Download The Hall fields and fast magnetic reconnection

PHYSICS OF PLASMAS 15, 042306 共2008兲
The Hall fields and fast magnetic reconnection
J. F. Drake,1,a兲 M. A. Shay,2 and M. Swisdak3
1
University of California, Berkeley, California 94720, USA
University of Delaware, Newark, Delaware 19716, USA
3
University of Maryland, College Park, Maryland 20742, USA
2
共Received 21 November 2007; accepted 3 March 2008; published online 15 April 2008兲
The results of large-scale, particle-in-cell simulations are presented on the role of Hall electric and
magnetic fields on the structure of the electron dissipation region and outflow exhaust during the
collisionless magnetic reconnection of antiparallel fields. The simulations reveal that the whistler
wave plays the key role in driving the electrons away from the magnetic x-line. Further downstream
the electron outflow exhaust consists of a narrow super-Alfvénic jet, which remains collimated far
downstream of the x-line, flanked by a pedestal whose width increases monotonically with
increasing distance downstream. The open outflow exhaust, which is required for fast reconnection
in large systems, is driven by the Hall electric and magnetic fields. Finally, it is the whistler that
ultimately facilitates fast reconnection by diverting the electrons flowing toward the current layer
into the outflow direction and thereby limiting the length of this layer. The results are contrasted
with reconnection in an electron-positron plasma where the Hall fields are absent. The consequence
of the expanding outflow exhaust is that, consistent with recent observations, the extended
super-Alfvénic electron outflow jet carries a smaller and smaller fraction of the outflowing electrons
with increasing distance downstream of the x-line. The results suggest that the structure of the
electron current layer and exhaust in simulations might be sensitive to boundary conditions unless
the simulation boundary along the outflow direction is sufficiently far from the x-line. © 2008
American Institute of Physics. 关DOI: 10.1063/1.2901194兴
I. INTRODUCTION
Magnetic reconnection drives the release of magnetic
energy in explosive events such as disruptions in laboratory
experiments, magnetic substorms in the Earth’s magnetosphere, and flares in the solar corona. The rate of reconnection is controlled by a narrow boundary layer, the “dissipation region,” where strong currents are driven and dissipative
processes enable magnetic field lines to reconnect. In the
Sweet–Parker model of reconnection the plasma resistivity ␩
provides the dissipation required to break magnetic field
lines and the tension associated with newly reconnected
magnetic field lines drives plasma away from the x-line at
the Alfvén speed cA.1,2 Since the 1950s, however, it was recognized that the long and narrow dissipation region that develops in the Sweet–Parker model throttles reconnection and
the resulting rate of reconnection is inconsistent with the
observed fast energy release. Computer simulations of reconnection based on the resistive MHD equations with constant
resistivity confirmed the macroscopic elongation of the current layer of the Sweet–Parker model and the resultant slow
rate of reconnection.3
On the other hand, in the high temperature plasmas
where magnetic reconnection is active, the width of the dissipation region in the Sweet–Parker model ␦SP becomes
comparable to kinetic scales such as the ion inertial length
c / ␻ pi, where the MHD description of the plasma dynamics
breaks down. At these small spatial scales electron and ion
a兲
Permanent address: University of Maryland, College Park, Maryland
20742.
1070-664X/2008/15共4兲/042306/10/$23.00
motion decouples and the Alfvén wave which drives the outflow from the x-line in the MHD model is replaced by the
whistler wave 共or the kinetic Alfvén wave in the case of
reconnection with a guide field兲. The dynamics and structure
of the outflow jet during reconnection is a complex nonlinear
problem. Nevertheless, Fig. 1 illustrates how the basics of
the outflow can be understood from a simple wave picture.
