Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-09T21:55:11.923Z Has data issue: false hasContentIssue false
  • English
  • Français

Critical balance in magnetohydrodynamic, rotating and stratified turbulence: towards a universal scaling conjecture

Published online by Cambridge University Press:  29 March 2011

SERGEI V. NAZARENKO  and
ALEXANDER A. SCHEKOCHIHIN
SERGEI V. NAZARENKO
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK Institut Henri Poincaré, Université Pierre et Marie Curie, 75231 Paris CEDEX 5, France
ALEXANDER A. SCHEKOCHIHIN*
Affiliation:
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK Institut Henri Poincaré, Université Pierre et Marie Curie, 75231 Paris CEDEX 5, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

It is proposed that critical balance – a scale-by-scale balance between the linear propagation and nonlinear interaction time scales – can be used as a universal scaling conjecture for determining the spectra of strong turbulence in anisotropic wave systems. Magnetohydrodynamic (MHD), rotating and stratified turbulence are considered under this assumption and, in particular, a novel and experimentally testable energy cascade scenario and a set of scalings of the spectra are proposed for low-Rossby-number rotating turbulence. It is argued that in neutral fluids the critically balanced anisotropic cascade provides a natural path from strong anisotropy at large scales to isotropic Kolmogorov turbulence at very small scales. It is also argued that the k−2⊥ spectra seen in recent numerical simulations of low-Rossby-number rotating turbulence may be analogous to the k−3/2⊥ spectra of the numerical MHD turbulence in the sense that they could be explained by assuming that fluctuations are polarised (aligned) approximately as inertial waves (Alfvén waves for MHD).

JFM classification

MHD and Electrohydrodynamics: MHD turbulence Turbulent Flows: Rotating turbulence Turbulent Flows: Stratified turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 677 , 25 June 2011 , pp. 134 - 153
Copyright
Copyright © Cambridge University Press 2011

