1063/1.1932309 doi: 10.1063/1.1932309
|
| [99] |
B. Rodenborn, D. Kiefer, H. P. Zhang, H. L. Swinney, Harmonic generation by reflecting internal waves, Phys. Fluids, 23 (2011), 026601. https://doi.org/10.1063/1.3553294 doi: 10.1063/1.3553294
|
| [100] |
T. Kataoka, T. R. Akylas, Viscous reflection of internal waves from a slope, Phys. Rev. Fluids, 5 (2020), 014803. https://doi.org/10.1103/PhysRevFluids.5.014803 doi: 10.1103/PhysRevFluids.5.014803
|
| [101] |
V. K. Chalamalla, S. Sarkar, PSI in the case of internal wave beam reflection at a uniform slope, J. Fluid Mech., 789 (2016), 347–367. https://doi.org/10.1017/jfm.2015.608 doi: 10.1017/jfm.2015.608
|
| [102] |
T. Gerkema, C. Staquet, P. Bouruet-Aubertot, Non-linear effects in internal-tide beams, and mixing, Ocean Model., 12 (2006), 302–318. https://doi.org/10.1016/j.ocemod.2005.06.001 doi: 10.1016/j.ocemod.2005.06.001
|
| [103] | I. Pairaud, C. Staquet, J. Sommeria, M. M. Mahdizadeh, Generation of harmonics and sub-harmonics from an internal tide in a uniformly stratified fluid: numerical and laboratory experiments, In: IUTAM symposium on turbulence in the atmosphere and oceans, Dordrecht: Springer, 2010, 51–62. https://doi.org/10.1007/978-94-007-0360-5_5 |
| [104] |
Q. Zhou, P. J. Diamessis, Reflection of an internal gravity wave beam off a horizontal free-slip surface, Phys. Fluids, 25 (2013), 036601. https://doi.org/10.1063/1.4795407 doi: 10.1063/1.4795407
|
| [105] |
K. G. Lamb, Nonlinear interaction among internal wave beams generated by tidal flow over supercritical topography, Geophys. Res. Lett., 31 (2004), L09313. https://doi.org/10.1029/2003GL019393 doi: 10.1029/2003GL019393
|
| [106] |
S. A. Thorpe, On wave interactions in a stratified fluid, J. Fluid Mech., 24 (1966), 737–751. https://doi.org/10.1017/S002211206600096X doi: 10.1017/S002211206600096X
|
| [107] |
S. Martin, W. F. Simmons, C. I. Wunsch, Resonant internal wave interactions, Nature, 224 (1969), 1014–1016. https://doi.org/10.1038/2241014a0 doi: 10.1038/2241014a0
|
| [108] |
D. Varma, M. Mathur, Internal wave resonant triads in finite-depth non-uniform stratifications, J. Fluid Mech., 824 (2017), 286–311. https://doi.org/10.1017/jfm.2017.343 doi: 10.1017/jfm.2017.343
|
| [109] |
R. E. Davis, A. Acrivos, The stability of oscillatory internal waves, J. Fluid Mech., 30 (1967), 723–736. https://doi.org/10.1017/S0022112067001727 doi: 10.1017/S0022112067001727
|
| [110] |
S. Joubaud, J. Munroe, P. Odier, T. Dauxois, Experimental parametric subharmonic instability in stratified fluids, Phys. Fluids, 24 (2012), 041703. https://doi.org/10.1063/1.4706183 doi: 10.1063/1.4706183
|
| [111] |
A. D. McEwan, Degeneration of resonantly-excited standing internal gravity waves, J. Fluid Mech., 50 (1971), 431–448. https://doi.org/10.1017/S0022112071002684 doi: 10.1017/S0022112071002684
|
| [112] |
A. D. McEwan, D. W. Mander, R. K. Smith, Forced resonant second-order interaction between damped internal waves, J. Fluid Mech., 55 (1972), 589–608. https://doi.org/10.1017/S0022112072002034 doi: 10.1017/S0022112072002034
|
| [113] |
