1

A current filamentation mechanism for breaking magnetic field lines during reconnection

H. Che1, J. F. Drake2, M. Swisdak2

1University of Colorado, Boulder, Colorado

2University of Maryland, College Park, MD 20742

During magnetic reconnection, the field lines must break and reconnect to release the energy that drives solar and stellar flares1, 2 and other explosive events in nature3 and the laboratory4. Specifically how this happens has been unclear since dissipation is needed to break magnetic field lines and classical collisions are typically weak. Ion-electron drag arising from turbulence5, dubbed ‘anomalous resistivity’, and thermal momentum transport6 are two mechanisms that have been widely invoked. Measurements of enhanced turbulence near reconnection sites in space7, 8 and in the laboratory9, 10 lend support to the anomalous resistivity idea but there has been no demonstration from measurements that this turbulence produces the necessary enhanced drag11. Here we report computer simulations that conclusively show that neither of the two previously favored mechanisms control how magnetic field lines reconnect in the plasmas of greatest interest, those in which the magnetic field dominates the energy budget. Rather, we find that when the current layers that form during magnetic reconnection become too intense, they disintegrate and spread into a complex web of filaments that causes the rate of reconnection to increase abruptly. This filamentary web can be explored in the laboratory or in space with satellites that can measure the resulting electromagnetic turbulence. Particle-in-cell (PIC) simulations reveal that at late time the rate of reconnection in a 3-D system jumps sharply above the rate measured in 2-D (Fig. 1a). The jump is a consequence of the development of turbulence in 3-D. To understand why this happens, it is necessary to explore how magnetic fields break and reconnect. Magnetic reconnection (Fig. 2a) produces large electric fields that are parallel to the local magnetic field in the vicinity of the x-line (Fig. 2b). In the absence of some dissipative process, such a parallel electric field would produce an infinite current and reconnection would cease. The mechanisms that limit the electron response to parallel electric fields therefore break field lines. They can be understood from the electron momentum equation in the direction perpendicular to the plane of reconnection (the z direction)12,

€

∂pez

∂t+ ∇ • p evez = −ene Ez +

1c j e × B ( )

z− ∇ • P ez , (1)

where

€

p e is the momentum density,

€

j e is the electron current density and

€

P ez =

P e • ˆ z with

€

P e the pressure tensor. In a 3-D system where the intense current layers

produced during reconnection (Fig. 2c) drive short-scale turbulence with wavevectors

2

along z , the impact of this turbulence can be quantified by averaging Eq. (1) over z to obtain a generalized Ohm’s law,

€

< Ez >= −1c

< v e⊥ > × <

B ⊥ >( )z

−

∇ ⊥• <

P ez >

< ne > e−

me

e∂ < vez >

∂t+ < v e⊥ > •

∇ ⊥ < vez >

⎛ ⎝ ⎜

⎞ ⎠ ⎟ + Dez +

∇ ⊥ •

T ez

where the brackets denote an average over z,

€

δf = f − < f > for any f, the subscript

€

⊥ denotes the x-y plane,

€

Dez =−1

< ne >< δneδEz > and

€

T ez = −

1< ne > e

< δ p e⊥ δvez −

emec

δAz

⎛

⎝ ⎜

⎞

⎠ ⎟ >

are the turbulent drag and momentum transport, respectively, and

€

δAz is the vector potential for

€

δ B ⊥ . This representation for

€

T ez is valid only in the strong guide field limit

(see the Supplementary Information). In a 2-D system Dez and

€

T ez are both zero and

€

v e⊥ is typically zero at the x-line. In steady state the inertia term is also zero so the only the thermal momentum transport arising from the xz and yz components of

€

P e can balance

the reconnection electric field. This was confirmed in earlier 2-D simulations when the ambient guide magnetic field Bz is small or absent6. However, in systems in which the magnetic energy exceeds the local electron plasma pressure so that

