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MHD Turbulence, Turbulent Dynamo and Applications

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Magnetic Fields in Diffuse Media

Part of the book series: Astrophysics and Space Science Library ((ASSL,volume 407))

Abstract

MHD Turbulence is common in many space physics and astrophysics environments. We first discuss the properties of incompressible MHD turbulence. A well-conductive fluid amplifies initial magnetic fields in a process called small-scale dynamo. Below equipartition scale for kinetic and magnetic energies the spectrum is steep (Kolmogorov \(-5/3\)) and is represented by critically balanced strong MHD turbulence. In this Chapter we report the basic reasoning behind universal nonlinear small-scale dynamo and the inertial range of MHD turbulence. We measured the efficiency of the small-scale dynamo C E  = 0. 05, Kolmogorov constant C K  = 4. 2 and anisotropy constant C A  = 0. 63 for MHD turbulence in high-resolution direct numerical simulations. We also discuss so-called imbalanced or cross-helical MHD turbulence which is relevant for in many objects, most prominently in the solar wind. We show that properties of incompressible MHD turbulence are similar to the properties of Alfvénic part of MHD cascade in compressible turbulence. The other parts of the cascade evolve according to their own dynamics. The slow modes are being cascaded by Alfvénic modes, while fast modes create an independent cascade. We show that different ways of decomposing compressible MHD turbulence into Alfvén, slow and fast modes provide consistent results and are useful in understanding not only turbulent cascade, but its interaction with fast particles.

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Notes

  1. 1.

    The anisotropy should be understood in terms of local magnetic field direction, i.e. the magnetic field direction at the given scale. The original treatment, e.g. the closure relations employed, in the Goldreich–Sridhar paper uses the global frame of reference which was noticed later in Lazarian et al. (1999) and used in the numerical works that validated the theory (Cho et al. 2000, 2002; Maron et al. 2001).

  2. 2.

    The latter, \(\int \mathbf{v} \cdot \mathbf{B}\,d^{3}x\) is a quantity conserved in the absence of dissipation.

  3. 3.

    We assume that imbalanced turbulence is “strong” as long as the applicability of weak Alfvénic turbulence breaks down. This requires that at least one component is perturbed strongly. In the imbalanced turbulence the amplitude of the dominant component is larger, so that in the transition to strong regime the applicability of weak cascading of the subdominant component breaks down first.

  4. 4.

    Throughout this Chapter we assume that w + is the larger-amplitude wave. This choice, however, is purely arbitrary and corresponds to the choice of positive versus negative total cross-helicity.

  5. 5.

    In the limiting case of compressibility going to zero, the fast modes are sound waves with phase speed going to infinity.

  6. 6.

    One way to remove the effect by the wandering of field lines is to drive turbulence anisotropically in such a way as \(k_{\perp,L}\delta V \sim k_{\|,L}V _{A}\), where k  ⊥ , L and \(k_{\|,L}\) stand for the wavelengths of the driving scale and δ V is the r.m.s. velocity. By increasing the \(k_{\perp,L}/k_{\|,L}\) ratio, we can reduce the degree of mixing of different wave modes.

  7. 7.

    A claim in the literature is that a strong coupling of incompressible and compressible motions is required to explain simulations that show fast decay of MHD turbulence. There is not true. The incompressible motions decay themselves in just one Alfvén crossing time.

  8. 8.

    In astrophysics spatial or temporal averaging is used.

  9. 9.

    In practical terms this means that instead of obtaining S p as a function of r, one gets S p as a function of S 3, which is nonlinear in a way to correct for the distortions of S p.

  10. 10.

    The cited paper introduces the model of compressible turbulence which it calls Kolmogorov–Burgers model. Within this model turbulence goes first along the Kolmogorov scaling and then, at small scales forms shocks. The model was motivated by the numerical measurements of the turbulence spectrum that indicated the index of supersonic turbulence close to \(-5/3\). This however was shown to be an artifact of numerical simulations with lower resolution. Simulations in Kritsuk et al. (2007) showed that the slope with \(-5/3\) is the result of the numerical bottleneck and the actual slope of the highly compressible turbulence is − 2, as was expected earlier.

  11. 11.

    The violation of frozen in condition in turbulence is implicit in LV99. It was stated explicitly in Vishniac et al. (1999) and discussed in terms of star formation in Lazarian (2005). The first formal quantitative study was performed in Eyink (2011).

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Acknowledgements

AB was supported by Humboldt Fellowship. AL acknowledges the support of the NSF grant AST-1212096, the Vilas Associate Award as well as the support of the NSF Center for Magnetic Self- Organization. In addition, AL thanks the International Institute of Physics (Natal, Brazil) and the Observatoire de Nice for its hospitality during writing this review. Discussions with Ethan Vishniac and Greg Eyink are acknowledged. We thank Siyao Xu for reading the manuscript.

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  1. Ruhr-Universität Bochum, 44780, Bochum, Germany

    Andrey Beresnyak

  2. University of Wisconsin-Madison, Madison, WI, USA

    Alex Lazarian

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Correspondence to Andrey Beresnyak .

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  1. Department of Astronomy, University of Wisconsin-Madison, Madison, Wisconsin, USA

    Alexander Lazarian

  2. Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de São Paulo, São Paulo, Brazil

    Elisabete M. de Gouveia Dal Pino

  3. Instituto de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de São Paulo, São Paulo, Brazil

    Claudio Melioli

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Beresnyak, A., Lazarian, A. (2015). MHD Turbulence, Turbulent Dynamo and Applications. In: Lazarian, A., de Gouveia Dal Pino, E., Melioli, C. (eds) Magnetic Fields in Diffuse Media. Astrophysics and Space Science Library, vol 407. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44625-6_8

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