P. Bouruet-Aubertot, J. Sommeria, C. Staquet, Breaking of standing internal gravity waves through two-dimensional instabilities, J. Fluid Mech., 285 (1995), 265–301. https://doi.org/10.1017/S0022112095000541 doi: 10.1017/S0022112095000541
|
| [114] |
D. Benielli, J. Sommeria, Excitation and breaking of internal gravity waves by parametric instability, J. Fluid Mech., 374 (1998), 117–144. https://doi.org/10.1017/S0022112098002602 doi: 10.1017/S0022112098002602
|
| [115] |
D. Varma, V. K. Chalamalla, M. Mathur, Spontaneous superharmonic internal wave excitation by modal interactions in uniform and nonuniform stratifications, Dynam. Atmos. Oceans, 91 (2020), 101159. https://doi.org/10.1016/j.dynatmoce.2020.101159 doi: 10.1016/j.dynatmoce.2020.101159
|
| [116] |
P. Husseini, D. Varma, T. Dauxois, S. Joubaud, P. Odier, M. Mathur, Experimental study on superharmonic wave generation by resonant interaction between internal wave modes, Phys. Rev. Fluids, 5 (2020), 074804. https://doi.org/10.1103/PhysRevFluids.5.074804 doi: 10.1103/PhysRevFluids.5.074804
|
| [117] |
Y. Liang, A. Zareei, M. R. Alam, Inherently unstable internal gravity waves due to resonant harmonic generation, J. Fluid Mech., 811 (2017), 400–420. https://doi.org/10.1017/jfm.2016.754 doi: 10.1017/jfm.2016.754
|
| [118] |
D. Broutman, J. W. Rottman, S. D. Eckermann, Ray methods for internal waves in the atmosphere and ocean, Annu. Rev. Fluid Mech., 36 (2004), 233–253. https://doi.org/10.1146/annurev.fluid.36.050802.122022 doi: 10.1146/annurev.fluid.36.050802.122022
|
| [119] |
D. C. Fritts, L. Yuan, An analysis of gravity wave ducting in the atmosphere: Eckart's resonances in thermal and Doppler ducts, J. Geophys. Res. Atmos., 94 (1989), 18455–18466. https://doi.org/10.1029/JD094iD15p18455 doi: 10.1029/JD094iD15p18455
|
| [120] |
Y. V. Kistovich, Y. D. Chashechkin, Linear theory of the propagation of internal wave beams in an arbitrarily stratified liquid, J. Appl. Mech. Tech. Phys., 39 (1998), 729–737. https://doi.org/10.1007/BF02468043 doi: 10.1007/BF02468043
|
| [121] |
J. T. Nault, B. R. Sutherland, Internal wave transmission in nonuniform flows, Phys. Fluids, 19 (2007), 016601. https://doi.org/10.1063/1.2424791 doi: 10.1063/1.2424791
|
| [122] |
M. Mathur, T. Peacock, Internal wave beam propagation in non-uniform stratifications, J. Fluid Mech., 639 (2009), 133–152. https://doi.org/10.1017/S0022112009991236 doi: 10.1017/S0022112009991236
|
| [123] |
S. J. Ghaemsaidi, H. V. Dosser, L. Rainville, T. Peacock, The impact of multiple layering on internal wave transmission, J. Fluid Mech., 789 (2016), 617–629. https://doi.org/10.1017/jfm.2015.682 doi: 10.1017/jfm.2015.682
|
| [124] |
B. R. Sutherland, Internal wave transmission through a thermohaline staircase, Phys. Rev. Fluids, 1 (2016), 013701. https://doi.org/10.1103/PhysRevFluids.1.013701 doi: 10.1103/PhysRevFluids.1.013701
|
| [125] |
R. Supekar, T. Peacock, Interference and transmission of spatiotemporally locally forced internal waves in non-uniform stratifications, J. Fluid Mech., 866 (2019), 350–368. https://doi.org/10.1017/jfm.2019.106 doi: 10.1017/jfm.2019.106
|
| [126] |
B. Gayen, S. Sarkar, Degradation of an internal wave beam by parametric subharmonic instability in an upper ocean pycnocline, J. Geophys. Res., 118 (2013), 4689–4698. https://doi.org/10.1002/jgrc.20321 doi: 10.1002/jgrc.20321