€

βe = 8πneTe /B2 is

small, such as in solar and stellar coronae and some regions of the Earth’s magnetosphere, we show that thermal momentum transport is weak so that other mechanisms for balancing the reconnection electric field need to be explored. In a 3-D system the strong currents driven by the reconnection electric field around the x-line could potentially drive turbulence, scattering electrons to produce an electron-ion drag or anomalous resistivity to balance the reconnection electric field5, 12-14. This turbulence is typically weak or absent in 2-D because the growth rates of instabilities peak when wavevectors have components along the direction of the current, which is strongest along z. The drag Dez in Eq. (2) measures the effect of this turbulence but neither simulations nor observations have established the viability of the mechanism. The narrow current layers that result from reconnection can also break up through self-driven turbulence. The resulting turbulent transport of electron momentum away from the x-line as described by Tez could balance the reconnection electric field and act to break field lines15, 16. This idea is explored here. Of course, at larger spatial scales the ambient turbulence, which is common in astrophysical plasmas, might enable fast reconnection by breaking up the large-scale current layers that normally characterize MHD reconnection17, 18. In such models, however, the question as to what provides the dissipation required to break field lines is not addressed. The development of turbulence during simulations of 3-D reconnection and its consequences are presented in Figs. 1-4. The comparison of the time-development of the rate of reconnection between 2-D and 3-D reveals that for a system that initially has low

€

βe the rate of reconnection rises in time with the two simulations matching until the rate in the 3-D case abruptly increases above that in 2-D (Fig. 1a). Coincident with the rate increase in 3-D is an increase in the turbulent-driven drag and momentum transport (Fig. 1b). The reconnection electric field

€

Ez drives a strong current around the x-line that strengthens and narrows in time (Fig. 1a). At late time the current layer in the 3-D case abruptly broadens as the nearly laminar current layer transitions to a filamented

3

current layer (Figs. 3a, b). The sharp jump in the strength of the magnetic field perturbations (Figs. 3c, d) and the wavelength of the electric field perturbations (Figs. 3e, f) indicate that this filamentation instability is distinct from instabilities explored earlier in observations7,8 and modeling19-23. To explore the role of this turbulence in reconnection and whether it facilitates the breaking of field lines, we evaluate Ohm’s law in Eq. (2) around the x-line (Fig. 4). In 2-D the reconnection electric field is balanced by the electron inertia even at late time. In 3-D the mechanisms controlling how the magnetic field lines break evolve in time: early in time it is the electron inertia, at intermediate times the drag reaches parity with the inertia, and at late time the turbulent momentum transport dominates. The broadening of the current layer (Fig. 1a) causes the electron streaming velocity, the inertia term in Ohm’s law and the turbulent drag to decrease sharply at the x-line (Fig. 1b). By the end of the simulation the momentum transport has also decreased because of the broadening of the current layer. In larger simulation domains, where more magnetic flux is available to reconnect, the process of current layer thinning and broadening goes through several cycles before coming to an approximate balance (see the movie in the Supplementary Information). What drives the filamentation of the current shown in Fig. 3? The time-dependence of the transverse electric fields reveals that it is a right-hand circularly polarized electromagnetic wave and hence is part of the whistler/electron-cyclotron branch. The spatial correlation between the current density and transport in Fig. 2 suggests that it is driven by the current density gradient and not by the relative streaming velocity of electrons and ions. 3-D PIC simulations of narrow current layers reveal that the filamentation instability is insensitive to the ion mass and remains robust even for stationary ions (see Supplementary Information). A simple fluid description of electromagnetic waves in this regime is given by the electron MHD equations24 and the possibility that gradients in the electron current could drive instability has been previously discussed16, 25. Linearization of the electron MHD equation in the presence of a local current density gradient

€

Jez '= dJez /dy yields an instability with a peak growth rate of

€

γ ~ Jez ' /(2ne) (see Supplementary Information). Of course, a kinetic treatment will be required to fully understand this instability. What is the range of parameters (e.g., the guide field) over which the instability is important? A series of 3-D PIC simulations of narrow current layers reveal a surprising result: the guide field has a destabilizing rather than a stabilizing influence (Fig. 1c). The instability requires a finite guide field to develop, is strongest for a guide field that is around twice the reconnecting magnetic field, but remains robust down to a guide field of 0.5 of the reconnecting field for current layer widths of 0.5de with de the electron skin depth (see also the Supplementary Information). During reconnection in a 2-D system with a guide field, the width of the current layer decreases to the electron Larmor radius

€

ρe = βe1/ 2de

6, 26. Thus, in the Earth’s magnetosphere reconnection driven current layers should filament for values of

€

βe below 0.25 and guide fields around 0.5. Satellites such as Cluster and THEMIS should be able to measure the predicted asymmetry in the distribution of turbulence between the two separatrices (Fig. 2c, d). Detailed measurements of de scale current layers must await the MMS mission.