The newly reconnected magnetic field line that drives the
outflow from the x-line can be treated as a half-wavelength
of a periodic standing wave. In the MHD description this
standing wave is a conventional Alfvén wave and from the
linear theory of the Alfvén wave the velocity vx that results
from the relaxation of the bent magnetic field is simply cAx
= Bx / 冑4␲min, with Bx the “upstream” magnetic field. This
velocity is the usual MHD condition on the outflow from the
x-line during reconnection. In the Hall reconnection model
with no guide field the Alfvén wave is replaced by the whistler wave in the region within an ion inertial scale of the
x-line. In this case the magnetic field is frozen-in 共or at least
partially frozen-in兲 to the electrons carrying the current supporting the magnetic field of the standing wave. Rather than
simply expanding outward, the in-plane magnetic field rotates out of the plane producing a Bz whose magnitude is
nearly equal to Bx. The electron current in the x direction
supporting Bz carries the field line away from the x-line.
When the width of the current channel is the electron skin
depth de = c / ␻ pe, which is universally seen in collisionless
reconnection simulations, the magnitude of the outflow velocity supported by the whistler is given by the electron
Alfvén speed cAe = Bx / 冑4␲men. In a Hall MHD model it was
shown that fast reconnection develops if and only if the
15, 042306-1
© 2008 American Institute of Physics
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042306-2
Phys. Plasmas 15, 042306 共2008兲
Drake, Shay, and Swisdak
Vin
Vout
y
x
FIG. 1. Schematic of the magnetic field lines and associated plasma flows
around a magnetic x-line undergoing reconnection. The newly reconnected
field lines can be represented by a half period of a simple standing wave to
understanding how these newly reconnected field lines relax their stress and
accelerate the plasma away from the x-line.
plasma dynamics near the x-line is controlled by dispersive
waves, either the whistler or kinetic Alfvén wave.4 In the
GEM Reconnection Challenge various simulation models
produced essentially identical reconnection rates as long as
the Hall term and the associated whistler dynamics were included in the model.5 Similarly, the rate of collisionless reconnection was found to be independent of the electron
mass,6,7 which is a consequence of the electron flux away
from the x-line ⌫ = cAede being independent of mass; for
smaller me the width of the outflow channel narrows but the
outflow velocity goes up to compensate. Finally, in the Hall
reconnection model the dissipation region is dramatically
shorter than in the MHD model and produces fast reconnection independent of the system size.8,9
There remains some disagreement about the critical role
that Hall physics plays in facilitating fast reconnection. It has
been suggested that “ion kinetics” alone can drive fast
reconnection.10 The authors found that even after eliminating
the Hall term in Ohm’s law, magnetic reconnection in a hybrid model remained fast. The authors therefore concluded
that “ion kinetics alone can give rise to fast reconnection”
and that the “quadratic property of whistlers is not critical to
fast reconnection in the kinetic regime.” However, in the
simulations that led to these conclusions the authors used a
spatially localized resistivity. It is well known that even in
the resistive MHD model a localized resistivity produces fast
reconnection.11 In recent hybrid simulations without the Hall
term and without a localized resistivity the rate of reconnection remained small.12
Recent kinetic simulations in open boundary configurations have also called into question the fundamental tenets of
the Hall reconnection model and therefore the current explanation of fast reconnection seen in nature.13,14 In these simulations the electron out-of-plane current layer stretches along
the outflow direction, leading to an electron flow bottleneck
which causes the rate of reconnection to drop. The development of secondary islands eventually halts the lengthening of
the current layer and the associated drop in the rate of reconnection. On the other hand, the authors see periods of steady
reconnection with no stretching of the electron current layer
and no secondary island formation.14 The authors conclude
that the Hall electric and magnetic fields do not play a sig-
nificant role in the dynamics of the electron current layer and
do not facilitate fast reconnection in large systems. In contrast, kinetic simulations carried out in very large systems to
test the assertion that the limited simulation domains in earlier models were responsible for fast reconnection, reproduced the earlier results that reconnection was fast and independent of the system size.15
In the present paper we explore more carefully the dynamics of the electron current layer and the role of the Hall
fields in controlling the structure of this layer and the outflow
exhaust during collisionless reconnection. The results from
large-scale particle-in-cell 共PIC兲 simulations demonstrate
that the whistler wave controls the outflow of electrons from
the x-line and that the opening of the electron outflow jet
from the x-line, necessary for fast reconnection in large systems, results from fast, nearly field-aligned propagation of
the Hall electric and magnetic field downstream of the x-line.
