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Flux-freezing breakdown in high-conductivity magnetohydrodynamic turbulence


The idea of ‘frozen-in’ magnetic field lines for ideal plasmas1 is useful to explain diverse astrophysical phenomena2, for example the shedding of excess angular momentum from protostars by twisting of field lines frozen into the interstellar medium. Frozen-in field lines, however, preclude the rapid changes in magnetic topology observed at high conductivities, as in solar flares2,3. Microphysical plasma processes are a proposed explanation of the observed high rates4,5,6, but it is an open question whether such processes can rapidly reconnect astrophysical flux structures much greater in extent than several thousand ion gyroradii. An alternative explanation7,8 is that turbulent Richardson advection9 brings field lines implosively together from distances far apart to separations of the order of gyroradii. Here we report an analysis of a simulation of magnetohydrodynamic turbulence at high conductivity that exhibits Richardson dispersion. This effect of advection in rough velocity fields, which appear non-differentiable in space, leads to line motions that are completely indeterministic or ‘spontaneously stochastic’, as predicted in analytical studies10,11,12,13. The turbulent breakdown of standard flux freezing at scales greater than the ion gyroradius can explain fast reconnection of very large-scale flux structures, both observed (solar flares and coronal mass ejections) and predicted (the inner heliosheath, accretion disks, γ-ray bursts and so on). For laminar plasma flows with smooth velocity fields or for low turbulence intensity, stochastic flux freezing reduces to the usual frozen-in condition.

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Figure 1: Stochastic flux-freezing for resistive MHD.
Figure 2: The archived MHD turbulence flow.
Figure 3: Stochastic flux freezing in MHD turbulence.
Figure 4: Evidence of Richardson dispersion.

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  1. Alfvén, H. On the existence of electromagnetic-hydrodynamic waves. Ark. Mat. Astron. Fys. 29, 1–7 (1942)

    MATH  Google Scholar 

  2. Kulsrud, R. Plasma Physics for Astrophysics (Princeton Univ. Press, 2005)

    Book  Google Scholar 

  3. Priest, E. R. & Forbes, T. G. Magnetic Reconnection: MHD Theory and Applications (Cambridge Univ. Press, 2000)

    Book  Google Scholar 

  4. Che, H., Drake, J. F. & Swisdak, M. A current filamentation mechanism for breaking magnetic field lines during reconnection. Nature 474, 184–187 (2011)

    Article  ADS  CAS  Google Scholar 

  5. Daughton, W. et al. Role of electron physics in the development of turbulent magnetic reconnection in collisionless plasmas. Nature Phys. 7, 539–542 (2011)

    Article  ADS  CAS  Google Scholar 

  6. Moser, A. L. & Bellan, P. M. Magnetic reconnection from a multiscale instability cascade. Nature 482, 379–381 (2012)

    Article  ADS  CAS  Google Scholar 

  7. Lazarian, A. & Vishniac, E. T. Reconnection in a weakly stochastic field. Astrophys. J. 517, 700–718 (1999)

    Article  ADS  Google Scholar 

  8. Eyink, G. L., Lazarian, A. & Vishniac, E. T. Fast magnetic reconnection and spontaneous stochasticity. Astrophys. J. 743, 51 (2011)

    Article  ADS  Google Scholar 

  9. Richardson, L. F. Atmospheric diffusion shown on distance-neighbor graph. Proc. R. Soc. Lond. A 110, 709–737 (1926)

    Article  ADS  Google Scholar 

  10. Bernard, D., Gawȩdzki, K. & Kupiainen, A. Slow modes in passive advection. J. Stat. Phys. 90, 519–569 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  11. E, W. & Vanden Eijnden, E. Vanden Eijnden, E. Generalized flows, intrinsic stochasticity, and turbulent transport. Proc. Natl Acad. Sci. USA 97, 8200–8205 (2000)

    Article  ADS  MathSciNet  CAS  Google Scholar 

  12. Chaves, M., Gawȩdzki, K., Horvai, P., Kupiainen, A. & Vergassola, M. Lagrangian dispersion in Gaussian self-similar velocity ensembles. J. Stat. Phys. 113, 643–692 (2003)

    Article  MathSciNet  Google Scholar 

  13. Eyink, G. L. Turbulent diffusion of lines and circulations. Phys. Lett. A 368, 486–490 (2007)

    Article  ADS  CAS  Google Scholar 

  14. Frisch, U. Turbulence 52–56 (Cambridge Univ. Press, 1995)

    Book  Google Scholar 

  15. Leamon, R. J., Smith, C. W., Ness, N. F., Matthaeus, W. H. & Wong, H.-K. Observational constraints on the dynamics of the interplanetary magnetic field dissipation range. J. Geophys. Res. 103, 4775–4787 (1998)

    Article  ADS  Google Scholar 

  16. Chepurnov, A. & Lazarian, A. Extending the big power law in the sky with turbulence spectra from Wisconsin Hα Mapper data. Astrophys. J. 710, 853–858 (2010)

