Skip to main content Accessibility help
×
×
Home

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

  • SERGEI V. NAZARENKO (a1) (a2) and ALEXANDER A. SCHEKOCHIHIN (a3) (a2)
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).

Copyright
Corresponding author
Email address for correspondence: a.schekochihin1@physics.ox.ac.uk
References
Hide All
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.
Bellet, F., Godeferd, F. S., Scott, J. F. & Cambon, C. 2006 Wave turbulence in rapidly rotating flows. J. Fluid Mech. 562, 83121.
Beresnyak, A. 2011 The spectral slope and Kolmogorov constant of MHD turbulence. Phys. Rev. Lett. (in press) arXiv:1011.2505.
Billant, P. & Chomaz, J.-M. 2001 Self-similarity of strongly stratified inviscid flows. Phys. Fluids 13, 16451651.
Biskamp, D. & Schwarz, E. 2001 On two-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 8, 32823292.
Biskamp, D., Schwarz, E., Zeiler, A., Celani, A. & Drake, J. F. 1999 Electron magnetohydrodynamic turbulence. Phys. Plasmas 6, 751758.
Biskamp, D. & Welter, H. 1989 Dynamics of decaying two-dimensional magnetohydrodynamic turbulence. Phys. Fluids B 1, 19641979.
Boldyrev, S. 2006 Spectrum of magnetohydrodynamic turbulence. Phys. Rev. Lett. 96, 115002.
Boldyrev, S., Mason, J. & Cattaneo, F. 2009 Dynamic alignment and exact scaling laws in magnetohydrodynamic turbulence. Astrophys. J. 699, L39L42.
Boldyrev, S. & Perez, J. C. 2009 Spectrum of weak magnetohydrodynamic turbulence. Phys. Rev. Lett. 103, 225001.
Bourouiba, L. 2008 Discreteness and resolution effects in rapidly rotating turbulence. Phys. Rev. E 78, 056309.
Cambon, C. 2001 Turbulence and vortex structures in rotating and stratified flows. Eur. J. Mech. B 20, 489510.
Canuto, V. M. & Dubovikov, M. S. 1997 A dynamical model for turbulence. V. The effect of rotation. Phys. Fluids 9, 21322140.
Cardy, J. L., Falkovich, G. & Gawedzki, K. 2008 Non-Equilibrium Statistical Mechanics and Turbulence. Cambridge University Press.
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.
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.
Cho, J. & Lazarian, A. 2004 The anisotropy of electron magnetohydrodynamic turbulence. Astrophys. J. 615, L41L44.
Cho, J. & Lazarian, A. 2009 Simulations of electron magnetohydrodynamic turbulence. Astrophys. J. 701, 236252.
Cho, J. & Vishniac, E. T. 2000 The anisotropy of magnetohydrodynamic Alfvénic turbulence. Astrophys. J. 539, 273282.
Davidson, P. A., Staplehurst, P. J. & Dalziel, S. B. 2006 On the evolution of eddies in a rapidly rotating system. J. Fluid Mech. 557, 135144.
Denissenko, P., Lukaschuk, S. & Nazarenko, S. 2007 Gravity wave turbulence in a laboratory flume. Phys. Rev. Lett. 99, 014501.
Dewan, E. 1997 Saturated-cascade similitude theory of gravity wave spectra. J. Geophys. Res. 102, 2979929818.
Dubrulle, B. & Valdettaro, L. 1992 Consequences of rotation in energetics of accretion disks. Astron. Astrophys. 263, 387400.
Elsasser, W. M. 1950 The hydromagnetic equations. Phys. Rev. 79, 183183.
Frieman, E. A. & Chen, L. 1982 Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria. Phys. Fluids 25, 502508.
Galtier, S. 2003 Weak inertial-wave turbulence theory. Phys. Rev. E 68, 015301.
Galtier, S., Nazarenko, S. V., Newell, A. C. & Pouquet, A. 2000 A weak turbulence theory for incompressible magnetohydrodynamics. J. Plasma Phys. 63, 447488.
Galtier, S., Pouquet, A. & Mangeney, A. 2005 On spectral scaling laws for incompressible anisotropic magnetohydrodynamic turbulence. Phys. Plasmas 12, 092310.
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, 115159.
Goldreich, P. & Sridhar, S. 1995 Toward a theory of interstellar turbulence. 2: Strong Alfvénic turbulence. Astrophys. J. 438, 763775.
Gómez, D. O., Mahajan, S. M. & Dmitruk, P. 2008 Hall magnetohydrodynamics in a strong magnetic field. Phys. Plasmas 15, 102303.
Higdon, J. C. 1984 Density fluctuations in the interstellar medium: evidence for anisotropic magnetogasdynamic turbulence. I – Model and astrophysical sites. Astrophys. J. 285, 109123.
Horbury, T. S., Forman, M. & Oughton, S. 2008 Anisotropic scaling of magnetohydrodynamic turbulence. Phys. Rev. Lett. 101, 175005.