|
| [127] |
B. Gayen, S. Sarkar, PSI to turbulence during internal wave beam refraction through the upper ocean pycnocline, Geophys. Res. Lett., 41 (2014), 8953–8960. https://doi.org/10.1002/2014GL061226 doi: 10.1002/2014GL061226
|
| [128] |
S. J. Ghaemsaidi, S. Joubaud, T. Dauxois, P. Odier, T. Peacock, Nonlinear internal wave penetration via parametric subharmonic instability, Phys. Fluids, 28 (2016), 011703. https://doi.org/10.1063/1.4939001 doi: 10.1063/1.4939001
|
| [129] |
N. Grisouard, C. Staquet, T. Gerkema, Generation of internal solitary waves in a pycnocline by an internal wave beam: a numerical study, J. Fluid Mech., 676 (2011), 491–513. https://doi.org/10.1017/jfm.2011.61 doi: 10.1017/jfm.2011.61
|
| [130] |
M. Mercier, M. Mathur, L. Gostiaux, T. Gerkema, J. M. Magalhaes, J. C. B. Da Silva, et al., Soliton generation by internal tidal beams impinging on a pycnocline: laboratory experiments, J. Fluid Mech., 704 (2012), 37–60. https://doi.org/10.1017/jfm.2012.191 doi: 10.1017/jfm.2012.191
|
| [131] |
P. J. Diamessis, S. Wunsch, I. Delwiche, M. P. Richter, Nonlinear generation of harmonics through the interaction of an internal wave beam with a model oceanic pycnocline, Dynam. Atmos. Oceans., 66 (2014), 110–137. https://doi.org/10.1016/j.dynatmoce.2014.02.003 doi: 10.1016/j.dynatmoce.2014.02.003
|
| [132] |
S. Wunsch, I. Delwiche, G. Frederick, A. Brandt, Experimental study of nonlinear harmonic generation by internal waves incident on a pycnocline, Exp. Fluids, 56 (2015), 87. https://doi.org/10.1007/s00348-015-1954-2 doi: 10.1007/s00348-015-1954-2
|
| [133] | I. Stakgold, Boundary value problems of mathematical physics: Volume 1, Society for Industrial and Applied Mathematics, 2000. https://doi.org/10.1137/1.9780898719888 |
| [134] |
S. Wunsch, Harmonic generation by nonlinear self-interaction of a single internal wave mode, J. Fluid Mech., 828 (2017), 630–647. https://doi.org/10.1017/jfm.2017.532 doi: 10.1017/jfm.2017.532
|
| [135] |
L. E. Baker, B. R. Sutherland, The evolution of superharmonics excited by internal tides in non-uniform stratification, J. Fluid Mech., 891 (2020), R1. https://doi.org/10.1017/jfm.2020.188 doi: 10.1017/jfm.2020.188
|
| [136] |
J. A. MacKinnon, K. B. Winters, Subtropical catastrophe: Significant loss of low‐mode tidal energy at $28.9^0$, Geophys. Res. Lett., 32 (2005), L15605. https://doi.org/10.1029/2005GL023376 doi: 10.1029/2005GL023376
|
| [137] |
O. Richet, J. M. Chomaz, C. Muller, Internal tide dissipation at topography: triadic resonant instability equatorward and evanescent waves poleward of the critical latitude, J. Geophys. Res.: Oceans, 123 (2018), 6136–6155. https://doi.org/10.1029/2017JC013591 doi: 10.1029/2017JC013591
|
| [138] |
T. Gerkema, C. Staquet, P. Bouruet‐Aubertot, Decay of semi‐diurnal internal‐tide beams due to subharmonic resonance, Geophys. Res. Lett., 33 (2006), L08604. https://doi.org/10.1029/2005GL025105 doi: 10.1029/2005GL025105
|
| [139] |
W. R. Young, Y. K. Tsang, N. J. Balmforth, Near-inertial parametric subharmonic instability, J. Fluid Mech., 607 (2008), 25–49. https://doi.org/10.1017/S0022112008001742 doi: 10.1017/S0022112008001742
