Hostname: page-component-7479d7b7d-wxhwt Total loading time: 0 Render date: 2024-07-11T11:58:05.691Z Has data issue: false hasContentIssue false
  • English
  • Français

Experimental study of the nonlinear saturation of the elliptical instability: inertial wave turbulence versus geostrophic turbulence

Published online by Cambridge University Press:  27 September 2019

Thomas Le Reun Open the ORCID record for Thomas Le Reun [Opens in a new window] ,
Benjamin Favier Open the ORCID record for Benjamin Favier [Opens in a new window]  and
Michael Le Bars
Thomas Le Reun*
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille, France
Benjamin Favier
Affiliation:
Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille, France
Michael Le Bars
Affiliation:
Aix Marseille Univ, 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

In this paper, we present an experimental investigation of the turbulent saturation of the flow driven by the parametric resonance of inertial waves in a rotating fluid. In our set-up, a half-metre wide ellipsoid filled with water is brought to solid-body rotation, and then undergoes sustained harmonic modulation of its rotation rate. This triggers the exponential growth of a pair of inertial waves via a mechanism called the libration-driven elliptical instability. Once the saturation of this instability is reached, we observe a turbulent state for which energy is injected into the resonant inertial waves only. Depending on the amplitude of the rotation rate modulation, two different saturation states are observed. At large forcing amplitudes, the saturation flow mainly consists of a steady, geostrophic anticyclone. Its amplitude vanishes as the forcing amplitude is decreased while remaining above the threshold of the elliptical instability. Below this secondary transition, the saturation flow is a superposition of inertial waves which are in weakly nonlinear resonant interaction, a state that could asymptotically lead to inertial wave turbulence. In addition to being a first experimental observation of a wave-dominated saturation in unstable rotating flows, the present study is also an experimental confirmation of the model of Le Reun et al. (Phys. Rev. Lett., vol. 119 (3), 2017, 034502) who introduced the possibility of these two turbulent regimes. The transition between these two regimes and their relevance to geophysical applications are finally discussed.

JFM classification

Geophysical and Geological Flows: Waves in rotating fluids Instability: Parametric instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 879 , 25 November 2019 , pp. 296 - 326
Check for updates
Copyright
© 2019 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