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, 590614.
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.
Iroshnikov, R. S. 1963 Turbulence of a conducting fluid in a strong magnetic field. Astron. Zh. 40, 742.
Jacquin, L., Leuchter, O., Cambon, C. & Mathieu, J. 1990 Homogeneous turbulence in the presence of rotation. J. Fluid Mech. 220, 152.
Julien, K. & Knobloch, E. 2007 Reduced models for fluid flows with strong constraints. J. Math. Phys. 48, 065405.
Julien, K., Knobloch, E. & Werne, J. 1998 A new class of equations for rotationally constrained flows. Theor. Comput. Fluid Dyn. 11, 251261.
Kadomtsev, B. B. 1965 Plasma Turbulence. Academic Press.
Kaneda, Y. & Yoshida, K. 2004 Small-scale anisotropy in stably stratified turbulence. New J. Phys. 6, 34.
Kraichnan, R. H. 1965 Inertial-range spectrum of hydromagnetic turbulence. Phys. Fluids 8, 13851387.
Kuznetsov, E. A. 1972 Turbulence of ion sound in a plasma located in a magnetic field. Sov. Phys. JETP 35, 310.
Kuznetsov, E. A. 2004 Turbulence spectra generated by singularities. JETP Lett. 80, 8389.
Laval, J.-P., McWilliams, J. C. & Dubrulle, B. 2003 Forced stratified turbulence: successive transitions with Reynolds number. Phys. Rev. E 68, 036308.
Lindborg, E. 2006 The energy cascade in a strongly stratified fluid. J. Fluid Mech. 550, 207242.
Lindborg, E. & Brethouwer, G. 2007 Stratified turbulence forced in rotational and divergent modes. J. Fluid Mech. 586, 83108.
Maron, J. & Goldreich, P. 2001 Simulations of incompressible magnetohydrodynamic turbulence. Astrophys. J. 554, 11751196.
Mason, J., Cattaneo, F. & Boldyrev, S. 2008 Numerical measurements of the spectrum in magnetohydrodynamic turbulence. Phys. Rev. E 77, 036403.
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.
Montgomery, D. & Turner, L. 1981 Anisotropic magnetohydrodynamic turbulence in a strong external magnetic field. Phys. Fluids 24, 825831.
Morize, C., Moisy, F. & Rabaud, M. 2005 Decaying grid-generated turbulence in a rotating tank. Phys. Fluids 17, 095105.
Nazarenko, S. 2007 2D enslaving of MHD turbulence. New J. Phys. 9, 307.
Newell, A. C. & Zakharov, V. E. 2008 The role of the generalized Phillips' spectrum in wave turbulence. Phys. Lett. A 372, 42304233.
Ozmidov, R. V. 1992 Length scales and dimensionless numbers in a stratified ocean. Oceanology 32, 259262.
Perez, J. C. & Boldyrev, S. A. 2008 On weak and strong magnetohydrodynamic turbulence. Astrophys. J. 672, L61L64.
Phillips, O. M. 1958 The equilibrium range in the spectrum of wind-generated waves. J. Fluid Mech. 4, 426434.
Phillips, O. M. 1985 Spectral and statistical properties of the equilibrium range in wind-generated gravity waves. J. Fluid Mech. 156, 505531.
Podesta, J. J. 2009 Dependence of solar-wind power spectra on the direction of the local mean magnetic field. Astrophys. J. 698, 986999.
Proment, D., Nazarenko, S. & Onorato, M. 2009 Quantum turbulence cascades in the Gross-Pitaevskii model. Phys. Rev. A 80, 051603.
Rhines, P. B. 1975 Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417443.
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.
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, 310377.
Schekochihin, A. A. & Nazarenko, S. V. 2011 Weak Alfvén-wave turbulence revisited. (submitted).
Smith, L. M. & Waleffe, F. 1999 Transfer of energy to two-dimensional large scales in forced, rotating three-dimensional turbulence. Phys. Fluids 11, 16081622.
Strauss, H. R. 1976 Nonlinear, three-dimensional magnetohydrodynamics of noncircular tokamaks. Phys. Fluids 19, 134140.
Thiele, M. & Müller, W. 2009 Structure and decay of rotating homogeneous turbulence. J. Fluid Mech. 637, 425442.
Vinen, W. F. 2000 Classical character of turbulence in a quantum liquid. Phys. Rev. B 61, 14101420.
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, L31L35.
Zakharov, V. E., L'vov, V. S. & Falkovich, G. 1992 Kolmogorov Spectra of Turbulence I: Wave turbulence. Springer.
Zeman, O. 1994 A note on the spectra and decay of rotating homogeneous turbulence. Phys. Fluids 6, 32213223.
Zhou, Y. 1995 A phenomenological treatment of rotating turbulence. Phys. Fluids 7, 20922094.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed