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Onset of three-dimensionality in rapidly rotating turbulent flows

Published online by Cambridge University Press:  28 August 2020

Kannabiran Seshasayanan Open the ORCID record for Kannabiran Seshasayanan [Opens in a new window]  and
Basile Gallet Open the ORCID record for Basile Gallet [Opens in a new window]
Kannabiran Seshasayanan*
Affiliation:
Service de Physique de l’État Condensé, CEA, CNRS UMR 3680, Université Paris-Saclay, CEA Saclay, 91191Gif-sur-Yvette, France
Basile Gallet
Affiliation:
Service de Physique de l’État Condensé, CEA, CNRS UMR 3680, Université Paris-Saclay, CEA Saclay, 91191Gif-sur-Yvette, France
*
†Email address for correspondence: s.kannabiran@gmail.com
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Abstract

Turbulent flows driven by a vertically invariant body force were proven to become exactly two-dimensional (2-D) above a critical rotation rate, using upper bound theory. This transition in dimensionality of a turbulent flow has key consequences for the energy dissipation rate. However, its location in parameter space is not provided by the bounding procedure. To determine this precise threshold between exactly two-dimensional and partially three-dimensional (3-D) flows, we perform a linear stability analysis over a fully turbulent 2-D base state. This requires integrating numerically a quasi-2-D set of equations over thousands of turnover times, to accurately average the growth rate of the 3-D perturbations over the statistics of the turbulent 2-D base flow. We leverage the capabilities of modern graphics processing units to achieve this task, which allows us to investigate the parameter space up to $Re=10^5$. At the Reynolds numbers typical of 3-D direct numerical simulations and laboratory experiments, $Re\in [10^2, 5\times 10^3]$, the turbulent 2-D flow becomes unstable to 3-D motion through a centrifugal-type instability. However, at even higher Reynolds numbers, another instability takes over. A candidate mechanism for the latter instability is the parametric excitation of inertial waves by the modulated 2-D flow, a phenomenon that we illustrate with an oscillatory 2-D Kolmogorov flow.

JFM classification

Turbulent Flows: Rotating turbulence Geophysical and Geological Flows: Geostrophic turbulence
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 901 , 25 October 2020 , R5
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Alexakis, A. 2015 Rotating Taylor–Green flow. J. Fluid Mech. 769, 46–78.10.1017/jfm.2015.82CrossRefGoogle Scholar
Alexakis, A. & Biferale, L. 2018 Cascades and transitions in turbulent flows. Phys. Rep. 767–769, 1–101.10.1016/j.physrep.2018.08.001CrossRefGoogle Scholar
Alexakis, A. & Doering, C. R. 2006 Energy and enstrophy dissipation in steady state 2D turbulence. Phys. Lett. A 359 (6), 652–657.10.1016/j.physleta.2006.07.048CrossRefGoogle Scholar
Bartello, P., Métais, O. & Lesieur, M. 1994 Coherent structures in rotating three-dimensional turbulence. J. Fluid Mech. 273, 1–29.10.1017/S0022112094001837CrossRefGoogle Scholar
Bellet, F., Godeferd, F. S., Scott, J. F. & Cambon, C. 2006 Wave turbulence in rapidly rotating flows. J. Fluid Mech. 562, 83–121.10.1017/S0022112006000929CrossRefGoogle Scholar
Benavides, S. J. & Alexakis, A. 2017 Critical transitions in thin layer turbulence. J. Fluid Mech. 822, 364–385.10.1017/jfm.2017.293CrossRefGoogle Scholar
van Bokhoven, L. J. A., Clercx, H. J. H., van Heijst, G. J. F. & Trieling, R. R. 2009 Experiments on rapidly rotating turbulent flows. Phys. Fluids 21 (9), 096601.10.1063/1.3197876CrossRefGoogle Scholar
Brunet, M., Gallet, B. & Cortet, P.-P. 2020 Shortcut to geostrophy in wave-driven rotating turbulence: the quartetic instability. Phys. Rev. Lett. 124 (12), 124501.CrossRefGoogle ScholarPubMed
Buzzicotti, M., Aluie, H., Biferale, L. & Linkmann, M. 2018 Energy transfer in turbulence under rotation. Phys. Rev. Fluids 3 (3), 034802.10.1103/PhysRevFluids.3.034802CrossRefGoogle Scholar
Campagne, A., Gallet, B., Moisy, F. & Cortet, P.-P. 2014 Direct and inverse energy cascades in a forced rotating turbulence experiment. Phys. Fluids 26 (12), 125112.CrossRefGoogle Scholar
Campagne, A., Gallet, B., Moisy, F. & Cortet, P.-P. 2015 Disentangling inertial waves from eddy turbulence in a forced rotating-turbulence experiment. Phys. Rev. E 91 (4), 043016.CrossRefGoogle Scholar
Campagne, A., Machicoane, N., Gallet, B., Cortet, P.-P. & Moisy, F. 2016 Turbulent drag in a rotating frame. J. Fluid Mech. 794, R5.10.1017/jfm.2016.214CrossRefGoogle Scholar
Chen, Q., Chen, S., Eyink, G. L. & Holm, D. D. 2005 Resonant interactions in rotating homogeneous three-dimensional turbulence. J. Fluid Mech. 542, 139–164.10.1017/S0022112005006324CrossRefGoogle Scholar
Deusebio, E., Boffetta, G., Lindborg, E. & Musacchio, S. 2014 Dimensional transition in rotating turbulence. Phys. Rev. E 90 (2), 023005.CrossRefGoogle ScholarPubMed
Dickinson, S. C. & Long, R. R. 1983 Oscillating-grid turbulence including effects of rotation. J. Fluid Mech. 126, 315–333.10.1017/S002211208300018XCrossRefGoogle Scholar
Gallet, B. 2015 Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows. J. Fluid Mech. 783, 412–447.10.1017/jfm.2015.569CrossRefGoogle Scholar
Gallet, B., Campagne, A., Cortet, P-P & Moisy, F. 2014 Scale-dependent cyclone-anticyclone asymmetry in a forced rotating turbulence experiment. Phys. Fluids 26 (3), 035108.10.1063/1.4867914CrossRefGoogle Scholar
Galtier, S. 2003 Weak inertial-wave turbulence theory. Phys. Rev. E 68 (1), 015301.10.1103/PhysRevE.68.015301CrossRefGoogle ScholarPubMed
Godeferd, F. S. & Lollini, L. 1999 Direct numerical simulations of turbulence with confinement and rotation. J. Fluid Mech. 393, 257–308.10.1017/S0022112099005637CrossRefGoogle Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press.Google Scholar
Hopfinger, E. J., Browand, F. K. & Gagne, Y. 1982 Turbulence and waves in a rotating tank. J. Fluid Mech. 125, 505–534.10.1017/S0022112082003462CrossRefGoogle Scholar
van Kan, A. & Alexakis, A. 2019 Condensates in thin-layer turbulence. J. Fluid Mech. 864, 490–518.10.1017/jfm.2019.29CrossRefGoogle Scholar
van Kan, A. & Alexakis, A. 2020 Critical transition in fast-rotating turbulence within highly elongated domains. J. Fluid Mech. 899, A33.10.1017/jfm.2020.443CrossRefGoogle Scholar
Kerswell, R. R. 1999 Secondary instabilities in rapidly rotating fluids: inertial wave breakdown. J. Fluid Mech. 382, 283–306.10.1017/S0022112098003954CrossRefGoogle Scholar