The critical role that the Hall fields play in opening the electron outflow exhaust and the rapid propagation of these fields
downstream suggests that care must be taken in any finite
simulation to ensure that the proximity of the boundary does
not adversely affect the structure of the Hall fields.
II. COMPUTATIONAL MODEL
Our simulations are performed with the particle-in-cell
code p3d.16 The electromagnetic fields are defined on grid
points and advanced in time with an explicit trapezoidalleapfrog method using second-order spatial derivatives. The
Lorentz equation of motion for each particle is evolved by a
Boris algorithm where the velocity v is accelerated by E for
half a time step, rotated by B, and accelerated by E for the
final half time step. To ensure that ⵱ · E = 4␲␳ a correction
electric field is calculated by inverting Poisson’s equation
with a multigrid algorithm. Although the code permits other
choices, we work with fully periodic boundary conditions.
The results are presented in normalized units: the magnetic field to the asymptotic value of the reversed field, the
density to the value at the center of the current sheet minus
the uniform background density, velocities to the Alfvén
speed vA, lengths to the ion inertial length di, times to the
inverse ion cyclotron frequency ⍀−1
ci , and temperatures to
mivA2 . We consider a system periodic in the x − y plane where
flows into and away from the x-line are parallel to ŷ and x̂,
respectively. The reconnection electric field is parallel to ẑ.
The initial equilibrium consists of two Harris current sheets
superimposed on a ambient population of uniform density.
The reconnection magnetic field is given by Bx = tanh关共y
− Ly / 4兲 / w0兴 − tanh关共y − 3Ly / 4兲 / w0兴 − 1, where w0 and Ly are
the half-width of the initial current sheets and the box size in
the ŷ direction. The electron and ion temperatures are initially uniform. The initial density profile is the usual Harris
form plus a uniform background of 0.2. The simulations presented here are two-dimensional, i.e., ⳵ / ⳵z = 0. Reconnection
is initiated with a small initial magnetic perturbation that
produces a single magnetic island on each current layer.
We present the results of simulations with Te = 1 / 12, Ti
= 5 / 12, and four mass ratios, mi / me = 1, 25, 100, and 400.
Data from all but the mass-ratio unity case were presented
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Phys. Plasmas 15, 042306 共2008兲
The Hall fields and fast magnetic reconnection
(a)
10
jez
8
6
-1
4
-2
30
40
45
50
40
45
50
(b)
10
8
6
4
30
35
x/di
10
Bz
y/di
8
0.2
6
0.0
4
-0.2
30
35
40
45
50
x/di
(d)
0.05
o
0.00
-0.05
Bz
We examine first the dynamics of the electric and magnetic fields and the associated motion of the electrons just
downstream of the x-line. Arguments about the role of the
whistler wave and its role in reconnection are based on
simple arguments based on one-dimensional waves such as
shown in Fig. 1. The structure of the x-line that develops
during reconnection is two-dimensional and possibly even
three-dimensional. Further, momentum transport in collisionless plasmas produces substantial dissipation 共the offdiagonal pressure tensor contribution to Ohm’s law兲,7,15
which can also affect the dynamics of the region around the
x-line. We now seek to establish more quantitatively the role
played by the dynamics of the whistler in the region around
the x-line based on direct analysis of simulation data.
Shown in Fig. 2 are plots of 共a兲 the out-of-plane electron
current jez, 共b兲 the contours of the flux function, and 共c兲 the
out-of-plane magnetic field Bz in a blowup around one of the
x-lines from the me = 1 / 400 simulation. The electron current
around the x-line is flowing in the negative z direction so Bx
is positive above the current layer and negative below. In the
conventional whistler picture the electrons flowing in the
positive z direction drag the newly reconnected field lines in
the positive z direction out of the plane producing the quadrupole field as shown in 共c兲.17,18 We now put this picture to
the test.
