Flux-freezing breakdown in high-conductivity magnetohydrodynamic turbulence

Journal name:
Nature
Volume:
497,
Pages:
466–469
Date published:
DOI:
doi:10.1038/nature12128
Received
Accepted
Published online

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.

At a glance

Figures

  1. Stochastic flux-freezing for resistive MHD.
    Figure 1: Stochastic flux-freezing for resistive MHD.

    Magnetic fields that solve the resistive induction equation are frozen-in to stochastic trajectories, , that satisfy . The field vectors, which start at τ = t0, are transported to τ = t along these random trajectories as ‘virtual’ magnetic fields, stretched and rotated by the plasma flow. The ensemble of virtual field vectors that arrive at the same final point, (x,t), must be averaged (that is, ‘glued’ together by resistivity) to give the physical magnetic field B(x,t) at that point17. To find the stochastic trajectories that arrive at (x,t), it is most efficient to integrate backwards in time. We use the velocity field, u, from the database to solve , starting with and integrating back to the initial time, t0. This step generates the N-sample ensemble of stochastic trajectories , n = 1,,N (a). We then retrieve from the database the magnetic field, B, at the random locations , n = 1,,N, and transport these vectors back along the stochastic trajectories to the point (x,t) using the frozen-in equation (b). This requires retrieving the velocity gradient, u, from the database at each forward time step. Finally, the virtual magnetic field vectors , n = 1,,N, are averaged over the N samples to recover the archived physical magnetic field, B(x,t), by choosing N to be sufficiently large (c). For more discussion, see Supplementary Information.

  2. The archived MHD turbulence flow.
    Figure 2: The archived MHD turbulence flow.

    The data were generated by a simulation of the incompressible MHD equations on a space grid of 1,0243 points. The magnetic Prandtl number is 1, with ν = λ = 1.1×10−4. The MHD momentum equation was driven by a body force F of Taylor–Green form at wavenumbers with magnitude kf = 2k0, twice the lowest value. It is important that no external source of electric field was added to the induction equation, so the stirring to generate turbulence does not directly break standard flux freezing. Once the simulation reached a statistical steady state, 1,024 time slices of data on the 1,0243 grid were stored. The 56 terabytes of data are public and can be accessed remotely through a modern web-services interface24, 25. a, Ohmic electric field EOhm = J/σ = c×B/4πσ normalized by the r.m.s. value, Emot, of the motional field, Emot = −v×B/c, for one 1,0243-point time slice of the archived data. The colour scale covers a range from 0.1 to 10 times the r.m.s. value, 1.68×10−2, of the normalized Ohmic field. The Ohmic electric field is negligible compared with the motional field, except in the most intense current sheets. b, Energy spectra normalized by total energy, Etot, for the velocity (Eu, red) and magnetic field (Eb, blue) of the flow, each with about a decade of −3/2-power-law range. See Supplementary Information for more details.

  3. Stochastic flux freezing in MHD turbulence.
    Figure 3: Stochastic flux freezing in MHD turbulence.

    a, Stochastic trajectories that arrive at point xf = (1.09, 2.42, 5.03) in the archived flow at time tf = 2.56. This final point is not in the vicinity of an intense current sheet, having a small Ohmic electric field just 1.3% of the r.m.s. motional field. The points along the stochastic trajectories are colour-coded red, green and blue from earlier to later times. The clouds of points indicate the spatial regions at those times that contribute significantly to B(xf,tf) when calculated as described in Fig. 1. We also estimate this magnetic field using conventional flux freezing, by transporting the initial magnetic field along the deterministic trajectory satisfying dx/dt = u(x,t) and x(tf) = xf (black). b, Errors for the magnetic field calculated from stochastic flux freezing, relative to the archived magnetic field, B(xf,tf). The errors for increasing N are small, of similar order as the interpolation error in calculating the velocity gradient. In contrast, the relative error for standard flux freezing (black) is very large. The results shown here for one point are representative (Supplementary Information).

  4. Evidence of Richardson dispersion.
    Figure 4: Evidence of Richardson dispersion.

    a, Mean squared dispersion of field lines backwards in time with the variable t* = tft, in directions both parallel (red) and perpendicular (blue) to the local magnetic field. Times are normalized by the inverse r.m.s. current, 1/jrms, and distances are normalized by the resistive length, . Error bars, s.e.m. The dashed line shows the conventional diffusive estimate, left fencer2(t*)right fence = 4λt*, and the solid line is t*8/3. The long-time power laws are left fenceri2(t)right fencegiLu2(ut*/Lu)8/3, i = ||, , with Lu the velocity integral length. The constant g|| = 0.086 in the direction parallel to the local field is greater than the constant g = 0.035 in the perpendicular direction. Both are substantially smaller than the corresponding constant in hydrodynamic turbulence, g0.5. To verify that we are seeing an MHD analogue of Richardson dispersion, we check for self-similarity of the PDFs of the pair separation. b, PDFs at 96 times in the interval where the t*8/3 dependence occurs, rescaled by r.m.s. values, with good collapse to the stretched-exponential form, Pexp(−Cr3/4), at less than four times the r.m.s. separation. The power in the exponent corresponds to h = 1/4, consistent with scaling of the energy spectrum and pair dispersion.

Videos

  1. Ohmic electric fields in the archived MHD simulation
    Video 1: 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.
  2. Stochastic flux-freezing for resistive MHD
    Video 2: 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.

References

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Author information

Affiliations

  1. Department of Applied Mathematics & Statistics, The Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218, USA

    • Gregory Eyink,
    • Cristian Lalescu &
    • Hussein Aluie
  2. Department of Physics & Astronomy, The Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218, USA

    • Gregory Eyink &
    • Alexander Szalay
  3. Department of Mechanical Engineering, The Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218, USA

    • Gregory Eyink &
    • Charles Meneveau
  4. Institute for Data Intensive Engineering & Science, The Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218, USA

    • Gregory Eyink,
    • Randal Burns,
    • Charles Meneveau &
    • Alexander Szalay
  5. Department of Physics and Engineering Physics, University of Saskatchewan, Saskatoon, Saskatchewan S7N 5E2, Canada

    • Ethan Vishniac
  6. Los Alamos National Laboratory, T-Division and Center for Nonlinear Studies, Los Alamos, New Mexico 87545, USA

    • Hussein Aluie
  7. Department of Computer Science, The Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218, USA

    • Kalin Kanov &
    • Randal Burns
  8. Fakultät für Informatik, Technische Universität München, Boltzmannstraße 3, D-85748 Garching bei München, Germany

    • Kai Bürger

Contributions

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.

Competing financial interests

The authors declare no competing financial interests.

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Supplementary information

Video

  1. Video 1: Ohmic electric fields in the archived MHD simulation (24,046 KB, Download)
    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.
  2. Video 2: Stochastic flux-freezing for resistive MHD (33,269 KB, Download)
    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.

PDF files

  1. Supplementary Information (1.1 MB)

    This file contains Supplementary Text and Data, Supplementary Figures 1-2 and Supplementary References.

Additional data