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

REFERENCES

Bale, S. D., Kellogg, P. J., Mozer, F. S., Horbury, T. S. & Reme, H. 2005 Measurement of the electric fluctuation spectrum of magnetohydrodynamic turbulence. Phys. Rev. Lett. 94, 215002.CrossRefGoogle ScholarPubMed
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
Beresnyak, A. 2011 The spectral slope and Kolmogorov constant of MHD turbulence. Phys. Rev. Lett. (in press) arXiv:1011.2505.CrossRefGoogle Scholar
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 1645–1651.CrossRefGoogle Scholar
Biskamp, D. & Schwarz, E. 2001 On two-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 8, 3282–3292.CrossRefGoogle Scholar
Biskamp, D., Schwarz, E., Zeiler, A., Celani, A. & Drake, J. F. 1999 Electron magnetohydrodynamic turbulence. Phys. Plasmas 6, 751–758.CrossRefGoogle Scholar
Biskamp, D. & Welter, H. 1989 Dynamics of decaying two-dimensional magnetohydrodynamic turbulence. Phys. Fluids B 1, 1964–1979.CrossRefGoogle Scholar
Boldyrev, S. 2006 Spectrum of magnetohydrodynamic turbulence. Phys. Rev. Lett. 96, 115002.CrossRefGoogle ScholarPubMed
Boldyrev, S., Mason, J. & Cattaneo, F. 2009 Dynamic alignment and exact scaling laws in magnetohydrodynamic turbulence. Astrophys. J. 699, L39–L42.CrossRefGoogle Scholar
Boldyrev, S. & Perez, J. C. 2009 Spectrum of weak magnetohydrodynamic turbulence. Phys. Rev. Lett. 103, 225001.CrossRefGoogle ScholarPubMed
Bourouiba, L. 2008 Discreteness and resolution effects in rapidly rotating turbulence. Phys. Rev. E 78, 056309.CrossRefGoogle ScholarPubMed
Cambon, C. 2001 Turbulence and vortex structures in rotating and stratified flows. Eur. J. Mech. B 20, 489–510.CrossRefGoogle Scholar
Canuto, V. M. & Dubovikov, M. S. 1997 A dynamical model for turbulence. V. The effect of rotation. Phys. Fluids 9, 2132–2140.CrossRefGoogle Scholar
Cardy, J. L., Falkovich, G. & Gawedzki, K. 2008 Non-Equilibrium Statistical Mechanics and Turbulence. Cambridge University Press.CrossRefGoogle Scholar
Chen, C. H. K., Horbury, T. S., Schekochihin, A. A., Wicks, R. T., Alexandrova, O. & Mitchell, J. 2010 Anisotropy of solar wind turbulence between ion and electron scales. Phys. Rev. Lett. 104, 255002.CrossRefGoogle ScholarPubMed
Chen, C. H. K., Mallet, A., Yousef, T. A., Schekochihin, A. A. & Horbury, T. S. 2011 Anisotropy of Alfvénic turbulence in the solar wind and numerical simulations. Mon. Not. R. Astron. Soc. (in press) arXiv:1009.0662.CrossRefGoogle Scholar
Cho, J. & Lazarian, A. 2004 The anisotropy of electron magnetohydrodynamic turbulence. Astrophys. J. 615, L41–L44.CrossRefGoogle Scholar
Cho, J. & Lazarian, A. 2009 Simulations of electron magnetohydrodynamic turbulence. Astrophys. J. 701, 236–252.CrossRefGoogle Scholar
Cho, J. & Vishniac, E. T. 2000 The anisotropy of magnetohydrodynamic Alfvénic turbulence. Astrophys. J. 539, 273–282.CrossRefGoogle Scholar
Davidson, P. A., Staplehurst, P. J. & Dalziel, S. B. 2006 On the evolution of eddies in a rapidly rotating system. J. Fluid Mech. 557, 135–144.CrossRefGoogle Scholar
Denissenko, P., Lukaschuk, S. & Nazarenko, S. 2007 Gravity wave turbulence in a laboratory flume. Phys. Rev. Lett. 99, 014501.CrossRefGoogle Scholar
Dewan, E. 1997 Saturated-cascade similitude theory of gravity wave spectra. J. Geophys. Res. 102, 29799–29818.CrossRefGoogle Scholar
Dubrulle, B. & Valdettaro, L. 1992 Consequences of rotation in energetics of accretion disks. Astron. Astrophys. 263, 387–400.Google Scholar
Elsasser, W. M. 1950 The hydromagnetic equations. Phys. Rev. 79, 183–183.CrossRefGoogle Scholar
Frieman, E. A. & Chen, L. 1982 Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria. Phys. Fluids 25, 502–508.CrossRefGoogle Scholar
Galtier, S. 2003 Weak inertial-wave turbulence theory. Phys. Rev. E 68, 015301.CrossRefGoogle ScholarPubMed
Galtier, S., Nazarenko, S. V., Newell, A. C. & Pouquet, A. 2000 A weak turbulence theory for incompressible magnetohydrodynamics. J. Plasma Phys. 63, 447–488.CrossRefGoogle Scholar
Galtier, S., Pouquet, A. & Mangeney, A. 2005 On spectral scaling laws for incompressible anisotropic magnetohydrodynamic turbulence. Phys. Plasmas 12, 092310.CrossRefGoogle Scholar
Godeferd, F. S. & Staquet, C. 2003 Statistical modelling and direct numerical simulations of decaying stably stratified turbulence. Part 2. Large-scale and small-scale anisotropy. J. Fluid Mech. 486, 115–159.CrossRefGoogle Scholar
Goldreich, P. & Sridhar, S. 1995 Toward a theory of interstellar turbulence. 2: Strong Alfvénic turbulence. Astrophys. J. 438, 763–775.CrossRefGoogle Scholar
Gómez, D. O., Mahajan, S. M. & Dmitruk, P. 2008 Hall magnetohydrodynamics in a strong magnetic field. Phys. Plasmas 15, 102303.CrossRefGoogle Scholar
Higdon, J. C. 1984 Density fluctuations in the interstellar medium: evidence for anisotropic magnetogasdynamic turbulence. I – Model and astrophysical sites. Astrophys. J. 285, 109–123.CrossRefGoogle Scholar
Horbury, T. S., Forman, M. & Oughton, S. 2008 Anisotropic scaling of magnetohydrodynamic turbulence. Phys. Rev. Lett. 101, 175005.CrossRefGoogle ScholarPubMed
Howes, G. G., Cowley, S. C., Dorland, W., Hammett, G. W., Quataert, E. & Schekochihin, A. A. 2006 Astrophysical gyrokinetics: basic equations and linear theory. Astrophys. J. 651, 590–614.CrossRefGoogle Scholar