In Fig. 3 we show the profile of Bx along a cut through
the x-line at x = 37.4. The electron current layer causes the
magnetic field to increase sharply to around 0.2, after which
Bx increases much more gradually, eventually approaching
unity. If the whistler picture is correct, this magnetic field
just upstream of the current layer should rotate out of the
plane until it is pointing in the z direction and the magnitude
of this z directed field should take on a value that is close to
the upstream Bx, in this case 0.2. For steady-state reconnection the rotation of the field occurs as the plasma convects
downstream. Shown in Fig. 3 is a vertical cut of Bz at the end
of the current layer. Bz peaks at around 0.2. To further check
that the upstream magnetic field Bxup rotates into the z direction as in a simple whistler, we show in Fig. 2共d兲 a hodogram
of Bx and Bz plotted along the top edge of the current layer
35
x/di
(c)
III. DYNAMICS OF THE INNER REGION
1
0
y/di
previously, where it was shown that the rates of reconnection
at late time were steady and close to 0.14 for all three mass
ratios and independent of the size of the computational domain 共for domain sizes Lx ⫻ Ly greater than 51.2⫻ 25.6兲 and
w0.15 All of the data shown is taken during times when the
rate of reconnection is steady. For mi / me = 25, the simulation
is on a computational domain of 204.8⫻ 102.4 with a grid
scale ⌬ = 0.05, w0 = 1.5 and the speed of light c = 15. For
mi / me = 100, the computational domain is 102.4⫻ 51.2 with
a grid ⌬ = 0.025, w0 = 1.1 and c = 20. For mi / me = 400, the
computational domain is 51.2⫻ 25.6 with ⌬ = 0.0125, w0
= 0.6 and c = 40. The mass unity run was on a domain of
200⫻ 100 with a grid of 0.195, had Te = Ti = 0.25, c = 5, w0
= 2.0 and a steady rate of reconnection of around 0.2.
y/di
042306-3
-0.10
-0.15
-0.20
-0.05 0.00
y = 6.54
0.05
0.10
0.15
0.20
Bx
FIG. 2. 共Color online兲 Blowup around the x-line of 共a兲 the electron outplane current Jez, 共b兲 the in-plane magnetic field lines 共Bx positive above the
current layer and negative below兲, 共c兲 out-of-plane magnetic field Bz and 共d兲
a hodogram of Bx and Bz along the white line in 共c兲, which lies just above
the electron current layer. The 0 in 共d兲 marks beginning of the hodogram
共close to the x-line兲.
along the horizontal line in Fig. 2共c兲 共37.4⬍ x ⬍ 45.0, y
= 6.54兲. The “0” marks the starting point just upstream of the
x-line where Bz = 0 and Bx has its maximum value. The magnetic field rotates in the right-hand sense 共By is positive兲 as in
0.8
0.6
0.4
Bz , x=40.0
0.2
0.0
-0.2
-0.4
Bx , x=37.4
-0.6
-0.8
4
5
6
7
8
y/di
FIG. 3. From the simulation shown in Fig. 2, Bx along a vertical cut through
the x-line at x = 37.4 and Bz along a vertical cut at x = 40.0, the downstream
edge of the out-of-plane current layer.
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042306-4
Phys. Plasmas 15, 042306 共2008兲
Drake, Shay, and Swisdak
Ex =
FIG. 4. 共Color兲 From the simulation shown in Fig. 2. In 共a兲 Ex close to the
x-line. In 共b兲 cuts through the x-line in the outflow direction of the x components of Ohm’s law: Ex in black, vezBy / c in red, −共1 / ne兲 ⳵ peyx / ⳵y in blue,
2
/ ⳵x in green. In 共c兲 cuts of
−共1 / ne兲 ⳵ pexx / ⳵x in yellow, and −共1 / 2e兲 ⳵ mevex
the field line velocity, vBx = −Ez / By in black and vBz = Ex / By in red.
a simple whistler. Further, as expected from the cuts shown
in Fig. 3, the maximum amplitudes of Bx and Bz are nearly
equal.
