Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-23T11:01:14.214Z Has data issue: false hasContentIssue false
  • English
  • Français

Parametric instability and wave turbulence driven by tidal excitation of internal waves

Published online by Cambridge University Press:  14 February 2018

Thomas Le Reun Open the ORCID record for Thomas Le Reun [Opens in a new window] ,
Benjamin Favier  and
Michael Le Bars
Thomas Le Reun*
Affiliation:
Aix Marseille University, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille, France
Benjamin Favier
Affiliation:
Aix Marseille University, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille, France
Michael Le Bars
Affiliation:
Aix Marseille University, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille, France
*
†Email address for correspondence: lereun@irphe.univ-mrs.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the stability of stratified fluid layers undergoing homogeneous and periodic tidal deformation. We first introduce a local model which allows us to study velocity and buoyancy fluctuations in a Lagrangian domain periodically stretched and sheared by the tidal base flow. While keeping the key physical ingredients only, such a model is efficient in simulating planetary regimes where tidal amplitudes and dissipation are small. With this model, we prove that tidal flows are able to drive parametric subharmonic resonances of internal waves, in a way reminiscent of the elliptical instability in rotating fluids. The growth rates computed via direct numerical simulations (DNSs) are in very good agreement with Wentzel–Kramers–Brillouin analysis and Floquet theory. We also investigate the turbulence driven by this instability mechanism. With spatio-temporal analysis, we show that it is weak internal wave turbulence occurring at small Froude and buoyancy Reynolds numbers. When the gap between the excitation and the Brunt–Väisälä frequencies is increased, the frequency spectrum of this wave turbulence displays a $-2$ power law reminiscent of the high-frequency branch of the Garett and Munk spectrum (Geophys. Fluid Dyn., vol. 3 (1), 1972, pp. 225–264) which has been measured in the oceans. In addition, we find that the mixing efficiency is altered compared to what is computed in the context of DNS of stratified turbulence excited at small Froude and large buoyancy Reynolds numbers and is consistent with a superposition of waves.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 840 , 10 April 2018 , pp. 498 - 529
Copyright
© 2018 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

Aspden, J. M. & Vanneste, J. 2009 Elliptical instability of a rapidly rotating, strongly stratified fluid. Phys. Fluids 21 (7), 074104.CrossRefGoogle Scholar
Aubourg, Q. & Mordant, N. 2015 Nonlocal resonances in weak turbulence of gravity-capillary waves. Phys. Rev. Lett. 114 (14), 144501.CrossRefGoogle ScholarPubMed
Bajars, J., Frank, J. & Maas, L. R. M. 2013 On the appearance of internal wave attractors due to an initial or parametrically excited disturbance. J. Fluid Mech. 714, 283–311.CrossRefGoogle Scholar
Barker, A. J. 2016 Non-linear tides in a homogeneous rotating planet or star: global simulations of the elliptical instability. Mon. Not. R. Astron. Soc. 459 (1), 939–956.CrossRefGoogle Scholar
Barker, A. J., Braviner, H. J. & Ogilvie, G. I. 2016 Non-linear tides in a homogeneous rotating planet or star: global modes and elliptical instability. Mon. Not. R. Astron. Soc. 459 (1), 924–938.CrossRefGoogle Scholar
Barker, A. J. & Lithwick, Y. 2013 Non-linear evolution of the tidal elliptical instability in gaseous planets and stars. Mon. Not. R. Astron. Soc. 435 (4), 3614–3626.CrossRefGoogle Scholar
Bartello, P. & Tobias, S. M. 2013 Sensitivity of stratified turbulence to the buoyancy Reynolds number. J. Fluid Mech. 725, 1–22.CrossRefGoogle Scholar