Howes, G. G., Dorland, W., Cowley, S. C., Hammett, G. W., Quataert, E., Schekochihin, A. A. & Tatsuno, T. 2008 Kinetic simulations of magnetized turbulence in astrophysical plasmas. Phys. Rev. Lett. 100, 065004.CrossRefGoogle ScholarPubMed
Iroshnikov, R. S. 1963 Turbulence of a conducting fluid in a strong magnetic field. Astron. Zh. 40, 742.Google Scholar
Jacquin, L., Leuchter, O., Cambon, C. & Mathieu, J. 1990 Homogeneous turbulence in the presence of rotation. J. Fluid Mech. 220, 1–52.CrossRefGoogle Scholar
Julien, K. & Knobloch, E. 2007 Reduced models for fluid flows with strong constraints. J. Math. Phys. 48, 065405.CrossRefGoogle Scholar
Julien, K., Knobloch, E. & Werne, J. 1998 A new class of equations for rotationally constrained flows. Theor. Comput. Fluid Dyn. 11, 251–261.CrossRefGoogle Scholar
Kadomtsev, B. B. 1965 Plasma Turbulence. Academic Press.Google Scholar
Kaneda, Y. & Yoshida, K. 2004 Small-scale anisotropy in stably stratified turbulence. New J. Phys. 6, 34.CrossRefGoogle Scholar
Kraichnan, R. H. 1965 Inertial-range spectrum of hydromagnetic turbulence. Phys. Fluids 8, 1385–1387.CrossRefGoogle Scholar
Kuznetsov, E. A. 1972 Turbulence of ion sound in a plasma located in a magnetic field. Sov. Phys. JETP 35, 310.Google Scholar
Kuznetsov, E. A. 2004 Turbulence spectra generated by singularities. JETP Lett. 80, 83–89.CrossRefGoogle Scholar
Laval, J.-P., McWilliams, J. C. & Dubrulle, B. 2003 Forced stratified turbulence: successive transitions with Reynolds number. Phys. Rev. E 68, 036308.CrossRefGoogle ScholarPubMed
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207–242.CrossRefGoogle Scholar
Lindborg, E. & Brethouwer, G. 2007 Stratified turbulence forced in rotational and divergent modes. J. Fluid Mech. 586, 83–108.CrossRefGoogle Scholar
Maron, J. & Goldreich, P. 2001 Simulations of incompressible magnetohydrodynamic turbulence. Astrophys. J. 554, 1175–1196.CrossRefGoogle Scholar
Mason, J., Cattaneo, F. & Boldyrev, S. 2008 Numerical measurements of the spectrum in magnetohydrodynamic turbulence. Phys. Rev. E 77, 036403.CrossRefGoogle ScholarPubMed
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, 015108.CrossRefGoogle Scholar
Montgomery, D. & Turner, L. 1981 Anisotropic magnetohydrodynamic turbulence in a strong external magnetic field. Phys. Fluids 24, 825–831.CrossRefGoogle Scholar
Morize, C., Moisy, F. & Rabaud, M. 2005 Decaying grid-generated turbulence in a rotating tank. Phys. Fluids 17, 095105.CrossRefGoogle Scholar
Nazarenko, S. 2007 2D enslaving of MHD turbulence. New J. Phys. 9, 307.CrossRefGoogle Scholar
Newell, A. C. & Zakharov, V. E. 2008 The role of the generalized Phillips' spectrum in wave turbulence. Phys. Lett. A 372, 4230–4233.CrossRefGoogle Scholar
Ozmidov, R. V. 1992 Length scales and dimensionless numbers in a stratified ocean. Oceanology 32, 259–262.Google Scholar
Perez, J. C. & Boldyrev, S. A. 2008 On weak and strong magnetohydrodynamic turbulence. Astrophys. J. 672, L61–L64.CrossRefGoogle Scholar
Phillips, O. M. 1958 The equilibrium range in the spectrum of wind-generated waves. J. Fluid Mech. 4, 426–434.CrossRefGoogle Scholar
Phillips, O. M. 1985 Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. J. Fluid Mech. 156, 505–531.CrossRefGoogle Scholar
Podesta, J. J. 2009 Dependence of solar-wind power spectra on the direction of the local mean magnetic field. Astrophys. J. 698, 986–999.CrossRefGoogle Scholar
Proment, D., Nazarenko, S. & Onorato, M. 2009 Quantum turbulence cascades in the Gross-Pitaevskii model. Phys. Rev. A 80, 051603.CrossRefGoogle Scholar
Rhines, P. B. 1975 Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417–443.CrossRefGoogle Scholar
Sahraoui, F., Goldstein, M. L., Robert, P. & Khotyaintsev, Y. V. 2009 Evidence of a cascade and dissipation of solar-wind turbulence at the electron gyroscale. Phys. Rev. Lett. 102, 231102.CrossRefGoogle ScholarPubMed
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Quataert, E. & Tatsuno, T. 2009 Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas. Astrophys. J. Suppl. 182, 310–377.CrossRefGoogle Scholar
Schekochihin, A. A. & Nazarenko, S. V. 2011 Weak Alfvén-wave turbulence revisited. (submitted).CrossRefGoogle Scholar
Smith, L. M. & Waleffe, F. 1999 Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence. Phys. Fluids 11, 1608–1622.CrossRefGoogle Scholar
Strauss, H. R. 1976 Nonlinear, three-dimensional magnetohydrodynamics of noncircular tokamaks. Phys. Fluids 19, 134–140.CrossRefGoogle Scholar
Thiele, M. & Müller, W. 2009 Structure and decay of rotating homogeneous turbulence. J. Fluid Mech. 637, 425–442.CrossRefGoogle Scholar
Vinen, W. F. 2000 Classical character of turbulence in a quantum liquid. Phys. Rev. B 61, 1410–1420.CrossRefGoogle Scholar
Wicks, R. T., Horbury, T. S., Chen, C. H. K. & Schekochihin, A. A. 2010 Power and spectral index anisotropy of the entire inertial range of turbulence in the fast solar wind. Mon. Not. R. Astron. Soc. 407, L31–L35.CrossRefGoogle Scholar
Zakharov, V. E., L'vov, V. S. & Falkovich, G. 1992 Kolmogorov Spectra of Turbulence I: Wave turbulence. Springer.CrossRefGoogle Scholar
Zeman, O. 1994 A note on the spectra and decay of rotating homogeneous turbulence. Phys. Fluids 6, 3221–3223.CrossRefGoogle Scholar
Zhou, Y. 1995 A phenomenological treatment of rotating turbulence. Phys. Fluids 7, 2092–2094.CrossRefGoogle Scholar