In an ideal whistler the out-of-plane electron flow vez is
produced by an electric field Ex = vezBy / c, which would be
positive 共negative兲 to the right 共left兲 of the x-line in Figs. 1
and 2. In Fig. 4共a兲 we show a blowup of Ex around the x-line
averaged over the time interval ⍀it = 41.0 to 41.2 from the
simulation of Fig. 2. Although noisy, Ex has the expected
symmetry. Ex is also shown in the black line in Fig. 4共b兲
along a cut through the x-line along the outflow direction.
The ridges of large Ex above and below the symmetry line in
Fig. 4共a兲 correspond to the x component of the traditional
Hall electric field, which is perpendicular to the magnetic
field and maps the magnetic separatrices during collisionless
reconnection.19
To understand the mechanism for the development of Ex,
it is helpful to look at the field line velocity vB in the vicinity
of the x-line. This velocity is notoriously difficult to define,
especially in a region where the parallel electric field E储
= b · E is nonzero. A line-preserving velocity cE ⫻ B / B2 can
be defined only if20
b ⫻ ⵱ ⫻ 共bE储兲 = 0.
共1兲
Because of the symmetry of antiparallel reconnection, this
relation is satisfied along the outflow symmetry line and the
components of the field-line velocity are given by
共−cEz / By , 0 , cEx / By兲. The relation
vBz
By
c
共2兲
means that Ex is simply the motional electric field associated
with the out-of-plane motion of By. The nonzero velocities
vBx 共black兲 and vBz 共red兲 are shown along the outflow direction in Fig. 4共c兲. The x component of the velocity of the field
line diverges at the x-line, which is required since the continuity of the flow of magnetic flux toward and away from the
x-line must be preserved, which along the outflow direction
requires that vBxBy remain finite. The out-of-plane motion of
the field lines vBz is nearly constant in a region around the
x-line 兵note that the apparent singularity close to the x-line is
an artifact of the noise in Ex in the simulation 关black curve in
Fig. 4共b兲兴其 since the symmetry of the various terms in Eq. 共3兲
for Ex below requires that Ex → 0 where By → 0. The key
result of Fig. 4共c兲 is that vBz is much less than the local
out-of-plane electron velocity vez ⬃ 8. The difference between these two velocities implies that the dynamics downstream of the x-line does not correspond to an ideal whistler,
for which the particle and field-line velocity would be equal.
The driving mechanism for Ex in the complex environment of the x-line can be understood by writing the x component of the electron equation of motion along the symmetry line of the outflow, which in steady state becomes
1
1 ⳵
1 ⳵
1 ⳵
2
peyx −
pexx −
mevex
.
Ex = vezBy −
c
ne ⳵ y
ne ⳵ x
2e ⳵ x
共3兲
The source terms for Ex are plotted along with Ex in Fig. 4共b兲
with vezBy / c in red, −共1 / ne兲 ⳵ peyx / ⳵y in blue,
2
/ ⳵x in green.
−共1 / ne兲 ⳵ pexx / ⳵x in yellow, and −共1 / 2e兲 ⳵ mevex
The dominant source terms on the right-hand side of Eq. 共3兲
are the electron motional electric field vezBy / c and the offdiagonal pressure tensor contribution, which corresponds to
a transport of x-directed momentum along y. The sum of all
of the contributions on the right-hand side of Eq. 共2兲 approximately matches Ex.
To understand the dynamics of the outflowing electrons,
consider their motion in a frame moving with their velocity
vez in the z direction. In this frame the electric field Ex⬘ seen
by the electrons is Ex⬘ = Ex − vezBy / c, which points toward the
x-line and accelerates the electrons outward from the x-line.
In the absence of inertia or other retarding force in the x
direction, Ex would have to increase until the field-line velocity in the z direction matched that of the electrons, forcing
Ex⬘ → 0. However, the y-directed flux of the electron momentum density nmevex, which is given by peyx 关convert Eq. 共3兲
into a momentum transport equation by multiplying through
by ne兴, effectively acts as a drag on vex. This drag reduces Ex
so that the electrons can move faster than the field lines in
the out-of-plane direction.