|
| [140] |
M. Nikurashin, S. Legg, A mechanism for local dissipation of internal tides generated at rough topography, J. Phys. Oceanogr., 41 (2011), 378–395. https://doi.org/10.1175/2010JPO4522.1 doi: 10.1175/2010JPO4522.1
|
| [141] |
H. H. Karimi, T. R. Akylas, Near-inertial parametric subharmonic instability of internal wave beams, Phys. Rev. Fluids, 2 (2017), 074801. https://doi.org/10.1103/PhysRevFluids.2.074801 doi: 10.1103/PhysRevFluids.2.074801
|
| [142] |
Y. Onuki, Y. Tanaka, Instabilities of finite‐amplitude internal wave beams, Geophys. Res. Lett., 46 (2019), 7527–7535. https://doi.org/10.1029/2019GL082570 doi: 10.1029/2019GL082570
|
| [143] |
P. Maurer, S. Joubaud, P. Odier, Generation and stability of inertia–gravity waves, J. Fluid Mech., 808 (2016), 539–561. https://doi.org/10.1017/jfm.2016.635 doi: 10.1017/jfm.2016.635
|
| [144] |
B. R. Sutherland, R. Jefferson, Triad resonant instability of horizontally periodic internal modes, Phys. Rev. Fluids, 5 (2020), 034801. https://doi.org/10.1103/PhysRevFluids.5.034801 doi: 10.1103/PhysRevFluids.5.034801
|
| [145] |
Y. Niwa, T. Hibiya, Nonlinear processes of energy transfer from traveling hurricanes to the deep ocean internal wave field, J. Geophys. Res.: Oceans, 102 (1997), 12469–12477. https://doi.org/10.1029/97JC00588 doi: 10.1029/97JC00588
|
| [146] |
G. Bordes, F. Moisy, T. Dauxois, P. P. Cortet, Experimental evidence of a triadic resonance of plane inertial waves in a rotating fluid, Phys. Fluids, 24 (2012), 014105. https://doi.org/10.1063/1.3675627 doi: 10.1063/1.3675627
|
| [147] |
D. O. Mora, E. Monsalve, M. Brunet, T. Dauxois, P. P. Cortet, Three-dimensionality of the triadic resonance instability of a plane inertial wave, Phys. Rev. Fluids, 6 (2021), 074801. https://doi.org/10.1103/PhysRevFluids.6.074801 doi: 10.1103/PhysRevFluids.6.074801
|
| [148] |
K. Ha, J. M. Chomaz, S. Ortiz, Transient growth, edge states, and repeller in rotating solid and fluid, Phys. Rev. E, 103 (2021), 033102. https://doi.org/10.1103/PhysRevE.103.033102 doi: 10.1103/PhysRevE.103.033102
|
| [149] |
R. Ferrari, C. Wunsch, Ocean circulation kinetic energy: Reservoirs, sources, and sinks, Annu. Rev. Fluid Mech., 41 (2009), 253–282. https://doi.org/10.1146/annurev.fluid.40.111406.102139 doi: 10.1146/annurev.fluid.40.111406.102139
|
| [150] |
M. P. Lelong, J. J. Riley, Internal wave—vortical mode interactions in strongly stratified flows, J. Fluid Mech., 232 (1991), 1–19. https://doi.org/10.1017/S0022112091003609 doi: 10.1017/S0022112091003609
|
| [151] |
W. R. Young, M. B. Jelloul, Propagation of near-inertial oscillations through a geostrophic flow, J. Mar. Res., 55 (1997), 735–766. https://doi.org/10.1357/0022240973224283 doi: 10.1357/0022240973224283
|
| [152] |
O. Bühler, Wave–vortex interactions in fluids and superfluids, Annu. Rev. Fluid Mech., 42 (2010), 205–228. https://doi.org/10.1146/annurev.fluid.010908.165251 doi: 10.1146/annurev.fluid.010908.165251
|
| [153] | W. H. Munk, Internal waves and small-scale processes, In: Evolution of physical oceanography: scientific surveys in honor of Henry Stommel, Cambridge, MA: MIT Press, 1981,264–291. |
| [154] |
