1
A current filamentation mechanism for breaking
magnetic field lines during reconnection
H. Che1, J. F. Drake2, M. Swisdak2
1
University of Colorado, Boulder, Colorado
2
University 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 3D 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  
1  
+ ∇ • p e v ez = −en e E z + je × B − ∇ • Pez ,
(1)
z
∂t
c


where pe is the momentum density, j e is the electron current density and



Pez = Pe • zˆ with Pe 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, 





∇ ⊥ • < Pez > me ⎛ ∂ < v ez >
1 

⎞
< E z >= − < v e⊥ > × < B⊥ > z −
−
+ < v e⊥ > •∇ ⊥ < v ez >⎟ + Dez + ∇ ⊥ • Tez
⎜
⎠
c
< ne > e
e ⎝ ∂t
where the brackets denote an average over z, δf = f − < f > for any f, the subscript ⊥
denotes the x-y plane,
⎞

1
e
−1
 ⎛
< δpe⊥⎜δv ez −
δAz ⎟ >
Dez =
< δn eδE z > and Tez = −
mec
< ne >
€
⎝
⎠
€ < ne > e
are the turbulent drag and momentum transport, respectively, and δAz is the vector


potential for δB⊥ . This representation for Tez is valid only in the strong guide field limit


(see the Supplementary
€ Information). In a 2-D system Dez and Tez 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 Pe 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 β e = 8πn e Te /B 2 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 selfdriven 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 β e 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 E z 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 J ez '= dJ ez /dy yields an instability with a peak growth
rate of γ ~ J ez ' /(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 ρ e = β1/e 2 de 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).
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5
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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:
[email protected]).
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) /w 0 ] − tanh[(y − 3Ly /4) /w 0 ] − 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/micA2 =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 Ω−1
i = m ic /eB0 , the Alfvén speed is
c A = B0 / 4πm i n 0 and the ion inertial length is di = c A /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 2D (dashed) and in (b) are the dominant components of Ohm’s law, the electron
inertia −(me /e)∂ < v ez > /∂t (dot-dashed orange), the drag Dz (dashed blue), the
€
€
€
8
€
y-directed turbulent momentum

 transport, Tyz (solid red), and the thermal
momentum transport, −∇ ⊥ • < Pez > / < n e > 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 Ω−1
e = m e c /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 = 3.75Ω−1
ci 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 ∇ • Tez 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
€ with the region of high current density and
transport. It does not overlap
therefore cannot be represented as an anomalous resistivity.
10
Figure 3: The filamentary structure of the electron current layer. In the y-z plane
−1
in a cut through the x-line at t = 3.0Ω−1
ci (a, c, e) and t = 3.75Ω ci (b, d, f) are the
electron current jez (a, b), the magnetic field perturbations δBx in (c, d) and the
electric field perturbations δE x in (e, f). The filamentation instability onsets
sharply at t = 3.5Ω−1
. This instability did not€appear in earlier simulations13,14, 28
ci€
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.
−1
−1
At t = 3.0Ω−1
ci , t = 3.35Ω ci and t = 3.75Ω ci from the 3-D simulation and
t = 3.75Ω−1
ci 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/ < n e > 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
€ a solution that is not consistent with steady
electric field at late time,
reconnection. In 3-D the turbulent drag and inertia dominate at t = 3.0Ω−1
ci . By
−1
t = 3.35Ωci the turbulent drag and the rate of reconnection both sharply
increase, suggesting a causal relation. By t = 3.75Ω−1
ci 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.