There has been discussion in the recent literature on the
possible role that Ex could play in trapping the electrons in
the vicinity of the x-line since Ex points outwards from the
x-line.14 However, it is clear that the effective electric field
seen by the moving electrons Ex⬘ is large and points inwards.
Thus, the bulk of the electrons will experience a repulsive
rather than a trapping force.
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Phys. Plasmas 15, 042306 共2008兲
The Hall fields and fast magnetic reconnection
Bz,out
␦ y / de
nout
vex
cAex
␦x
vey
veyc
25
0.35
0.36
3.3
0.14
4.3
4.5
7.2
0.36
0.4
100
400
0.23
0.20
0.21
0.18
2.8
3.0
0.11
0.11
6.5
10.0
7.1
11.4
3.5
2.0
0.55
0.88
0.52
0.75
The electron outflow velocity vex develops as the
z-directed flow of the electrons is rotated by the reconnected
magnetic field By into the x direction. Since the electron
outflow velocity also produces the current Jex = −nevex and
therefore the magnetic field Bz, the electron outflow velocity
is constrained by Bz, i.e., the rotation is a collective wave
phenomenon and not simply single particle motion. Since we
have already shown from the simulation data that Bz and Bx
are linked we establish the usual Alfvénic relation for the
electron outflow velocity, vex = cAexde / ␦y, where cAex is the
electron Alfvén speed based on Bx and ␦y is the half width of
the electron current layer. Since the current layer width
scales with de 共as long as the electron upstream ␤ is less than
unity兲, the electron outflow velocity scales with the electron
Alfvén speed. This was demonstrated earlier in PIC and
finite-electron-mass hybrid simulations.18,21 In the present
simulation vex = 10.0 at x = 40.0, which with a density of 0.12
and ␦y = 6.54− 6.4= 0.14 from Ampère’s law yields a magnetic field of 0.17, consistent with the amplitude of Bz.
In Table I we summarize the data from the proximity of
the x-line from the mi / me = 400 simulation along with similar
data from mi / me = 100 and 25. Shown are the values of the
magnetic field just upstream 共a distance ␦y兲 of the electron
current layer Bx,up, the out-of-plane magnetic field Bz,out
evaluated just downstream of the electron out-of-plane current layer 共where By reaches a plateau15 and a distance ␦y
above the symmetry line兲, the density in the outflow channel
nout, the electron outflow velocity vex and the electron Alfvén
speed cAex based on Bx,up. In all cases the magnetic fields
Bx,up and Bz,out match, consistent with the rotation of the Bx,up
field into the out-of-plane direction although the value of
both of these fields decreases with decreasing electron mass.
The electron outflow velocity very closely tracks the electron
Alfvén speed. These results confirm the idea that the whistler
drives the electron outflow from the x-line just as the Alfvén
wave drives the Alfvénic outflow downstream of the ion dissipation region.
IV. OPENING OF THE ELECTRON
OUTFLOW EXHAUST
Fast reconnection requires a mechanism for efficiently
breaking the frozen-in condition and for opening the outflow
exhaust downstream from the x-line so that the flow into and
y/di
vez
18
16
14
12
10
8
6
4
2
0
18
16
14
12
10
8
6
18
16
14
12
10
8
6
18
16
14
12
10
8
6
90
100
6
4
2
0
-2
-4
-6
70
80
90
100
x/di
v(ExB)x
4
2
0
-2
-4
70
80
90
100
x/di
1.5
Ey
1.0
0.5
0.0
-0.5
-1.0
-1.5
60
(e) 20
80
vex
60
(d) 20
70
x/di
60
(c) 20
8
6
60
y/di
Bx,up
18
16
14
12
10
8
6
(b) 20
y/di
mi / me
(a) 20
y/di
TABLE I. Parameters near the x-line: the mass ratio mi / me, the magnetic
field just upstream of the electron current layer Bx,up 共a distance ␦y兲, the
magnetic field just downstream of the electron out-of-plane current layer
Bz,out 共also evaluated a distance ␦y above the symmetry line兲, the half-width
along y of the electron out-of-plane current layer ␦y 共normalized to the
electron skin depth de to emphasize its well-known scaling with this parameter兲, the electron outflow velocity vex, the electron Alfvén speed based on
Bx,up, the half-length along the outflow direction ␦x of the electron current
layer, the electron inflow velocity vey, and the inflow velocity expected from
flow continuity veyc.
