Galtier, S., Nazarenko, Sergey, Newell, Alan C., 1941- and Pouquet, A.. (2000) A weak turbulence theory for incompressible magnetohydrodynamics. Journal of Plasma Physics, Vol.63 (No.5). pp. 447-488. ISSN 0022-3778

Preview

PDF WRAP_Galtier_weak_turbulence.pdf - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (1067Kb)

Abstract

We derive a weak turbulence formalism for incompressible magnetohydrodynamics. Three-wave interactions lead to a system of kinetic equations for the spectral densities of energy and helicity. The kinetic equations conserve energy in all wavevector planes normal to the applied magnetic field B0ê[parallel R: parallel]. Numerically and analytically, we find energy spectra E± [similar] kn±[bot bottom], such that n+ + n− = −4, where E± are the spectra of the Elsässer variables z± = v ± b in the two-dimensional case (k[parallel R: parallel] = 0). The constants of the spectra are computed exactly and found to depend on the amount of correlation between the velocity and the magnetic field. Comparison with several numerical simulations and models is also made.

Item Type:	Journal Article
Subjects:	Q Science > QC Physics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Turbulence, Magnetohydrodynamics, Wave mechanics, Magnetic fields
Journal or Publication Title:	Journal of Plasma Physics
Publisher:	Cambridge University Press
ISSN:	0022-3778
Date:	June 2000
Volume:	Vol.63
Number:	No.5
Page Range:	pp. 447-488
Identification Number:	10.1017/S0022377899008284
Status:	Peer Reviewed
Access rights to Published version:	Open Access
References:	Achterberg, A. 1979 The energy spectrum of electrons accelerated by weak magnetohydrodynamic turbulence. Astron. Astrophys. 76, 276{286. Akhiezer, A., Akhiezer, I. A., Polovin, R.V., Sitenko, A.G. and Stepanov, K.N. 1975 Plasma Electrodynamics II: Non-linear Theory and Fluctuations. Pergamon Press, Oxford. Balk, A. M., Nazarenko, S.V. and Zakharov, V.E. 1990a On the nonlocal turbulence of drift type waves. Phys. Lett. 146A, 217{221. Balk, A. M., Zakharov, V.E. and Nazarenko, S.V. 1990b Nonlocal turbulence of drift waves. Soviet Phys. JETP 71, 249{260. Bavassano, B., Dobrowolny, M., Fanfoni, G., Mariani, F. and Ness, N.F. 1982 Statistical properties of MHD fluctuations associated with high-speed streams from Helios-2 observations. Solar Phys. 78, 373{384. Benney, J. and Newell, A. C. 1969 Random wave closures. Stud. Applied Maths 48, 29{53. Bhattacharjee, A., Ng, C. S. and Spangler, S.R. 1998 Weakly compressible magnetohydrodynamic turbulence in the solar wind and the interstellar medium. Astrophys. J. 494, 409{418. Bieber, J.W., Wanner, W. and Matthaeus, W.H. 1996 Dominant two-dimensional solar wind turbulence with implications for cosmic ray transport. J. Geophys. Res. 101, 2511{2522. Cambon, C. and Jacquin, L. 1989 Spectral approach to non-isotropic turbulence subjected to rotation. J. Fluid Mech. 202, 295{317. Cambon, C., Mansour, N.N. and Godeferd, F. S. 1997 Energy transfer in rotating turbulence. J. Fluid Mech. 337, 303{332. Craya, A. 1958 Contribution à l'analyse de la turbulence associée à des vitesses moyennes. P.S.T. Ministére de l'Air 345. Falkovich, G. E. and Shafarenko, A.V. 1991 Nonstationary wave turbulence. J. Nonlinear Sci. 1, 457{480. Falgarone, E., Puget, J.-L. and Pérault, M. 1992 The small-scale density and velocity structure of quiescent molecular clouds. Astron. Astrophys. 