Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T22:11:52.707Z Has data issue: false hasContentIssue false
  • English
  • Français

Turbulent resistivity in wavy two-dimensional magnetohydrodynamic turbulence

Published online by Cambridge University Press:  08 January 2008

SHANE R. KEATING  and
P. H. DIAMOND
SHANE R. KEATING
Affiliation:
Center for Astrophysics and Space Sciences, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA
P. H. DIAMOND
Affiliation:
Center for Astrophysics and Space Sciences, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The theory of turbulent resistivity in ‘wavy’ magnetohydrodynamic turbulence in two dimensions is presented. The goal is to explore the theory of quenching of turbulent resistivity in a regime for which the mean field theory can be rigorously constructed at large magnetic Reynolds number Rm. This is achieved by extending the simple two-dimensional problem to include body forces, such as buoyancy or the Coriolis force, which convert large-scale eddies into weakly interacting dispersive waves. The turbulence-driven spatial flux of magnetic potential is calculated to fourth order in wave slope – the same order to which one usually works in wave kinetics. However, spatial transport, rather than spectral transfer, is the object here. Remarkably, adding an additional restoring force to the already tightly constrained system of high Rm magnetohydrodynamic turbulence in two dimensions can actually increase the turbulent resistivity, by admitting a spatial flux of magnetic potential which is not quenched at large Rm, although it is restricted by the conditions of applicability of weak turbulence theory. The absence of Rm-dependent quenching in this wave-interaction-driven flux is a consequence of the presence of irreversibility due to resonant nonlinear three-wave interactions, which are independent of collisional resistivity. The broader implications of this result for the theory of mean field electrodynamics are discussed.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 595 , 25 January 2008 , pp. 173 - 202
Copyright
Copyright © Cambridge University Press 2008