y/di
042306-5
70
80
90
100
x/di
0.4
Bz
0.2
0.0
-0.2
-0.4
60
70
80
90
100
x/di
FIG. 5. 共Color online兲 Blowup around the x-line of 共a兲 the electron outplane velocity vez, 共b兲 the electron outflow velocity vex, 共c兲 the x component
of the E ⫻ B velocity v共E⫻B兲x and the Hall fields Ey and Bz from a simulation
with mi / me = 100.
out of the dissipation region does not act as a bottleneck,
constraining the rate of reconnection. The momentum transport associated with the off-diagonal pressure tensor easily
balances the reconnection electric field in collisionless
plasma and therefore does not appear to constrain the rate of
reconnection.7 The opening of the outflow exhaust downstream of the x-line was central to Petschek’s original idea
for fast reconnection in the MHD model. In the case of collisionless reconnection this requirement extends to both species and was the key ingredient in the argument that Hall
reconnection remains fast even in very large systems.8,9
Whether the electron outflow exhaust opens up downstream
is also central to the recent discussion of whether collisionless reconnection in large systems remains fast or is constrained as in the MHD model.13–15 Thus, the question is
whether the electron outflow jet opens up downstream of the
x-line and whether it is the Hall generated fields or another
mechanism that opens the exhaust.
In Fig. 5 we show data in the vicinity of the x-line from
the simulation with mi / me = 100, including vez, vex, the x
component of the E ⫻ B drift v共E⫻B兲x and the Hall fields Ey
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042306-6
Phys. Plasmas 15, 042306 共2008兲
Drake, Shay, and Swisdak
y/di
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1.0
18
16
14
12
10
8
6
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0.0
-0.5
-1.0
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70
80
90
100
x/di
(b)
1.0
jex
0.5
0.0
-0.5
8
10
12
14
16
y/di
FIG. 6. 共Color online兲 From the same simulation as in Fig. 5 the electron
outflow current jex in 共a兲. The vertical lines 共in the y direction兲 at intervals of
4di mark the location of cuts of jex that appear in 共b兲.
and Bz. The out-of-plane electron velocity forms a localized
current layer around the x-line. The electron velocity vex
consists of a narrow super-Alfvénic jet flowing away from
the x-line13–15 flanked by a slower-moving pedestal that becomes broader with increasing distance downstream. The
structure of this outflow jet is shown in greater detail in Fig.
6共a兲, where vertical 共y direction兲 lines at intervals of 4di in a
plot of jex mark the location of cuts of jex that appear in Fig.
6共b兲. The outflow exhaust consists of the central high-speed
jet flanked by a widening pedestal. The reversal of jex at the
flanks of the pedestal marks the magnetic separatrix.
We now explore in detail what controls the structure of
the outflow exhaust and in particular the broad pedestal. In
Fig. 5 the spatial structure of the outflow exhaust in 共b兲
matches the x component of the E ⫻ B drift in 共c兲. In Fig.
7共a兲 we show the x components of the perpendicular electron
velocity v⬜ex 共solid兲 and the E ⫻ B drift 共dashed兲 along vertical cuts through the outflow exhaust at x = 65.0 in Fig. 5.
The two velocities are essentially identical except at the location of the jet in the center of the current sheet. The electrons making up the jet decouple from the local magnetic
field and stream faster than the local field lines.15 Thus,
within the jet the electron motion is not given by the local
E ⫻ B drift. In Fig. 7共b兲 we show similar cuts of Ey and Bz.
Except around the symmetry line at y = 12.8di the Hall fields
Ey and Bz within the exhaust typically exceed their counterparts Ez ⬃ 0.14 and By ⬃ 0.07 by a large margin so the E
⫻ B drift along x is dominated by the Hall fields.