Afanasyev, Y. D. & Peltier, W. R. 1998 Three-dimensional instability of anticyclonic swirling flow in rotating fluid: laboratory experiments and related theoretical predictions. Phys. Fluids 10 (12), 3194–3202.10.1063/1.869846Google Scholar
Aldridge, K. D. & Toomre, A. 1969 Axisymmetric inertial oscillations of a fluid in a rotating spherical container. J. Fluid Mech. 37 (2), 307–323.Google Scholar
Aubourg, Q. & Mordant, N. 2015 Nonlocal resonances in weak turbulence of gravity-capillary waves. Phys. Rev. Lett. 114 (14), 144501.Google Scholar
Bak, P., Tang, C. & Wiesenfeld, K. 1987 Self-organized criticality: an explanation of the 1/f noise. Phys. Rev. Lett. 59 (4), 381–384.Google 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.Google 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.Google 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.Google Scholar
Barker, A. J. & Lithwick, Y. 2014 Non-linear evolution of the elliptical instability in the presence of weak magnetic fields. Mon. Not. R. Astron. Soc. 437 (1), 305–315.Google Scholar
Bellet, F., Godeferd, F. S., Scott, J. F. & Cambon, C. 2006 Wave turbulence in rapidly rotating flows. J. Fluid Mech. 562, 83–121.Google Scholar
Bordes, G., Moisy, F., Dauxois, T & Cortet, P.-P. 2012 Experimental evidence of a triadic resonance of plane inertial waves in a rotating fluid. Phys. Fluids 24 (1), 014105.Google Scholar
Brouzet, C., Ermanyuk, E. V., Joubaud, S., Sibgatullin, I. & Dauxois, T. 2016 Energy cascade in internal-wave attractors. Europhys. Lett. 113 (4), 44001.Google Scholar
Busse, F. H. 2010 Mean zonal flows generated by librations of a rotating spherical cavity. J. Fluid Mech. 650, 505–512.Google Scholar
Campagne, A., Gallet, B., Moisy, F & Cortet, P.-P. 2014 Disentangling inertial waves from eddy turbulence in a forced rotating-turbulence experiment. Phys. Fluids 26 (12), 125112.Google Scholar
Cébron, D., Le Bars, M., Moutou, C. & Le Gal, P. 2012a Elliptical instability in terrestrial planets and moons. Astron. Astrophys. 539, A78.Google Scholar
Cébron, D., Le Bars, M., Noir, J. & Aurnou, J. M. 2012b Libration driven elliptical instability. Phys. Fluids 24 (6), 061703.Google Scholar
Cébron, D., Vantieghem, S. & Herreman, W. 2014 Libration-driven multipolar instabilities. J. Fluid Mech. 739, 502–543.10.1017/jfm.2013.623Google Scholar
Clark di Leoni, P., Cobelli, P. J., Mininni, P. D., Dmitruk, P. & Matthaeus, W. H. 2014 Quantification of the strength of inertial waves in a rotating turbulent flow. Phys. Fluids 26 (3), 035106.Google Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Düring, G., Josserand, C. & Rica, S. 2006 Weak turbulence for a vibrating plate: Can one hear a Kolmogorov spectrum? Phys. Rev. Lett. 97 (2), 025503.Google Scholar
Dwyer, C. A., Stevenson, D. J. & Nimmo, F. 2011 A long-lived lunar dynamo driven by continuous mechanical stirring. Nature 479 (7372), 212–214.10.1038/nature10564Google Scholar
Eloy, C., Le Gal, P. & Le Dizès, S. 2000 Experimental study of the multipolar vortex instability. Phys. Rev. Lett. 85 (16), 3400–3403.Google 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), 845–860.10.1093/mnras/stu003Google Scholar
Favier, B., Cambon, C. & Godeferd, F. 2010 On space and time correlations of isotropic and rotating turbulence. Phys. Fluids 22 (1), 015101.Google 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.Google Scholar
Galtier, S. 2003 Weak inertial-wave turbulence theory. Phys. Rev. E 68 (1), 015301.Google Scholar
Godeferd, F. S. & Moisy, F. 2015 Structure and dynamics of rotating turbulence: a review of recent experimental and numerical results. Appl. Mech. Rev. 67 (3), 030802.Google Scholar
Goodman, J. & Lackner, C. 2009 Dynamical tides in rotating planets and stars. Astrophys. J. 696 (2), 2054.Google 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.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Greenspan, H. P. 1969 On the non-linear interaction of inertial modes. J. Fluid Mech. 36 (02), 257–264.Google Scholar
Hough, S. S. 1895 XII. The oscillations of a rotating ellipsoidal shell containing fluid. Phil. Trans. R. Soc. Lond. A 186, 469–506.Google Scholar
Jones, E., Oliphant, T., Peterson, P. et al. 2001 SciPy: Open source scientific tools for Python. Tech. Rep.Google Scholar
Jouve, L. & Ogilvie, G. I. 2014 Direct numerical simulations of an inertial wave attractor in linear and nonlinear regimes. J. Fluid Mech. 745, 223–250.10.1017/jfm.2014.63Google Scholar
Kartashova, E. 2009 Discrete wave turbulence. Europhys. Lett. 87 (4), 44001.Google Scholar
Kerswell, R. R. 1999 Secondary instabilities in rapidly rotating fluids: inertial wave breakdown. J. Fluid Mech. 382, 283–306.Google Scholar
Kerswell, R. R. 2002 Elliptical instability. Annu. Rev. Fluid Mech. 34 (1), 83–113.Google Scholar
Kerswell, R. R. & Malkus, W. V. R. 1998 Tidal instability as the source for Io’s magnetic signature. Geophys. Res. Lett. 25 (5), 603–606.Google 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.Google Scholar