Bellet, F., Godeferd, F. S., Scott, J. F. & Cambon, C. 2006 Wave turbulence in rapidly rotating flows. J. Fluid Mech. 562, 83–121.CrossRefGoogle Scholar
Bender, C. M. & Orszag, S. A. 1978 Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill.Google Scholar
Benielli, D. & Sommeria, J. 1996 Excitation of internal waves and stratified turbelence by parametric instability. Dyn. Atmos. Oceans 23 (1), 335–343.CrossRefGoogle Scholar
Benielli, D. & Sommeria, J. 1998 Excitation and breaking of internal gravity waves by parametric instability. J. Fluid Mech. 374, 117–144.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13 (6), 1645–1651.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Brethouwer, G., Billant, P., Lindborg, E. & Chomaz, J.-M. 2007 Scaling analysis and simulation of strongly stratified turbulent flows. J. Fluid Mech. 585, 343.CrossRefGoogle Scholar
Brouzet, C.2016 Internal wave attractors: from geometrical focusing to non-linear energy cascade and mixing. PhD thesis, Université de Lyon.Google Scholar
Brouzet, C., Ermanyuk, E., Joubaud, S., Pillet, G. & Dauxois, T. 2017 Internal wave attractors: different scenarios of instability. J. Fluid Mech. 811, 544–568.CrossRefGoogle Scholar
Brouzet, C., Ermanyuk, E. V., Joubaud, S., Sibgatullin, I. & Dauxois, T. 2016 Energy cascade in internal-wave attractors. Europhys. Lett. 113 (4), 44001.CrossRefGoogle Scholar
Buffett, B. 2014 Geomagnetic fluctuations reveal stable stratification at the top of the Earth’s core. Nature 507 (7493), 484–487.CrossRefGoogle ScholarPubMed
Bühler, O. & Holmes-Cerfon, M. 2011 Decay of an internal tide due to random topography in the ocean. J. Fluid Mech. 678, 271–293.CrossRefGoogle Scholar
Cambon, C. 2001 Turbulence and vortex structures in rotating and stratified flows. Eur. J. Mech. (B/Fluids) 20, 489–510.CrossRefGoogle Scholar
Cébron, D., Le Bars, M., Moutou, C. & Le Gal, P. 2012 Elliptical instability in terrestrial planets and moons. Astron. Astrophys. 539, A78.CrossRefGoogle Scholar
Cébron, D., Maubert, P. & Le Bars, M. 2010 Tidal instability in a rotating and differentially heated ellipsoidal shell. Geophys. J. Int. 182 (3), 1311–1318.CrossRefGoogle Scholar
Cébron, D., Vantieghem, S. & Herreman, W. 2014 Libration-driven multipolar instabilities. J. Fluid Mech. 739, 502–543.CrossRefGoogle Scholar
Favier, B., Barker, A. J., Baruteau, C. & Ogilvie, G. I. 2014 Non-linear evolution of tidally forced inertial waves in rotating fluid bodies. Mon. Not. R. Astron. Soc. 439 (1), 845–860.CrossRefGoogle Scholar
Favier, B., Grannan, A. M., Le Bars, M. & Aurnou, J. M. 2015 Generation and maintenance of bulk turbulence by libration-driven elliptical instability. Phys. Fluids 27 (6), 066601.CrossRefGoogle Scholar
Galtier, S. 2003 Weak inertial-wave turbulence theory. Phys. Rev. E 68 (1), 015301(R).Google ScholarPubMed
Gamba, I. M., Smith, L. M. & Tran, M.-B.2017. On the wave turbulence theory for stratified flows in the ocean. arXiv:1709.08266.Google Scholar
Garrett, C. & Munk, W. 1972 Space–time scales of internal waves. Geophys. Fluid Dyn. 3 (1), 225–264.CrossRefGoogle Scholar
Garrett, C. & Munk, W. 1975 Space–time scales of internal waves: a progress report. J. Geophys. Res. 80 (3), 291–297.CrossRefGoogle Scholar
Garrett, C. & Munk, W. 1979 Internal waves in the ocean. Annu. Rev. Fluid Mech. 11 (1), 339–369.CrossRefGoogle Scholar
Gelash, A. A., L’vov, V. S. & Zakharov, V. E. 2017 Complete Hamiltonian formalism for inertial waves in rotating fluids. J. Fluid Mech. 831, 128–150.CrossRefGoogle Scholar
Godoy-Diana, R., Chomaz, J.-M. & Billant, P. 2004 Vertical length scale selection for pancake vortices in strongly stratified viscous fluids. J. Fluid Mech. 504, 229–238.CrossRefGoogle Scholar