R. Maugé, T. Gerkema, Generation of weakly nonlinear nonhydrostatic internal tides over large topography: a multi-modal approach, Nonlin. Processes Geophys., 15 (2008), 233–244. https://doi.org/10.5194/npg-15-233-2008 doi: 10.5194/npg-15-233-2008
|
| [155] |
S. Galtier, Weak inertial-wave turbulence theory, Phys. Rev. E, 68 (2003), 015301. https://doi.org/10.1103/PhysRevE.68.015301 doi: 10.1103/PhysRevE.68.015301
|
| [156] | S. Nazarenko, Wave turbulence, Berlin, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-15942-8 |
| [157] |
T. Le Reun, B. Favier, A. J. Barker, M. Le Bars, Inertial wave turbulence driven by elliptical instability, Phys. Rev. Lett., 119 (2017), 034502. https://doi.org/10.1103/PhysRevLett.119.034502 doi: 10.1103/PhysRevLett.119.034502
|
| [158] |
T. Le Reun, B. Favier, M. Le Bars, Parametric instability and wave turbulence driven by tidal excitation of internal waves, J. Fluid Mech., 840 (2018), 498–529. https://doi.org/10.1017/jfm.2018.18 doi: 10.1017/jfm.2018.18
|
| [159] |
E. Monsalve, M. Brunet, B. Gallet, P. P. Cortet, Quantitative experimental observation of weak inertial-wave turbulence, Phys. Rev. Lett., 125 (2020), 254502. https://doi.org/10.1103/PhysRevLett.125.254502 doi: 10.1103/PhysRevLett.125.254502
|
| [160] |
A. D. McEwan, R. A. Plumb, Off-resonant amplification of finite internal wave packets, Dynam. Atmos. Oceans, 2 (1977), 83–105. https://doi.org/10.1016/0377-0265(77)90017-3 doi: 10.1016/0377-0265(77)90017-3
|
| [161] |
S. Gururaj, A. Guha, Energy transfer in resonant and near-resonant internal wave triads for weakly non-uniform stratifications. Part 1. Unbounded domain, J. Fluid Mech., 899 (2020), A6. https://doi.org/10.1017/jfm.2020.431 doi: 10.1017/jfm.2020.431
|
| [162] |
B. Fan, T. R. Akylas, Effect of background mean flow on PSI of internal wave beams, J. Fluid Mech., 869 (2019), R1. https://doi.org/10.1017/jfm.2019.247 doi: 10.1017/jfm.2019.247
|
| [163] |
B. Fan, T. R. Akylas, Near-inertial parametric subharmonic instability of internal wave beams in a background mean flow, J. Fluid Mech., 911 (2021), R3. https://doi.org/10.1017/jfm.2020.1130 doi: 10.1017/jfm.2020.1130
|
| [164] |
T. Jamin, T. Kataoka, T. Dauxois, T. R. Akylas, Long-time dynamics of internal wave streaming, J. Fluid Mech., 907 (2021), A2. https://doi.org/10.1017/jfm.2020.806 doi: 10.1017/jfm.2020.806
|
| [165] |
K. Hasselmann, Feynman diagrams and interaction rules of wave‐wave scattering processes, Rev. Geophys., 4 (1966), 1–32. https://doi.org/10.1029/RG004i001p00001 doi: 10.1029/RG004i001p00001
|
| [166] |
D. J. Olbers, Models of the oceanic internal wave field, Rev. Geophys., 21 (1983), 1567–1606. https://doi.org/10.1029/RG021i007p01567 doi: 10.1029/RG021i007p01567
|
| [167] |
Y. Onuki, T. Hibiya, Decay rates of internal tides estimated by an improved wave–wave interaction analysis, J. Phys. Oceanogr., 48 (2018), 2689–2701. https://doi.org/10.1175/JPO-D-17-0278.1 doi: 10.1175/JPO-D-17-0278.1
|
| [168] |
C. H. McComas, F. P. Bretherton, Resonant interaction of oceanic internal waves, J. Geophys. Res., 82 (1977), 1397–1412. https://doi.org/10.1029/JC082i009p01397 doi: 10.1029/JC082i009p01397
|
| [169] |