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

Benney, D. J. & Newell, A. C. 1969 Random wave closures. Stud. Appl. Maths 48, 29.CrossRefGoogle Scholar
Blackman, E. G. & Field, G. B. 2000 Constraints on the magnitude of α in dynamo theory. Astrophys. J. 534, 984–988.CrossRefGoogle Scholar
Blackman, E. G. & Field, G. B. 2002 New dynamical mean-field dynamo theory and closure approach. Phys. Rev. Lett. 89, 265007.CrossRefGoogle ScholarPubMed
Bracco, A., Provenzale, A., Spiegel, E. A. & Yecko, P. 1998 Spotted discs. In Theory of Black Hole Accretion Disks (ed. Abramowicz, M. A., Bjornsson, G. & Pringle, J. E.), pp. 254–270.Google Scholar
Cattaneo, F. 1994 On the effects of a weak magnetic field on turbulent transport. Astrophys. J. 434, 200–205.Google Scholar
Cattaneo, F. & Vaǐnshteǐn, S. I. 1991 Suppression of turbulent transport by a weak magnetic field. Astrophys. J. 376, L21–L24.CrossRefGoogle Scholar
Davidson, R. C. 1972 Methods in Nonlinear Plasma Theory. Academic.Google Scholar
Diamond, P. H., Itoh, S.-I., Itoh, K. & Hahm, T. S. 2005 Zonal flows in plasmas – a review. Plasma Phys. Control. Fusion 47, R35–R161.CrossRefGoogle Scholar
Diamond, P. H., Itoh, S.-I., Itoh, K. & Silvers, L. 2007 β-plane magnetohydrodynamic turbulence and dissipation in the solar tachocline. In Proc. of the Tachocline Dynamics Workshop (Nov. 2004, Isaac Newton Institute, Cambridge, UK) (ed. Hughes, D. W., Rosner, R. & Weiss, N. O.). Cambridge University Press.Google Scholar
Fyfe, D. & Montgomery, D. 1976 High-beta turbulence in two-dimensional magnetohydrodynamics. J. Plasma Phys. 16, 181–191.CrossRefGoogle Scholar
Fyfe, D., Montgomery, D. & Joyce, G. 1977 Dissipative, forced turbulence in two-dimensional magnetohydrodynamics. J. Plasma Phys. 17, 369–398.CrossRefGoogle Scholar
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
Gruzinov, A. V. & Diamond, P. H. 1994 Self-consistent theory of mean-field elctrodynamics. Phys. Rev. Lett. 72, 1651–1653.CrossRefGoogle Scholar
Gruzinov, A. V. & Diamond, P. H. 1996 Nonlinear mean-field elctrodynamics of turbulent dynamos. Phys. Plasmas 3 (5), 1853–1857.CrossRefGoogle Scholar
Hasselmann, K. 1966 Feynmann diagrams and interaction rules of wave–wave scattering processes. Rev. Geophys. 4, 1–32.CrossRefGoogle Scholar
Kaburaki, O. & Uchida, Y. 1971 Magnetohydrodynamic wave-mode coupling. Quantum field-theoretical approach to weakly non-linear case with application to solar coronal heating. Publ. Astr. Soc. Japan 23, 405.Google Scholar
Kleeorin, N. I. & Ruzmaikin, A. A. 1982 Dynamics of the average turbulent helicity in a magnetic field. Magnetohydrodynamics 19, 116–122, transl. from Magnitaya Gidrodinamika.Google Scholar
Kleeorin, N. I., Rogachevskii, I. V. & Ruzmaikin, A. A. 1990 Magnetic force reversal and instability in a plasma with advanced magnetohydrodynamic turbulence. Sov. Phys. J. Exp. Theor. Phys. 70, 878.Google Scholar
Krommes, J. A. 2002 Fundamental statistical descriptions of plamsa turbulence in magnetic fields. Phys. Rep. 360, 1–352.CrossRefGoogle Scholar
Kulsrud, R. M. & Anderson, S. W. 1992 The spectrum of random magnetic fields in the mean field dynamo theory of the galactic magnetic field. Astrophys. J. 396, 606–630.CrossRefGoogle Scholar
McComas, C. H. & Bretherton, F. P. 1977 Resonant interaction of oceanic internal waves. J. Geophys. Res. 82, 1397–1412.Google Scholar
Miesch, M. S. 2005 Large-scale dynamics of the convection zone and tachocline. Living Rev. Solar Phys. 2, 1.CrossRefGoogle Scholar
Mikhailovskii, A. B., Novakovskaia, E. A., Lakhin, V. P., Novakovskii, S. V. & Onishchenko, O. G. 1989 A contribution to the theory of weakly turbulent Kolmogorov spectra of a homogeneous magnetized plasma. Sov. Phys., J. Exp. Theor. Phys. 95, 1598–1613.Google Scholar
Moffatt, H. K. 1970 Dynamo action associated with random inertial waves in a rotating conducting fluid. J. Fluid Mech. 44, 705–719.Google Scholar
Moffatt, H. K. 1972 An approach to a dynamic theory of dynamo action in a rotating conduting fluid. J. Fluid Mech. 53, 385–399.CrossRefGoogle Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Ng, C. S. & Bhattacharjee, A. 1997 Scaling of anisotropic spectra due to the weak interaction of shear-Alfvén wave packets. Phys. Plasmas 4, 605–610.Google Scholar
Ozmidov, R. V. 1965 On the turbulent exchange in a stably stratified ocean. Izv. Acad. Sci. USSR, Atmos. Ocean. Phys. 1, 853–860.Google Scholar
Peierls, R. E. 1929 Zur kinetischen theorie der wärmeleitungen in kristallen. Ann. Phys. 3, 1055–1101.Google Scholar
Phillips, O. M. 1960 On the dynamics of unsteady gravity waves of finite amplitude. Part 1. J. Fluid Mech. 9, 193–217.Google Scholar
Pouquet, A. 1978 On two-dimensional magnetohydrodynamic turbulence. J. Fluid Mech. 88, 1–16.CrossRefGoogle Scholar
Pouquet, A., Frisch, U. & Leorat, J. 1976 Strong magnetohydrodynamic helical turbulence and the nonlinear dynamo effect. J. Fluid Mech. 77, 321–354.CrossRefGoogle Scholar
Rhines, P. B. 1975 Waves and turbulence on a β-plane. J. Fluid Mech. 69, 417–443.CrossRefGoogle Scholar
Sagdeev, R. Z. & Galeev, A. A. 1969 Nonlinear Plasma Theory. W. A. Benjamin.Google Scholar
Silvers, L. J. 2005 Dynamic effects of a magnetic field on diffusion in a chaotic flow. Phys. Lett. A 334, 400–405.CrossRefGoogle Scholar
Silvers, L. J., Keating, S. R. & Diamond, P. H. 2008 On cross-phase and the quenching of the turbulent diffusion of magnetic fields in two dimensions. Astrophys. J. Lett. (submitted).Google Scholar
Spiegel, E. A. & Zahn, J.-P. 1992 The solar tachocline. Astron. Astrophys. 265, 106–114.Google Scholar
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34, 559–593.Google Scholar
Steenbeck, M., Krause, F. & Rädler, K. H. 1966 A calculation of the mean electromotive force in an electrically conducting fluid in turbulent motion under the influence of coriolis forces. Z. Naturf. A 21, 369–376.Google Scholar
Ting, A. C., Montgomery, D. & Matthaeus, W. H. 1986 Turbulent relaxation processes in magnetohydrodynamics. Phys. Fluids 29, 3261–3274.Google Scholar
Tobias, S. M. 2005 The solar tachocline: formation, stability and its role in the solar dynamo. In Fluid Dynamics and Dynamos in Astrophysics and Geophysics (ed. Soward, A. M., Jones, C. A., Hughes, D. W. & Weiss, N. O.), pp. 193–234. CRC Press.Google Scholar
Tsinober, A. B. 1975 Magnetohydrodynamic turbulence. Magnitnaia Gidrodinamika 1, 7–22.Google Scholar
Vaǐnshteǐn, S. I. & Zel'dovich, Y. B. 1972 Origin of magnetic fields in astrophysics. Sov. Phys. Usp. 15, 159.CrossRefGoogle Scholar
Wersinger, J. M., Finn, J. M. & Ott, E. 1980 Bifurcation and ‘strange’ behavior in instability saturation by nonlinear three-wave coupling. Phys. Fluids 23 (6), 1142–1154.Google Scholar
Zakharov, V. E., L'vov, V. X. & Falkovich, G. 1992 Kolmogorov Spectra of Turbulence I. Springer.Google Scholar
Zel'dovich, Ya. B. 1957 The magnetic field in the two-dimensional motion of a conducting turbulent liquid. Sov. Phys. J. Exp. Theor. Phys. 4, 460–462.Google Scholar
Zel'dovich, Ya. B., Ruzmaikin, A. A. & Sokoloff, D. D. 1983 Magnetic Fields in Astrophysics. Gordon & Breach.Google Scholar