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Magnetic Reconnection in Astrophysical Environments

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Magnetic Fields in Diffuse Media

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

Abstract

Magnetic reconnection is a process that changes magnetic field topology in highly conducting fluids. Traditionally, magnetic reconnection was associated mostly with solar flares. In reality, the process must be ubiquitous as astrophysical fluids are magnetized and motions of fluid elements necessarily entail crossing of magnetic frozen in field lines and magnetic reconnection. We consider magnetic reconnection in realistic 3D geometry in the presence of turbulence. This turbulence in most astrophysical settings is of pre-existing nature, but it also can be induced by magnetic reconnection itself. In this situation turbulent magnetic field wandering opens up reconnection outflow regions, making reconnection fast. We discuss Lazarian and Vishniac (1999) model of turbulent reconnection, its numerical and observational testings, as well as its connection to the modern understanding of the Lagrangian properties of turbulent fluids. We show that the predicted dependences of the reconnection rates on the level of MHD turbulence make the generally accepted Goldreich and Sridhar (1995) model of turbulence self-consistent. Similarly, we argue that the well-known Alfvén theorem on flux freezing is not valid for the turbulent fluids and therefore magnetic fields diffuse within turbulent volumes. This is an element of magnetic field dynamics that was not accounted by earlier theories. For instance, the theory of star formation that was developing assuming that it is only the drift of neutrals that can violate the otherwise perfect flux freezing, is affected and we discuss the consequences of the turbulent diffusion of magnetic fields mediated by reconnection. Finally, we briefly address the first order Fermi acceleration induced by magnetic reconnection in turbulent fluids.

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Notes

  1. 1.

    The basic idea of the model was first discussed by Sweet and the corresponding paper by Parker refers to the model as “Sweet model”.

  2. 2.

    The power-law ranges that are universal features of high-Reynolds-number turbulence can be inferred to be present from enhanced rates of dissipation and mixing (Eyink 2008) even when they are not seen.

  3. 3.

    In Sect. 12.7.1 we discuss the modification of the frozen in concept in the presence of turbulence. This is not important for the present discussion, however.

  4. 4.

    We should stress that weak and strong are not the characteristics of the amplitude of turbulent perturbations, but the strength of non-linear interactions (see more discussion in Cho et al. (2003)) and small scale Alfvénic perturbations can correspond to a strong Alfvénic cascade.

  5. 5.

    In a more recent work Shibata and Tanuma (2001) extended the concept suggesting that tearing may result in fractal reconnection taking place on very small scales.

  6. 6.

    Also earlier works suggest such a transfer (Dahlburg et al. 1992; Dahlburg and Karpen 1994; Dahlburg 1997; Ferraro and Rogers 2004).

  7. 7.

    Incidentally, this can explain the formation of density fluctuations on scales of thousands of Astronomical Units, that are observed in the ISM.

  8. 8.

    For instance, the increase of \(\varDelta\) increases the Reynolds number of the outflow, making the outflow more turbulent.

  9. 9.

    A similar process takes place in the case of molecular diffusivity in turbulent hydrodynamic flows. The result for the latter flows is well known: in the turbulent regime, molecular diffusivity is irrelevant for the turbulent transport. The process is called therefore “turbulent diffusivity” without adding the superfluous and inappropriate word “molecular”.

  10. 10.

    The First-Order Fermi acceleration is a process in which the energy gain is proportional to the first order of the ratio of the shock velocity to that of light. It should be distinguished from the stochastic Second-Order Fermi acceleration which is proportional to the square of this ratio.

  11. 11.

    Indeed, within the GS95 picture the reconnection happens with nearly parallel lines with magnetic pressure gradient \(V _{A}^{2}/l_{\|}\) being reduced by a factor \(l_{\perp }^{2}/l_{\|}^{2}\), since only reversing component is available for driving the outflow. At the same time the length of the contracted magnetic field lines is also reduced from l  ⊥  but \(l_{\perp }^{2}/l_{\|\vert }\). Therefore the acceleration is \(\tau _{eject}^{-2}l_{\perp }^{2}/l_{\|\vert }\). As a result, the Newtons’ law gives \(V _{A}^{2}l_{\perp }^{2}/l_{\|}^{3} \approx \tau _{eject}^{-2}l_{\perp }^{2}/l_{\|\vert }\). This provides the result for the ejection rate \(\tau _{eject}^{-1} \approx V _{A}/l_{\|}\). The length over which the magnetic eddies intersect is l  ⊥  and the rate of reconnection is V rec ∕l  ⊥ . For the stationary reconnection this gives V rec  ≈ V A l  ⊥ ∕l }, which provides the reconnection rate \(V _{A}/l_{\|}\), which is exactly the rate of the eddy turnovers in GS95 turbulence.

  12. 12.

    In fact, this unsatisfactory situation with the theory of turbulence motivated some of us to work seriously on testing turbulence models (see Cho and Vishniac 2000; Cho et al. 2002; Cho and Lazarian 2002, 2003)

  13. 13.

    The largest-scale Hall MHD simulations performed to date (Huang et al. 2011) do show somewhat higher reconnection rates for laminar X-point solutions than for plasmoid unstable regimes, but the X-point solutions lose stability and seem to have lower reconnection rates with decreasing ratios \(\delta _{i}/L_{x}.\)

  14. 14.

    Because the Hall MHD equations have played a prominent role in magnetic reconnection research of the past decade (Shay et al. 1998, 1999; Wang et al. 2000; Birn et al. 2001; Drake 2001; Malakit et al. 2009; Cassak et al. 2010), it is worth remarking that those equations are essentially never applicable in astrophysical environments. A derivation of Hall MHD based on collisionality requires that the ion skin-depth \(\delta _{i}\) must satisfy the conditions \(\delta _{i} \gg L \gg \ell_{\mathit{mfp},i}\). The second inequality is needed so that a two-fluid description is valid at the scales L of interest, while the first inequality is needed so that the Hall term remains significant at those scales. However, substituting \(\delta _{i} =\rho _{i}/\sqrt{\beta _{i}}\) into (12.4) yields the result

    $$\displaystyle{\frac{\ell_{\mathit{mfp},i}} {\delta _{i}} \propto \frac{\varLambda } {\ln \varLambda }\frac{v_{\mathit{th},i}} {c}.}$$

    The ratio v th, i ∕c is generally small in astrophysical plasmas, but the plasma parameter Λ is usually large by even much, much more (see Table 12.1). Thus, it is usually the case that \(\ell_{\mathit{mfp},i} \gg \delta _{i},\) unless the ion temperature is extremely low. A collisionless derivation of Hall MHD from gyrokinetics requires also a restrictive condition of cold ions (Schekochihin et al. 2009, Appendix E). Thus, Hall MHD is literally valid only for cold, dense plasmas like those produced in some laboratory experiments, such as the MRX reconnection experiment (Yamada 1999; Yamada et al. 2010).

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Acknowledgements

A.L. research is supported by the NSF grant AST 1212096, Vilas Associate Award as well as the support 1098 from the NSF Center for Magnetic Self-Organization. The research is supported by the Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas. Stimulating environment provided by Humboldt Award at the Universities of Cologne and Bochum, as well as a Fellowship at the International Institute of Physics (Brazil) is acknowledged. G.K. acknowledges support from FAPESP (projects no. 2013/04073-2 and 2013/18815-0). Part of the computations were performed using supercomputer RANGER (Teragrid AST080005N, TACC, USA, https://www.xsede.org/tg-archives/), supercomputer GALERA (ACK TASK, Poland, http://www.task.gda.pl/), and supercomputer ALPHACRUCIS (LAi, IAG-USP, Brazil, http://lai.iag.usp.br/). We thank Andrey Beresnyak for useful discussions of the generation of turbulence in the process of magnetic reconnection.

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  1. Department of Astronomy, University of Wisconsin, 475 North Charter Street, Madison, WI, 53706, USA

    Alex Lazarian

  2. Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, MD, 21218, USA

    Gregory L. Eyink

  3. Department of Physics and Astronomy, McMaster University, 1280 Main Street West, Hamilton, ON, L8S 4M1, Canada

    Ethan T. Vishniac

  4. Escola de Artes, Ciências e Humanidades, Universidade de São Paulo, Av. Arlindo Béttio, 1000 Ermelino Matarazzo, CEP, 03828-000, São Paulo, Brazil

    Grzegorz Kowal

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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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Lazarian, A., Eyink, G.L., Vishniac, E.T., Kowal, G. (2015). Magnetic Reconnection in Astrophysical Environments. 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_12

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