Goodman, J. 1993 A local tidal instability of tidally distorted accretion disks. Astrophys. J. 406, 596.CrossRefGoogle Scholar
Goodman, J. & Lackner, C. 2009 Dynamical tides in rotating planets and stars. Astrophys. J. 696 (2), 2054.CrossRefGoogle Scholar
Grannan, A. M., Favier, B., Le Bars, M. & Aurnou, J. M. 2017 Tidally forced turbulence in planetary interiors. Geophys. J. Intl 208 (3), 1690–1703.Google Scholar
Grannan, A. M., Le Bars, M., Cébron, D. & Aurnou, J. M. 2014 Experimental study of global-scale turbulence in a librating ellipsoid. Phys. Fluids 26 (12), 126601.CrossRefGoogle Scholar
Guimbard, D., Le Dizès, S., Le Bars, M., Le Gal, P. & Leblanc, S. 2010 Elliptic instability of a stratified fluid in a rotating cylinder. J. Fluid Mech. 660, 240–257.CrossRefGoogle Scholar
Herbert, C., Marino, R., Rosenberg, D. & Pouquet, A. 2016 Waves and vortices in the inverse cascade regime of stratified turbulence with or without rotation. J. Fluid Mech. 806, 165–204.CrossRefGoogle Scholar
Kerswell, R. R. 1993 Elliptical instabilities of stratified, hydromagnetic waves. Geophys. Astrophys. Fluid Dyn. 71 (1–4), 105–143.CrossRefGoogle Scholar
Kerswell, R. R. 1999 Secondary instabilities in rapidly rotating fluids: inertial wave breakdown. J. Fluid Mech. 382, 283–306.CrossRefGoogle Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34 (1), 83–113.CrossRefGoogle Scholar
Labrosse, S. 2015 Thermal evolution of the core with a high thermal conductivity. Phys. Earth Planet. Inter. 247, 36–55.CrossRefGoogle Scholar
Le Bars, M., Cébron, D. & Le Gal, P. 2015 Flows driven by libration, precession, and tides. Annu. Rev. Fluid Mech. 47 (1), 163–193.CrossRefGoogle Scholar
Le Dizès, S. 2000 Three-dimensional instability of a multipolar vortex in a rotating flow. Phys. Fluids 12 (11), 2762–2774.CrossRefGoogle Scholar
Le Reun, T., Favier, B., Barker, A. J. & Le Bars, M. 2017 Inertial wave turbulence driven by elliptical instability. Phys. Rev. Lett. 119 (3), 034502.CrossRefGoogle ScholarPubMed
Lesur, G. & Longaretti, P.-Y. 2005 On the relevance of subcritical hydrodynamic turbulence to accretion disk transport. Astron. Astrophys. 444 (1), 25–44.CrossRefGoogle Scholar
Levine, M. D. 2002 A modification of the Garrett–Munk internal wave spectrum. J. Phys. Oceanogr. 32 (11), 3166–3181.2.0.CO;2>CrossRefGoogle Scholar
Lifschitz, A. & Hameiri, E. 1991 Local stability conditions in fluid dynamics. Phys. Fluids A 3 (11), 2644–2651.CrossRefGoogle Scholar
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.CrossRefGoogle Scholar
Lindborg, E. & Brethouwer, G. 2008 Vertical dispersion by stratified turbulence. J. Fluid Mech. 614, 303–314.CrossRefGoogle Scholar
Maas, L. R. M. & Lam, F.-P. A. 1995 Geometric focusing of internal waves. J. Fluid Mech. 300, 1–41.CrossRefGoogle Scholar
MacKinnon, J. A. & Winters, K. B. 2005 Subtropical catastrophe: significant loss of low-mode tidal energy at 28.9. Geophys. Res. Lett. 32 (15), L15605.CrossRefGoogle Scholar
Maffioli, A., Brethouwer, G. & Lindborg, E. 2016 Mixing efficiency in stratified turbulence. J. Fluid Mech. 794, R3.CrossRefGoogle Scholar
Maffioli, A. & Davidson, P. A. 2016 Dynamics of stratified turbulence decaying from a high buoyancy Reynolds number. J. Fluid Mech. 786, 210–233.CrossRefGoogle Scholar
Marino, R., Mininni, P. D., Rosenberg, D. & Pouquet, A. 2013 Inverse cascades in rotating stratified turbulence: fast growth of large scales. Europhys. Lett. 102 (4), 44006.CrossRefGoogle Scholar
McEwan, A. D. & Robinson, R. M. 1975 Parametric instability of internal gravity waves. J. Fluid Mech. 67 (4), 667–687.CrossRefGoogle Scholar
McWilliams, J. C. & Yavneh, I. 1998 Fluctuation growth and instability associated with a singularity of the balance equations. Phys. Fluids 10 (10), 2587–2596.CrossRefGoogle Scholar
Miyazaki, T. 1993 Elliptical instability in a stably stratified rotating fluid. Phys. Fluids A 5 (11), 2702–2709.CrossRefGoogle Scholar
Miyazaki, T. & Fukumoto, Y. 1992 Three dimensional instability of strained vortices in a stably stratified fluid. Phys. Fluids A 4 (11), 2515–2522.CrossRefGoogle Scholar
Morize, C., Le Bars, M., Le Gal, P. & Tilgner, A. 2010 Experimental determination of zonal winds driven by tides. Phys. Rev. Lett. 104 (21), 214501.CrossRefGoogle ScholarPubMed
Nazarenko, S. 2011 Wave Turbulence. Springer.CrossRefGoogle Scholar
Ogilvie, G. I. 2014 Tidal dissipation in stars and giant planets. Annu. Rev. Astron. Astrophys. 52 (1), 171–210.CrossRefGoogle Scholar
Ogilvie, G. I. & Lin, D. N. C. 2004 Tidal dissipation in rotating giant planets. Astrophys. J. 610 (1), 477.CrossRefGoogle Scholar
Osborn, T. R. 1980 Estimates of the local rate of vertical diffusion from dissipation measurements. J. Phys. Oceanogr. 10 (1), 83–89.2.0.CO;2>CrossRefGoogle Scholar
Peltier, W. R. & Caulfield, C. P. 2003 Mixing efficiency in stratified shear flows. Annu. Rev. Fluid Mech. 35 (1), 135–167.CrossRefGoogle Scholar
Rogallo, R. S. 1981 Numerical experiments in homogeneous turbulence. NASA Tech. Memorandum. NASA Ames Research Center.Google Scholar
Rorai, C., Mininni, P. D. & Pouquet, A. 2015 Stably stratified turbulence in the presence of large-scale forcing. Phys. Rev. E 92 (1), 043002.Google ScholarPubMed
Salehipour, H. & Peltier, W. R. 2015 Diapycnal diffusivity, turbulent Prandtl number and mixing efficiency in boussinesq stratified turbulence. J. Fluid Mech. 775, 464–500.CrossRefGoogle Scholar
Salehipour, H., Peltier, W. R., Whalen, C. B. & MacKinnon, J. A. 2016 A new characterization of the turbulent diapycnal diffusivities of mass and momentum in the ocean. Geophys. Res. Lett. 43 (7), 2016GL068184.CrossRefGoogle Scholar
Sauret, A., Le Bars, M. & Le Gal, P. 2014 Tide-driven shear instability in planetary liquid cores. Geophys. Res. Lett. 41 (17), 6078–6083.CrossRefGoogle Scholar
Schaeffer, N. & Le Dizès, S. 2010 Nonlinear dynamics of the elliptic instability. J. Fluid Mech. 646, 471–480.CrossRefGoogle Scholar
Scolan, H., Ermanyuk, E. & Dauxois, T. 2013 Nonlinear fate of internal wave attractors. Phys. Rev. Lett. 110 (23), 234501.CrossRefGoogle ScholarPubMed
Scott, J. F. 2014 Wave turbulence in a rotating channel. J. Fluid Mech. 741, 316–349.CrossRefGoogle Scholar
Smith, L. M. & Lee, Y. 2005 On near resonances and symmetry breaking in forced rotating flows at moderate Rossby number. J. Fluid Mech. 535, 111–142.CrossRefGoogle Scholar
Sridhar, S. & Tremaine, S. 1992 Tidal disruption of viscous bodies. Icarus 95 (1), 86–99.CrossRefGoogle Scholar
St. Laurent, L. & Garrett, C. 2002 The role of internal tides in mixing the deep ocean. J. Phys. Oceanogr. 32 (10), 2882–2899.2.0.CO;2>CrossRefGoogle Scholar
Stanley, S. & Mohammadi, A. 2008 Effects of an outer thin stably stratified layer on planetary dynamos. Phys. Earth Planet. Inter. 168 (3–4), 179–190.CrossRefGoogle Scholar
Thomas, P. C., Tajeddine, R., Tiscareno, M. S., Burns, J. A., Joseph, J., Loredo, T. J., Helfenstein, P. & Porco, C. 2016 Enceladus’s measured physical libration requires a global subsurface ocean. Icarus 264, 37–47.CrossRefGoogle Scholar
Tilgner, A. 2007 Zonal wind driven by inertial modes. Phys. Rev. Lett. 99 (19), 194501.CrossRefGoogle ScholarPubMed
Tyler, R. H. 2009 Ocean tides heat Enceladus. Geophys. Res. Lett. 36 (15), L15205.CrossRefGoogle Scholar
Yarom, E. & Sharon, E. 2014 Experimental observation of steady inertial wave turbulence in deep rotating flows. Nat. Phys. 10 (7), 510–514.CrossRefGoogle Scholar
Zakharov, V. E., L’vov, V. S. & Falkovich, G. 1992 Kolmogorov Spectra of Turbulence I: Wave Turbulence. Springer.CrossRefGoogle Scholar