Le Bars, M., Lacaze, L., Le Dizès, S., Le Gal, P. & Rieutord, M. 2010 Tidal instability in stellar and planetary binary systems. Phys. Earth Planet. Inter. 178 (1–2), 48–55.Google Scholar
Le Bars, M., Le Dizès, S. & Le Gal, P. 2007 Coriolis effects on the elliptical instability in cylindrical and spherical rotating containers. J. Fluid Mech. 585, 323.Google Scholar
Le Bars, M., Wieczorek, M. A., Karatekin, Ö, Cébron, D. & Laneuville, M. 2011 An impact-driven dynamo for the early Moon. Nature 479 (7372), 215–218.Google Scholar
Le Dizès, S. 2000 Three-dimensional instability of a multipolar vortex in a rotating flow. Phys. Fluids 12 (11), 2762–2774.Google 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.Google Scholar
Le Reun, T., Favier, B. & Le Bars, M. 2018 Parametric instability and wave turbulence driven by tidal excitation of internal waves. J. Fluid Mech. 840, 498–529.Google Scholar
Lemasquerier, D., Grannan, A. M., Vidal, J., Cébron, D., Favier, B., Bars, M. L. & Aurnou, J. M. 2017 Libration-driven flows in ellipsoidal shells. J. Geophys. Res. 122 (9), 1926–1950.Google Scholar
Lin, Y., Noir, J. & Jackson, A. 2014 Experimental study of fluid flows in a precessing cylindrical annulus. Phys. Fluids 26 (4), 046604.Google Scholar
Malkus, W. V. R. 1989 An experimental study of global instabilities due to the tidal (elliptical) distortion of a rotating elastic cylinder. Geophys. Astrophys. Fluid Dyn. 48 (1–3), 123–134.10.1080/03091928908219529Google Scholar
Miquel, B. & Mordant, N. 2011 Nonstationary wave turbulence in an elastic plate. Phys. Rev. Lett. 107 (3), 034501.Google 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), 214501.10.1103/PhysRevLett.104.214501Google Scholar
Nazarenko, S. 2011 Wave Turbulence, Lecture Notes in Physics, vol. 825. Springer.10.1007/978-3-642-15942-8Google Scholar
Noir, J., Cébron, D., Bars, M. L., Sauret, A. & Aurnou, J. M. 2012 Experimental study of libration-driven zonal flows in non-axisymmetric containers. Phys. Earth Planet. Inter. 204–205, 1–10.Google Scholar
Noir, J., Hemmerlin, F., Wicht, J., Baca, S. M. & Aurnou, J. M. 2009 An experimental and numerical study of librationally driven flow in planetary cores and subsurface oceans. Phys. Earth Planet. Inter. 173 (1–2), 141–152.Google Scholar
Ogilvie, G. I. & Lin, D. N. C. 2004 Tidal dissipation in rotating giant planets. Astrophys. J. 610 (1), 477.Google Scholar
Oks, D., Mininni, P. D., Marino, R. & Pouquet, A. 2017 Inverse cascades and resonant triads in rotating and stratified turbulence. Phys. Fluids 29 (11), 111109.Google Scholar
Poincaré, H. 1885 Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation. Acta Mathematica 7 (1), 259–380.Google Scholar
Reddy, S. K., Favier, B. & Le Bars, M. 2018 Turbulent kinematic dynamos in ellipsoids driven by mechanical forcing. Geophys. Res. Lett. 45 (4), 1741–1750.Google Scholar
Salehipour, H., Peltier, W. R. & Caulfield, C. P. 2018 Self-organized criticality of turbulence in strongly stratified mixing layers. J. Fluid Mech. 856, 228–256.Google Scholar
Sauret, A., CéBron, D., Morize, C. & Le Bars, M. 2010 Experimental and numerical study of mean zonal flows generated by librations of a rotating spherical cavity. J. Fluid Mech. 662, 260–268.Google 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.Google Scholar
Sauret, A. & Le Dizès, S. 2013 Libration-induced mean flow in a spherical shell. J. Fluid Mech. 718, 181–209.Google 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.Google Scholar
Smith, L. M. & Waleffe, F. 1999 Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence. Phys. Fluids 11 (6), 1608–1622.10.1063/1.870022Google Scholar
Tilgner, A. 2007 Zonal wind driven by inertial modes. Phys. Rev. Lett. 99 (19), 194501.Google Scholar
Vanneste, J. 2005 Wave interactions. In Nonlinear Waves in Fluids: Recent Advances and Modern Applications (ed. Grimshaw, R.), pp. 69–94. Springer.Google Scholar
Vidal, J. & Cébron, D. 2017 Inviscid instabilities in rotating ellipsoids on eccentric Kepler orbits. J. Fluid Mech. 833, 469–511.Google Scholar
Vidal, J., Cébron, D. & Schaeffer, N.2017 Diffusionless hydromagnetic modes in rotating ellipsoids: a road to weakly nonlinear models? arXiv:1702.07198.Google Scholar
Vidal, J., Cébron, D., Schaeffer, N. & Hollerbach, R. 2018 Magnetic fields driven by tidal mixing in radiative stars. Mon. Not. R. Astron. Soc. 475 (4), 4579–4594.10.1093/mnras/sty080Google Scholar
Weber, M. J. 2018 Handbook of Optical Materials. CRC Press.Google Scholar
Yarom, E., Salhov, A. & Sharon, E. 2017 Experimental quantification of nonlinear time scales in inertial wave rotating turbulence. Phys. Rev. Fluids 2 (12), 122601.Google Scholar
Yarom, E. & Sharon, E. 2014 Experimental observation of steady inertial wave turbulence in deep rotating flows. Nat. Phys. 10 (7), 510–514.Google Scholar
Yokoyama, N. & Takaoka, M. 2017 Hysteretic transitions between quasi-two-dimensional flow and three-dimensional flow in forced rotating turbulence. Phys. Rev. Fluids 2 (9), 092602.Google Scholar