T. N. Stevenson, Axisymmetric internal waves generated by a travelling oscillating body, J. Fluid Mech., 35 (1969), 219–224. https://doi.org/10.1017/S0022112069001078 doi: 10.1017/S0022112069001078
|
| [170] |
B. King, H. P. Zhang, H. L. Swinney, Tidal flow over three-dimensional topography in a stratified fluid, Phys. Fluids, 21 (2009), 116601. https://doi.org/10.1063/1.3253692 doi: 10.1063/1.3253692
|
| [171] |
N. D. Shmakova, J. B. Flór, Nonlinear aspects of focusing internal waves, J. Fluid Mech., 862 (2019), R4. https://doi.org/10.1017/jfm.2018.1020 doi: 10.1017/jfm.2018.1020
|
| [172] |
S. Boury, T. Peacock, P. Odier, Experimental generation of axisymmetric internal wave super-harmonics, Phys. Rev. Fluids, 6 (2021), 064801. https://doi.org/10.1103/PhysRevFluids.6.064801 doi: 10.1103/PhysRevFluids.6.064801
|
| [173] |
R. Grimshaw, Resonant wave interactions in a stratified shear flow, J. Fluid Mech., 190 (1988), 357–374. https://doi.org/10.1017/S0022112088001351 doi: 10.1017/S0022112088001351
|
| [174] |
J. Vanneste, F. Vial, On the nonlinear interactions of geophysical waves in shear flows, Geophys. Astro. Fluid, 78 (1994), 115–141. https://doi.org/10.1080/03091929408226575 doi: 10.1080/03091929408226575
|
| [175] |
R. Patibandla, M. Mathur, A. Roy, Triadic resonances in internal wave modes with background shear, J. Fluid Mech., 929 (2021), A10. https://doi.org/10.1017/jfm.2021.847 doi: 10.1017/jfm.2021.847
|
| [176] |
J. Vanneste, Balance and spontaneous wave generation in geophysical flows, Annu. Rev. Fluid Mech., 45 (2013), 147–172. https://doi.org/10.1146/annurev-fluid-011212-140730 doi: 10.1146/annurev-fluid-011212-140730
|
| [177] |
T. Gerkema, J. T. F. Zimmerman, L. R. M. Maas, H. Van Haren, Geophysical and astrophysical fluid dynamics beyond the traditional approximation, Rev. Geophys., 46 (2008), RG2004. https://doi.org/10.1029/2006RG000220 doi: 10.1029/2006RG000220
|
| [178] |
G. N. Ivey, K. B. Winters, J. R. Koseff, Density stratification, turbulence, but how much mixing?, Annu. Rev. Fluid Mech., 40 (2008), 169–184. https://doi.org/10.1146/annurev.fluid.39.050905.110314 doi: 10.1146/annurev.fluid.39.050905.110314
|
| [179] |
S. Legg, Mixing by oceanic lee waves, Annu. Rev. Fluid Mech., 53 (2021), 173–201. https://doi.org/10.1146/annurev-fluid-051220-043904 doi: 10.1146/annurev-fluid-051220-043904
|
| [180] |
W. G. Large, J. C. McWilliams, S. C. Doney, Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization, Rev. Geophys., 32 (1994), 363–403. https://doi.org/10.1029/94RG01872 doi: 10.1029/94RG01872
|
| [181] |
S. A. Smith, D. C. Fritts, T. E. Vanzandt, Evidence for a saturated spectrum of atmospheric gravity waves, J. Atmos. Sci., 44 (1987), 1404–1410. https://doi.org/10.1175/1520-0469(1987)044<1404:EFASSO>2.0.CO;2 doi: 10.1175/1520-0469(1987)044<1404:EFASSO>2.0.CO;2
|
| [182] | G. I. Ogilvie, Internal waves and tides in stars and giant planets, In: Fluid mechanics of planets and stars, Cham: Springer, 2020, 1–30. https://doi.org/10.1007/978-3-030-22074-7_1 |
Figures(10)
Dheeraj Varma, Manikandan Mathur, Thierry Dauxois. Instabilities in internal gravity waves[J]. Mathematics in Engineering, 2023, 5(1): 1-34. doi: 10.3934/mine.2023016
DownLoad: