Hostname: page-component-76fb5796d-2lccl Total loading time: 0 Render date: 2024-04-27T04:43:08.334Z Has data issue: false hasContentIssue false
  • English
  • Français

Quantifying resonant and near-resonant interactions in rotating turbulence

Published online by Cambridge University Press:  18 November 2016

Patricio Clark di Leoni  and
Pablo D. Mininni
Patricio Clark di Leoni*
Affiliation:
Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IFIBA, CONICET, Ciudad Universitaria, 1428 Buenos Aires, Argentina
Pablo D. Mininni
Affiliation:
Departamento de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IFIBA, CONICET, Ciudad Universitaria, 1428 Buenos Aires, Argentina
*
†Email address for correspondence: clark@df.uba.ar
Get access Rights & Permissions [Opens in a new window]

Abstract

Nonlinear triadic interactions are at the heart of our understanding of turbulence. In flows where waves are present, modes must not only be in a triad to interact, but their frequencies must also satisfy an extra condition: the interactions that dominate the energy transfer are expected to be resonant. We derive equations that allow direct measurement of the actual degree of resonance of each triad in a turbulent flow. We then apply the method to the case of rotating turbulence, where eddies coexist with inertial waves. We show that for a range of wavenumbers, resonant and near-resonant triads are dominant, the latter allowing a transfer of net energy towards two-dimensional modes that would be inaccessible otherwise. The results are in good agreement with approximations often done in theories of rotating turbulence, and with the mechanism of parametric instability proposed to explain the development of anisotropy in such flows. We also observe that, at least for the moderate Rossby numbers studied here, marginally near-resonant and non-resonant triads play a non-negligible role in the coupling of modes.

JFM classification

Turbulent Flows: Rotating turbulence Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 809 , 25 December 2016 , pp. 821 - 842
Check for updates
Copyright
© 2016 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

Alexakis, A. 2015 Rotating Taylor–Green flow. J. Fluid Mech. 769, 46–78.CrossRefGoogle Scholar
Aluie, H. & Eyink, G. L. 2009 Localness of energy cascade in hydrodynamic turbulence. II. sharp spectral filter. Phys. Fluids 21, 115108.Google Scholar
Aubourg, Q. & Mordant, N. 2015 Nonlocal resonances in weak turbulence of gravity–capillary waves. Phys. Rev. Lett. 114, 144501.Google Scholar
Aubourg, Q. & Mordant, N. 2016 Investigation of resonances in gravity–capillary wave turbulence. Phys. Rev. Fluids 1, 023701.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
Bewley, G. P., Lathrop, D. P., Maas, L. R. M. & Sreenivasan, K. R. 2007 Inertial waves in rotating grid turbulence. Phys. Fluids 19 (7), 071701.Google Scholar
Biferale, L., Musacchio, S. & Toschi, F. 2013 Split energy-helicity cascades in three-dimensional homogeneous and isotropic turbulence. J. Fluid Mech. 730, 309–327.Google Scholar
van Bokhoven, L. J. A., Cambon, C., Liechtenstein, L., Godeferd, F. S. & Clercx, H. J. H. 2008 Refined vorticity statistics of decaying rotating three-dimensional turbulence. J. Turbul. 9, N6.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.CrossRefGoogle Scholar
Bourouiba, Lydia 2008 Discreteness and resolution effects in rapidly rotating turbulence. Phys. Rev. E 78 (5), 056309.Google Scholar
Cambon, C. & Jacquin, L. 1989 Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295–317.Google Scholar
Cambon, C., Mansour, N. N. & Godeferd, F. S. 1997 Energy transfer in rotating turbulence. J. Fluid Mech. 337, 303–332.Google Scholar
Cambon, C., Rubinstein, R. & Godeferd, F. S. 2004 Advances in wave turbulence: rapidly rotating flows. New J. Phys. 6, 73.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.Google 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.Google Scholar
Chen, S. & Kraichnan, R. H. 1989 Sweeping decorrelation in isotropic turbulence. Phys. Fluids A 1 (12), 2019.Google Scholar
Cheung, L. C. & Zaki, T. A. 2014 An exact representation of the nonlinear triad interaction terms in spectral space. J. Fluid Mech. 748, 175–188.Google Scholar
Domaradzki, J. A. & Rogallo, R. S. 1990 Local energy transfer and nonlocal interactions in homogeneous, isotropic turbulence. Phys. Fluids A 2 (3), 413–426.Google Scholar
Eyink, G. L. & Aluie, H. 2009 Localness of energy cascade in hydrodynamic turbulence. I. smooth coarse graining. Phys. Fluids 21, 115107.Google Scholar
Favier, B., Godeferd, F. S. & Cambon, C. 2010 On space and time correlations of isotropic and rotating turbulence. Phys. Fluids 22 (1), 015101.CrossRefGoogle Scholar
Gallet, B. 2015 Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows. J. Fluid Mech. 783, 412–447.Google Scholar
Galtier, S. 2003 Weak inertial-wave turbulence theory. Phys. Rev.  E 68, 015301.Google Scholar
Gómez, D. O., Mininni, P. D. & Dmitruk, P. 2005 MHD simulations and astrophysical applications. Adv. Space Res. 35, 899–907.CrossRefGoogle Scholar
Haudin, F., Cazaubiel, A., Deike, L., Jamin, T., Falcon, E. & Berhanu, M. 2016 Experimental study of three-wave interactions among capillary-gravity surface waves. Phys. Rev. E 93 (4), 043110.Google Scholar
Hernandez-Duenas, G., Smith, L. M. & Stechmann, S. N. 2014 Investigation of Boussinesq dynamics using intermediate models based on wave vortical interactions. J. Fluid Mech. 747, 247–287.CrossRefGoogle Scholar
Horne, E. & Mininni, P. D. 2013 Sign cancellation and scaling in the vertical component of velocity and vorticity in rotating turbulence. Phys. Rev. E 88 (1), 013011.Google Scholar
Kraichnan, R. H. 1958 Irreversible statistical mechanics of incompressible hydromagnetic turbulence. Phys. Rev. 109 (5), 1407–1422.CrossRefGoogle Scholar
Lamriben, C., Cortet, P.-P., Moisy, F. & Maas, L. R. M. 2011 Excitation of inertial modes in a closed grid turbulence experiment under rotation. Phys. Fluids 23 (1), 015102.Google Scholar
Lee, J. 1975 The triad interaction representation of homogeneous turbulence. J. Math. Phys. 16 (7), 1359–1366.Google Scholar
Clark di Leoni, P., Cobelli, P. J. & Mininni, P. D. 2015 The spatio-temporal spectrum of turbulent flows. Euro. Phys. J. E 38 (12), 1–10.Google 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
Mininni, P. D. 2011 Scale interactions in magnetohydrodynamic turbulence. Annu. Rev. Fluid Mech. 43 (1), 377–397.Google Scholar
Mininni, P. D., Alexakis, A. & Pouquet, A. 2006 Large-scale flow effects, energy transfer, and self-similarity on turbulence. Phys. Rev. E 74 (1), 061303.Google Scholar
Mininni, P. D., Alexakis, A. & Pouquet, A. 2008 Nonlocal interactions in hydrodynamic turbulence at high Reynolds numbers: the slow emergence of scaling laws. Phys. Rev. E 77 (3), 036306.Google Scholar
Mininni, P. D., Rosenberg, D. & Pouquet, A. 2012 Isotropization at small scales of rotating helically driven turbulence. J. Fluid Mech. 699, 263–279.CrossRefGoogle Scholar
Mininni, P. D., Rosenberg, D., Reddy, R. & Pouquet, A. 2011 A hybrid MPI–OpenMP scheme for scalable parallel pseudospectral computations for fluid turbulence. Parallel Comput. 37, 316–326.Google Scholar
Moffatt, H. K. 2014 Note on the triad interactions of homogeneous turbulence. J. Fluid Mech. 741, R3.Google Scholar
Müller, W.-C. & Thiele, M. 2007 Scaling and energy transfer in rotating turbulence. Europhys. Lett. 77, 34003.Google Scholar
Nazarenko, S. 2011 Wave Turbulence, 1st Edn. Springer.CrossRefGoogle Scholar
Nazarenko, S. V. & Schekochihin, A. A. 2011 Critical balance in magnetohydrodynamic, rotating and stratified turbulence: towards a universal scaling conjecture. J. Fluid Mech. 677, 134–153.Google Scholar
Newell, A. C. 1969 Rossby wave packet interactions. J. Fluid Mech. 35 (02), 255–271.Google Scholar
Newell, A. C. & Rumpf, B. 2011 Wave turbulence. Annu. Rev. Fluid Mech. 43 (1), 59–78.Google Scholar
Pouquet, A. & Mininni, P. D. 2010 The interplay between helicity and rotation in turbulence: implications for scaling laws and small-scale dynamics. Phil. Trans. R. Soc. Lond. A 368, 1635–1662.Google Scholar
Remmel, M., Sukhatme, J. & Smith, L. M. 2010 Nonlinear inertia-gravity wave-mode interactions in three dimensional rotating stratified flows. Commun. Math. Sci. 8 (2), 357–376.Google Scholar
Rieutord, M., Triana, S. A., Zimmerman, D. S. & Lathrop, D. P. 2012 Excitation of inertial modes in an experimental spherical Couette flow. Phys. Rev. E 86 (2), 026304.Google Scholar
Sen, A., Mininni, P. D., Rosenberg, D. & Pouquet, A. 2012 Anisotropy and nonuniversality in scaling laws of the large-scale energy spectrum in rotating turbulence. Phys. Rev. E 86, 036319.Google Scholar
Servidio, S., Carbone, V., Dmitruk, P. & Matthaeus, W. H. 2011 Time decorrelation in isotropic magnetohydrodynamic turbulence. Europhys. Lett. 96 (5), 55003.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
Staplehurst, P. J., Davidson, P. A. & Dalziel, S. B. 2008 Structure formation in homogeneous freely decaying rotating turbulence. J. Fluid Mech. 598, 81–105.Google Scholar
Waleffe, F. 1992 The nature of triad interactions in homogeneous turbulence. Phys. Fluids A 4 (2), 350–363.Google Scholar
Waleffe, F. 1993 Inertial transfers in the helical decomposition. Phys. Fluids A 5 (3), 677.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
Zakharov, V. E., Lvov, V. S. & Falkovic, G. 1992 Kolmogorov Spectra of Turbulence I Wave Turbulence. Springer.Google Scholar
Zhou, Y. 1995 A phenomenological treatment of rotating turbulence. Phys. Fluids 7, 2092–2094.Google Scholar