257, 715{730. Goldreich, P. and Sridhar, S. 1995 Toward a theory of interstellar turbulence. II. Strong Alfvénic turbulence. Astrophys. J. 438, 763{775. Goldreich, P. and Sridhar, S. 1997 Magnetohydrodynamic turbulence revisited. Astrophys. J. 485, 680{688. Heiles, C., Goodman, A. A., McKee, C.F. and Zweibel, E.G. 1993 Magnetic fields in starforming regions: observations. In: Protostars and Planets III (ed. E. H. Levy, J. I. Lunine, M. Guerr and M. S. Matthews), pp. 279{326. University of Arizona Press, Tucson. Herring, J. 1974 Approach of axisymmetric turbulence to isotropy. Phys. Fluids 17, 859{872. Horbury, T., Balogh, A., Forsyth, R. J. and Smith, E. J. 1995 Anisotropy of inertial range turbulence in the polar heliosphere. Geophys. Rev. Lett. 22, 3405{3408. Iroshnikov, P. 1963 Turbulence of a conducting uid in a strong magnetic field. Soviet Astron. 7, 566{571. Kaburaki, O. and Uchida, Y. 1971 Magnetohydrodynamic wave-mode coupling. Quantum field-theoretical approach to weakly non-linear case with application to solar corona heating. Publ. Astr. Soc. Japan 23, 405{423. Kinney, R. M. and McWilliams, J. C. 1998 Turbulent cascades in anisotropic magnetohydrodynamic. Phys. Rev. E57, 7111{7121. Kraichnan, R. H. 1965 Inertial-range spectrum of hydromagnetic turbulence. Phys. Fluids 8, 1385{1387. Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417{1423. Kuznetsov, E. A. 1972 Turbulence of ion sound in a plasma located in a magnetic field. J. Soviet Phys. JETP 35, 310{314. Kuznetsov, E. A. 1973Weak turbulence of MHD waves. Institute of Nuclear Physics Preprint 81{73. Nobosibirsk. Landau, L.D. and Lifshitz, E. M. 1968 Statistical Physics II. Pergamon Press, Oxford. Larson, R.B. 1981 Turbulence and star formation in molecular clouds. Mon. Not. R. Astron. Soc. 194, 809{826. Leith, C. E. and Kraichnan, R. H. 1972 Predictability of turbulent flows. J. Atmos. Sci. 29, 1041{1058. Lesieur, M. 1990 Turbulence in Fluids, 2nd edn. Kluwer, Dordrecht. McIvor, I. 1977 The inertial range of weak magnetohydrodynamic turbulence in the interstellar medium. Mon. Not. R. Astron. Soc. 178, 85{99. Marsch, E. and Tu, C.Y. 1994, Non{Gaussian probability distributions of solar wind fluctuations. Ann. Geophys. 12, 1127{1138. Matthaeus, W.H. and Montgomery, D. 1980 Selective decay hypothesis at high mechanical and magnetic Reynolds numbers. Ann. NY Acad. Sci. 357, 203. Matthaeus, W.H. and Goldstein, M. L. 1982 Measurement of the rugged invariants of magnetohydrodynamic turbulence in the solar wind. J. Geophys. Res. 87A, 6011{6028. Matthaeus, W. H., Goldstein, M. L. and Roberts, D.A. 1990 Evidence for the presence of quasi-two-dimensional nearly incompressible fluctuations in the solar wind. J. Geophys. Res. 95, 20673{20683. Matthaeus, W. H., Ghosh, S., Oughton, S. and Roberts, D.A. 1996 Anisotropic three dimensional MHD turbulence. J. Geophys. Res. 101, 7619{7629. Mikhailovskii, A.B., Novakovskaya, E. A., Lakhin, V.P., Novakovskii, S.V., Onishchenko, O.G. and Smolyakov, A. I. 1989 Contribution to the theory of weakly turbulent Kolmogorov spectra of a homogeneous magnetized plasma. Soviet Phys. JETP 68, 924{932. Montgomery, D. C. and Matthaeus, W.H. 1995 Anisotropic modal energy transfer in interstellar turbulence. Astrophys. J. 447, 706{707. Montgomery, D. and Turner, L. 1981 Anisotropic magnetohydrodynamic turbulence in a strong external magnetic field. Phys. Fluids 24, 825{831. Nazarenko, S.V. and Laval, J.