4

Filamentation of current layers in the VTF reconnection experiment at MIT has already been observed (J. Egedal, private communication). 1. Tsuneta, S. Heating and acceleration processes in hot thermal and impulsive solar flares. Astrophys. J. 290, 353-358 (1996). 2. Priest, E. R. & Forbes, T. G. Magnetic Reconnection: MHD Theory and Applications (Cambridge University Press, Cambridge, 2000). 3. Baker, D. N., Pulkkinen, T. I., Angelopoulos, V., Baumjohann, W., and McPherron, R. L. Neutral line model of substorms: Past results and present view. J. Geophys. Res. 101, 12975 (1996). 4. Yamada, M., et al. Investigation of magnetic reconnection during a sawtooth crash in a high-temperature tokamak plasma. Phys. Plasmas 1, 3269-3276 (1994). 5. Galeev, A. A. & Sagdeev, R. Z. Theory of weakly turbulent plasma. in Basic Plasma Physics (eds Galeev, A. A. & Sudan, R. N.) 677-731 (Vol. 1, North Holland Publishing Company, Amsterdam, 1983). 6. Hesse, M., Kuznetsova, M. & Hoshino, M. The structure of the dissipation region for component reconnection: Particle simulations. Geophys. Res. Lett. 29, 1563 (2002). 7. Matsumoto, H., Deng, X. H., Kojima, H. & Anderson, R. R. Geophys. Res. Lett. 30, 1326, (2003). 8. Cattell, C. et al. Cluster observations of electron holes in association with magnetotail reconnection and comparison to simulations. J. Geophys. Res. 110, A01211 (2005). 9. Ji, H., et al. Electromagnetic Fluctuations during Fast Reconnection in a Laboratory Plasma. Phys. Rev. Lett. 92, 115001 (2004). 10. Fox, W. et al. Laboratory Observation of Electron Phase-Space Holes during Magnetic Reconnection. Phys. Rev. Lett. 101, 255003:1-4 (2008). 11. Eastwood, J., Phan, T. D., Bale, S. D. & Tjulin, A. Observsations of Turbulence Generated by Magnetic Reconnection, Phys. Rev. Lett. 102, 035001:1-4 (2009). 12. Vasyliunas, V. M. Theoretical models of magnetic field line merging, 1. Rev. Geophys. Space Phys. 113, 303-336 (1975). 13. Drake, J. F. et al. Formation of electron holes and particle energization during magnetic reconnection. Science 299, 873-877 (2003). 14. Che, H., Drake, J. F., Swisdak, M. & Yoon, P. H. Electron holes and heating in the reconnection dissipation region. Geophys. Res. Lett. 37, L11105 (2010).

5

15. Kaw, P. K., Valeo, E. J. & Rutherford, P. H. Tearing modes in a plasma with magnetic braiding. Phys. Rev. Lett. 43, 1398-1401 (1979). 16. Drake, J. F., Kleva, R. G. & Mandt, M. E. Structure of thin current layers: implications for magnetic reconnection. Phys. Rev. Lett. 73, 1251-1254 (1994). 17. Lazarian, A. & Vishniac, E. Reconnection in a weakly stochastic magnetic field. Astrophys. J. 517, 700 (1999). 18. Kowal, G., Lazarian, A., Vishniac, E. & Otmianowska-Mazur, K. Reconnection in a weakly stochastic magnetic field. Astrophys. J. 700, 63-85 (2009). 19. Openheim, M. M., Vetoulis, G., Newman, D. L. & Goldman, M. V. Evolution of electron phase-space holes in 3-D. Geophys. Res. Lett. 28, 1891-1894 (2001). 20. Omura, Y., Matsumoto, H., Miyake, T. & Kojima, H. Electron beam instabilities as a generation mechanism of electrostatic solitary waves in the magnetotail. J. Geophys. Res. 101, 2685-2698 (1996). 21. McMillan, B. F. & Cairns, I. H. Lower hybrid turbulence driven by parallel currents and associated electron energization, Phys. Plasmas 13, 052104:1-13 (2006). 22. Goldman, M. V., Newman, D. L. & Pritchett, P. Vlasov simulations of electron holes driven by particle distribution from PIC reconnection simulations with a guide field. Geophys. Res. Lett. 35, L22109 (2008). 23. Che, H., Drake, J. F., Swisdak, M. & Yoon, P. H. Nonlinear Development of Streaming Instabilities in Strongly Magnetized Plasma. Phys. Rev. Lett. 102, 145004:1-4 (2009). 24. Kingsep, A. S., Chukbar & Yan’kov, Y. Y. Electron magnetohydrodynamics. in Reviews of Plasma Physics (ed Kadomtsev, B. B.), 243-288 (Vol. 16, Consultants Bureau, New York, 1990). 25. Ferraro, N. M. & Rogers, B. N. Turbulence in low-

€

β Reconnection. Phys. Plasmas 11, 4382-4389 (2004). 26. Ricci, Paolo, Brackbill, J. U., Daughton, W. & Lapenta, Giovanni Collisionless magnetic reconnection in the prescence of a guide field. Phys. Plasmas 11, 4102-4114 (2004). 27. Zeiler, A. et al. Three-dimensional particle simulations of collisionless magnetic reconnection. J. Geophys. Res.107, 1230:1-9 (2002). 28. Pritchett, P. & Coroniti, F. V. Three-dimensional collisionless magnetic reconnection in the presence of a guide field. J. Geophys. Res. 109, A01220:1-17 (2004). Supplementary Information is linked to the online version of the paper at www.nature.com/nature.

6

Acknowledgements This work has been supported by the NSF/DOE program in plasma science and by NASA through the Supporting Research and Technology Program and the Magnetospheric Multiscale Mission Science Team. Computations were carried out in part at the National Energy Research Scientific Computing Center. Author Contributions All of the authors made significant contributions to this work. H. C. carried out the particle simulations of reconnection. M. S. carried out simulations of isolated electron current layers. H. C., J. F. D. and M. S. analyzed the data from the simulations. All of the authors discussed the results and commented on the paper. Author Information The authors declare that they have no competing financial interests. Correspondence and requests for materials should be addressed to H. C. (email: Haihong.Che@colorado.edu).

7

Figure 1: The time evolution of reconnection and the development of turbulence. Particle-in-cell simulations using the p3d code27 are performed in doubly periodic 2-D and 3-D geometries starting with two force-free current sheets. The reconnection magnetic field is

€

Bx /B0 = tanh[(y − Ly /4) /w0]− tanh[(y − 3Ly /4) /w0]−1, where the current layer width w0 = 0.5di and the box size is

€

Lx × Ly × Lz = 4di × 2di × 8di. The electron and ion temperatures, Te/micA

2 =Ti/micA2 = 0.04, and density n0 are initially

uniform. The initial out-of-plane “guide” field Bz/B0 is 5.0 outside of the current layers and increases within the current layers so that the total magnetic field B is a constant. The cyclotron time is

€

Ωi−1 = mic /eB0 , the Alfvén speed is

€

cA = B0 / 4πmin0 and the ion inertial length is

€

di = cA /Ωi. The electron mass is 0.01mi and the velocity of light c = 20cA. The 3-D spatial grid consists of 512 x 256 x 1024 cells with 20 particles per cell while in 2-D it is 2048 x 1024 with 100 particles per cell. Reconnection is initiated with a large-scale magnetic perturbation. In (a) are the rates of reconnection <Ez> and the half-widths of the z-averaged electron current layer at the x-line (see Fig. 2a) in 3-D (solid) and 2-D (dashed) and in (b) are the dominant components of Ohm’s law, the electron inertia

€

−(me /e)∂ < vez > /∂t (dot-dashed orange), the drag Dz (dashed blue), the

8

y-directed turbulent momentum transport, Tyz (solid red), and the thermal momentum transport,

€

− ∇ ⊥• <

P ez > / < ne > e (triple dots-dashed green). In 3-D

simulations with four times the initial electron and ion temperatures (not shown), no turbulence develops and thermal momentum transport balances <Ez>. Finally, in (c) are the magnetic field variances

€

< δBx2 > versus time from a series

of 3-D simulations of narrow current layers (no reconnection) with various guide fields Bz/B0 (5.0 in black, 2.5 in green, 1.0 in blue and 0.5 in red) and stationary ions, where

€

Ωe−1 = mec /eB0. The simulation domains

are

€

Lx × Ly × Lz =10de ×10de × 80de and all but the dot-dashed red have current layer widths of de. All but the guide field of 0.5 break up but reducing the layer width in this case from 1.0de to 0.5de (dot-dashed red) again leads to break-up. Current layers with Harris equilibria also break up (see the Supplementary Information).

9

Figure 2: The geometry of magnetic reconnection at late time. In the x-y plane at

€

t = 3.75Ωci−1 from the 3-D simulation of Fig. 1. In (a) the reconnected magnetic

field lines (averaged over z). In (b) the field line motion toward and away from the x-line induces an electric field that produces the parallel (to B) electric field

€

< E|| > that drives the intense electron current layer around the x-line in (c). The irregular structure of the current layer indicates that it is turbulent (see also Fig. 3). The filamentation of the current layer transports electron momentum pez away from the center of the current layer. The rate of momentum transport Tyz is shown in (d). The positive peak in

€

∇ • T ez with negative values upstream

demonstrates that momentum is being transported away from the x-line. The turbulence also produces a net electron-ion drag Dz shown in (e). Note that the spatial structure of the drag is very different from the turbulent momentum transport. It does not overlap with the region of high current density and therefore cannot be represented as an anomalous resistivity.

10

Figure 3: The filamentary structure of the electron current layer. In the y-z plane in a cut through the x-line at

€

t = 3.0Ωci−1 (a, c, e) and

€

t = 3.75Ωci−1 (b, d, f) are the

electron current jez (a, b), the magnetic field perturbations

€

δBx in (c, d) and the electric field perturbations

€

δEx in (e, f). The filamentation instability onsets sharply at

€

t = 3.5Ωci−1. This instability did not appear in earlier simulations13,14, 28

because the computational domains along z were too short to capture its long wavelength parallel to B (in Ref. 28 the computational domain along z based on the upstream density was only 2.86di). The measured phase speed of the filaments is close to the electron drift speed, suggesting that the coupling to the ions is weak.

11

Figure 4: Breaking magnetic field lines: the dominant components of Ohm’s law. At

€

t = 3.0Ωci−1,

€

t = 3.35Ωci−1 and

€

t = 3.75Ωci−1 from the 3-D simulation and

€

t = 3.75Ωci−1 from the 2-D simulation, the dominant contributions to Ohm’s law in

cuts along the inflow direction (y) through the x-line. Shown are <Ez> (solid black), -<vey><Bx>/c (dashed black), electron inertia (dot-dashed black), the thermal momentum transport

€

−(1/ < ne > e)∂ < Peyz > /∂y (dot-dashed green), the drag Dz (solid blue) and turbulent transport Tyz (dashed red). The solid grey line shows the sum of all of the contributions to Ohm’s law, which should lie on top of <Ez>. In 2-D the electron inertia continues to balance the reconnection electric field at late time, a solution that is not consistent with steady reconnection. In 3-D the turbulent drag and inertia dominate at

€

t = 3.0Ωci−1. By

€

t = 3.35Ωci−1 the turbulent drag and the rate of reconnection both sharply

increase, suggesting a causal relation. By

€

t = 3.75Ωci−1 the current layer is

becoming filamentary and turbulent momentum transport completely dominates force balance at the x-line. Momentum is transported upstream away from the x-line, producing a positive Tyz at the x-line and negative values upstream corresponding to a momentum transfer and not a momentum sink. At the x-line the drag drops sharply as the dynamics of the filaments dominates.