Kloosterziel, R. C. & Van Heijst, G. J. F. 1991 An experimental study of unstable barotropic vortices in a rotating fluid. J. Fluid Mech. 223, 1–24.10.1017/S0022112091001301CrossRefGoogle Scholar
Le Dizes, S. 2000 Three-dimensional instability of a multipolar vortex in a rotating flow. Phys. Fluids 12 (11), 2762–2774.10.1063/1.1289774CrossRefGoogle Scholar
Le Dizes, S. & Eloy, C. 1999 Short-wavelength instability of a vortex in a multipolar strain field. Phys. Fluids 11 (2), 500–502.10.1063/1.869872CrossRefGoogle Scholar
Le Reun, T. 2019 Régimes asymptotiques des écoulements en rotation excités par forçage mécanique dans les noyaux planétaires: saturation turbulente et organisation à grande échelle. Thèse de doctorat, Aix-Marseille University.Google Scholar
Le Reun, T., Favier, B. & Le Bars, M. 2019 Experimental study of the nonlinear saturation of the elliptical instability: inertial wave turbulence versus geostrophic turbulence. J. Fluid Mech. 879, 296–326.CrossRefGoogle Scholar
Le Reun, T., Gallet, B., Favier, B. & Le Bars, M. 2020 Near-resonant instability of geostrophic modes: beyond Greenspan's theorem. J. Fluid Mech. 900, R2.CrossRefGoogle Scholar
Mininni, P. D., Alexakis, A. & Pouquet, A. 2009 Scale interactions and scaling laws in rotating flows at moderate Rossby numbers and large Reynolds numbers. Phys. Fluids 21 (1), 015108.10.1063/1.3064122CrossRefGoogle Scholar
Mininni, P. D. & Pouquet, A. 2010 Rotating helical turbulence. Part 1. Global evolution and spectral behavior. Phys. Fluids 22 (3), 035105.10.1063/1.3358466CrossRefGoogle Scholar
Moisy, F, Morize, C, Rabaud, M & Sommeria, J 2011 Decay laws, anisotropy and cyclone–anticyclone asymmetry in decaying rotating turbulence. J. Fluid Mech. 666, 5–35.10.1017/S0022112010003733CrossRefGoogle Scholar
Morize, C. & Moisy, F. 2006 Energy decay of rotating turbulence with confinement effects. Phys. Fluids 18 (6), 065107.CrossRefGoogle Scholar
Pestana, T. & Hickel, S. 2019 Regime transition in the energy cascade of rotating turbulence. Phys. Rev. E 99 (5), 053103.10.1103/PhysRevE.99.053103CrossRefGoogle ScholarPubMed
Rocha, C. B., Wagner, G. L. & Young, W. R. 2018 Stimulated generation: extraction of energy from balanced flow by near-inertial waves. J. Fluid Mech. 847, 417–451.10.1017/jfm.2018.308CrossRefGoogle Scholar
Seshasayanan, K. & Alexakis, A. 2018 Condensates in rotating turbulent flows. J. Fluid Mech. 841, 434–462.10.1017/jfm.2018.106CrossRefGoogle Scholar
Seshasayanan, K., Gallet, B. & Alexakis, A. 2017 Transition to turbulent dynamo saturation. Phys. Rev. Lett. 119 (20), 204503.10.1103/PhysRevLett.119.204503CrossRefGoogle ScholarPubMed
Sipp, D. & Jacquin, L. 2000 Three-dimensional centrifugal-type instabilities of two-dimensional flows in rotating systems. Phys. Fluids 12 (7), 1740–1748.10.1063/1.870424CrossRefGoogle Scholar
Sipp, D., Lauga, E. & Jacquin, L. 1999 Vortices in rotating systems: centrifugal, elliptic and hyperbolic type instabilities. Phys. Fluids 11 (12), 3716–3728.10.1063/1.870180CrossRefGoogle 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), 1608–1622.10.1063/1.870022CrossRefGoogle Scholar
Staplehurst, P. J., Davidson, P. A. & Dalziel, S. B. 2008 Structure formation in homogeneous freely decaying rotating turbulence. J. Fluid Mech. 598, 81–105.10.1017/S0022112007000067CrossRefGoogle Scholar
Thiele, M. & Müller, W.-C. 2009 Structure and decay of rotating homogeneous turbulence. J. Fluid Mech. 637, 425–442.10.1017/S002211200999067XCrossRefGoogle Scholar
Vanneste, J. 2013 Balance and spontaneous wave generation in geophysical flows. Annu. Rev. Fluid Mech. 45 (1), 147–172.10.1146/annurev-fluid-011212-140730CrossRefGoogle Scholar
Yarom, E. & Sharon, E. 2014 Experimental observation of steady inertial wave turbulence in deep rotating flows. Nat. Phys. 10, 510–514.10.1038/nphys2984CrossRefGoogle Scholar
Yarom, E., Vardi, Y. & Sharon, E. 2013 Experimental quantification of inverse energy cascade in deep rotating turbulence. Phys. Fluids 25 (8), 085105.CrossRefGoogle Scholar
Yeung, P. K. & Zhou, Y. 1998 Numerical study of rotating turbulence with external forcing. Phys. Fluids 10 (11), 2895–2909.10.1063/1.869810CrossRefGoogle 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.CrossRefGoogle Scholar