-P. 2000 Non-local 2D turbulence and Batchelor's regime for passive scalars. J. Fluid Mech. (in press). Ng, C. S. and Bhattacharjee, A. 1996 Interaction of shear-Alfvén wave packets: implication for weak magnetohydrodynamic turbulence in astrophysical plasmas. Astrophys. J. 465, 845{854. Ng, C. S. and Bhattacharjee, A. 1997 Scaling of anisotropic spectra due to the weak interaction of shear-Alfvén wave packets. Phys. Plasmas 4, 605{610. Oughton, S., Priest, E. R. and Matthaeus,W.H. 1994 The influence of a mean magnetic field on three{dimensional magnetohydrodynamic turbulence. J. Fluid Mech. 280, 95{117. Parker, E.N. 1994 Spontaneous Current Sheets in Magnetic Fields with Applications to Stellar X-Rays. Oxford University Press. Priest, E. R. 1982 Solar Magnetohydrodynamics. Reidel, Dordrecht. Riley, J., Metcalfe, R. and Weissman, M. 1981 In: Nonlinear Properties of Internal Waves (ed. B. West). AIP Conf. Proc. 76, 79. Robinson, D. and Rusbridge, M. 1971 Structure of turbulence in the zeta plasma. Phys. Fluids 14, 2499{2511. Sagdeev, R. and Galeev, A. 1969 Nonlinear Plasma Theory. Benjamin, New York. Scalo, J.M. 1984 Turbulent velocity structure in interstellar clouds. Astrophys. J. 277, 556{561. Shebalin, J.V., Matthaeus, W.H. and Montgomery, D. 1983 Anisotropy in MHD turbulence due to a mean magnetic field. J. Plasma Phys. 29, 525{547. Sridhar, S. and Goldreich, P. 1994 Toward a theory of interstellar turbulence. I. Weak Alfvénic turbulence. Astrophys. J. 432, 612{621. Strauss, H.R. 1976 Nonlinear three-dimensional magnetohydrodynamics of noncircular tokamaks. Phys.Fluids 19, 134{140. Stribling, T., Matthaeus, W.H. and Ghosh, S. 1994 Nonlinear decay of magnetic helicity in magnetohydrodynamic turbulence with a mean magnetic field. J. Geophys. Res. 99, 2567{2576. Taylor, J.B. 1986 Relaxation and magnetic reconnection in plasmas. Rev. Mod. Phys. 58, 741{763. Taylor, J.B. 1993 Filamentation, current profiles, and transport in a tokamak. Phys. Fluids B5, 4378{4383. Tsytovich, V.N. 1970 Nonlinear Effects in Plasma. Plenum Press, New York. Vedenov, A. A. 1967 Theory of weakly turbulent plasma. In: Reviews of Plasma Physics, Vol. 3 (ed. M. A. Leontovich), pp. 229{276. New York, Consultants Bureau. Vedenov, A. A. 1968 Theory of Turbulent Plasma. Iliffe Books, London/American Elsevier, New York. Waleffe, F. 1992 The nature of triad interactions in homogeneous turbulence. Phys. Fluids A4, 350{363. Wild, N., Gekelman, W. and Stenzel, R. L. 1981 Resistivity and energy flow in a plasma undergoing magnetic field line reconnection. Phys. Rev. Lett. 46, 339{342. Woltjer, L. 1958 On hydromagnetic equilibrium. Proc. Natl Acad. Sci. USA 44, 833{841. Zakharov, V.E. 1974 The Hamiltonian formalism for waves in nonlinear media with dispersion. Izv. Vyssh. Uchebn. Radioz. [Radiophys. Quantum Electron.] 17, 431{453. Zakharov, V.E. 1984 Kolmogorov spectra in weak turbulence problems. In: Basic Plasma Physics II (ed. A. A. Galeev and R. Sudan), Chap. 5.3, p. 48. North-Holland, Amsterdam. Zakharov, V. E., L'vov, V. and Falkovich, G. E. 1992 Kolmogorov Spectra of Turbulence I: Wave Turbulence. Springer-Verlag, Berlin. Zweibel, E.G. and Heiles, C. 1997 Magnetic fields in galaxies and beyond. Nature 385, 131{136.
URI:	http://wrap.warwick.ac.uk/id/eprint/843

Data sourced from Thomson Reuters' Web of Knowledge

Request changes to a record

Actions (login required)

View Item