    Article  ADS  CAS  Google Scholar 

  17. Eyink, G. L. Stochastic line motion and stochastic flux conservation for nonideal hydromagnetic models. J. Math. Phys. 50, 083102 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  18. Eyink, G. L. Fluctuation dynamo and turbulent induction at small Prandtl number. Phys. Rev. E 82, 046314 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  19. Sawford, B. L., Yeung, P. K. & Hackl, J. F. Reynolds number dependence of relative dispersion statistics in isotropic turbulence. Phys. Fluids 20, 065111 (2008)

    Article  ADS  Google Scholar 

  20. Eyink, G. L. Stochastic flux freezing and magnetic dynamo. Phys. Rev. E 83, 056405 (2011)

    Article  ADS  Google Scholar 

  21. Busse, A., Müller, W.-C., Homann, H. & Grauer, R. Statistics of passive tracers in three-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 14, 122303 (2007)

    Article  ADS  Google Scholar 

  22. Boldyrev, S. Spectrum of magnetohydrodynamic turbulence. Phys. Rev. Lett. 96, 115002 (2006)

    Article  ADS  Google Scholar 

  23. Beresnyak, A. Spectral slope and Kolmogorov constant of MHD turbulence. Phys. Rev. Lett. 106, 075001 (2011)

    Article  ADS  CAS  Google Scholar 

  24. Li, Y. et al. A public turbulence database cluster and applications to study Lagrangian evolution of velocity increments in turbulence. J. Turbul. 9, N31 (2008)

    Article  ADS  Google Scholar 

  25. The JHU Turbulence Database Cluster. (2012)

  26. Goldreich, P. & Sridhar, S. Toward a theory of interstellar turbulence. 2: Strong Alfvénic turbulence. Astrophys. J. 438, 763–775 (1995)

    Article  ADS  Google Scholar 

  27. Hentschel, H. G. E. & Procaccia, I. Relative diffusion in turbulent media: the fractal dimension of clouds. Phys. Rev. A 29, 1461–1470 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  28. Kowal, G., Lazarian, A., Vishniac, E. T. & Otmianowska-Mazur, K. Numerical tests of fast reconnection in weakly stochastic magnetic fields. Astrophys. J. 700, 63–85 (2009)

    Article  ADS  Google Scholar 

  29. Schekochihin, A. A. et al. Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas. Astrophys. J. 182 (suppl.). 310–377 (2009)

    Article  Google Scholar 

  30. Ciaravella, A. & Raymond, J. C. The current sheet associated with the 2003 November 4 coronal mass ejection: density, temperature, thickness, and line width. Astrophys. J. 686, 1372–1382 (2008)

    Article  ADS  CAS  Google Scholar 

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The work of the group at the Johns Hopkins University was supported by the US NSF grant CDI-II: CMMI 0941530, and the database infrastructure was supported by US NSF grant OCI-108849 and JHU’s Institute for Data Intensive Engineering & Science. The work of E.V. was supported by the National Science and Engineering Research Council of Canada. The authors thank R. Westermann for his contributions to the visualization tool and A. Lazarian for discussions of the science.

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Authors and Affiliations



All of the authors made significant contributions to this work. H.A. carried out the simulations of MHD turbulence. K.K., R.B., A.S. and C.M. were primarily responsible for the construction of the MHD database and online analysis tools. G.E. designed the study and developed the numerical algorithms for stochastic flux freezing. C.L. generated the simulation results using the database. K.B. developed the visualization of the archived MHD data. G.E., E.V., C.L. and C.M. analysed the simulation results and were primarily responsible for writing the paper. All authors discussed the results and commented on the paper.

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Correspondence to Gregory Eyink.

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The authors declare no competing financial interests.

Supplementary information

Supplementary Information

This file contains Supplementary Text and Data, Supplementary Figures 1-2 and Supplementary References. (PDF 1227 kb)

Ohmic electric fields in the archived MHD simulation

The magnitude of the Ohmic electric field EOhm=J/σ=c×B/4πσ is plotted normalized by the rms value E'mot of the motional field Emot=-v×B/c, for one 10243 time-slice of the archived MHD turbulence simulation. This is the same field plotted in panel a of Fig.2, with the same volume rendering and color coding, but at higher resolution and in a rotating frame. The transparency and rotation provide a three-dimensional view, showing EOhm / E'mot is negligible outside thin, intense current sheets sparsely distributed over the volume. Richardson dispersion occurs at points throughout the flow and is not associated with the strong current sheets. (MOV 24045 kb)

Stochastic flux-freezing for resistive MHD

Shown is an animation of Figure 1, illustrating the numerical evaluation of the pointwise magnetic field via stochastic flux-freezing. First, stochastic trajectories are integrated backward to the starting time, when initial field vectors are sampled. Second, the initial vectors are transported along the trajectories to the final point. Third, these “virtual” vectors arriving at that point are averaged to obtain the actual magnetic field. Physically, all of the dynamics is forward in time and the first backward-integration step is only a convenient algorithm to obtain the ensemble of stochastic trajectories which arrive simultaneously at the chosen final point. (MOV 33269 kb)

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Eyink, G., Vishniac, E., Lalescu, C. et al. Flux-freezing breakdown in high-conductivity magnetohydrodynamic turbulence. Nature 497, 466–469 (2013).

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