Thus, the opening of the electron outflow exhaust downstream seen in Fig. 5 is produced by the Hall fields Ey and
Bz. As discussed earlier, these fields are generated by the
cross field motion of electrons around the x-line and rapidly
propagate parallel to the magnetic field away from the x-line.
As can be seen in Fig. 2, the peaks of both jez and Bz closely
map the magnetic separatrix. The proximity of the upstream
FIG. 7. Along a vertical cut at x = 65.0 from the simulation in Fig. 5, we
show in 共a兲 the x components of the perpendicular electron velocity vēx
共solid兲 and the E ⫻ B drift 共dashed兲 and in 共b兲 the Hall fields Ey and Bz. The
dotted lines mark the separatrix locations.
edge of the Hall fields to the separatrix can be seen more
clearly in cuts of Ey and Bz in Fig. 7, where the vertical
dotted lines mark the location of the separatrices. We can
obtain an estimate of the parallel propagation speed of the
Hall fields by examining the separation between the separatrix and the Hall field structure downstream: this separation
increases with distance downstream at a rate that is controlled by the propagation speed of the fields—higher propagation speeds yield smaller separations. In Fig. 8 we show a
2D plot of Ey around the x-line and separatrix from the
mi / me = 400 run. This run produces a larger separation in
velocities between dispersive waves and the Alfvén wave
than the mass-ratio 100 and 25 runs. The separatrix, shown
as a white line in this figure, lies slightly upstream from the
boundary layer formed by Ey. The spatial separation ⌬y of
the separatrix and the first peak in Ey along the left-hand
boundary of Fig. 8 is 0.39, which since Bx ⯝ 0.75, corresponds to a magnetic flux separation of 0.29. The rate of
reconnection at the time shown is steady and is close to
FIG. 8. 共Color online兲 The structure of the Hall field Ey around the x-line
and separatrix 共white line兲 from the simulation of Fig. 2.
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Phys. Plasmas 15, 042306 共2008兲
The Hall fields and fast magnetic reconnection
(a)
0.8
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60
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80
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0.14,15 so the field line corresponding to the peak of Ey reconnected at a time 2.1 earlier. The distance along the field
line from the x-line to the left boundary is 28.3 so the mean
parallel propagation speed of the Hall fields away from the
x-line is around 13.
The Hall fields propagate along the magnetic field either
as a whistler or a kinetic Alfvén disturbance. The total pressure, including magnetic and plasma, is nearly constant
across the boundary layer, which eliminates the whistler as
the propagation mechanism, since it is on the fast-mode
branch of the dispersion curve. The kinetic Alfvén wave has
a peak parallel propagation speed given by the electron
sound speed cse = 冑共Te + Ti兲 / me ⬃ 15, close to the observed
propagation rate. Thus, it is the kinetic Alfvén wave, whose
speed greatly exceeds the inflow velocities associated with
reconnection, that is responsible for the near field aligned
structure of Ey and Bz. Because these fields also drive the
electron exhaust, the boundaries of the exhaust also closely
map the separatrices.
The broad electron outflow exhaust and associated Hall
fields were seen in observations of a magnetopause reconnection event in Polar spacecraft data.22 This event was a
crossing of the ion diffusion region so the ion motion within
the crossing was distinctly sub-Alfvénic. The E ⫻ B motion,
which was dominated by the Hall fields, produced the electron outflow, and the associated current supported the Hall
out-of-plane magnetic field. No super-Alfvénic electron jet
was seen in these data, perhaps because the jet does not
extend the full length of the ion diffusion region or because
the irregular motion of the magnetopause 共which is evident
in the data兲 masked the signatures of the jet. In reconnection
simulations density depletion layers are seen close to the
separatrices during antiparallel reconnection.18 These dips
are evident in the Polar data so we take these dips as a proxy
for the location of the separatrices. The dips lie just upstream
from the Hall fields and the associated electron exhaust. The
electron outflow exhaust, including the localized superAlfvénic jet and broader pedestal, was recently seen in Cluster data from a magnetosheath reconnection event.23 In these
data, which were estimated to be 63di downstream from the
x-line, only 4% of the electron flux in the exhaust was carried by the super-Alfvénic jet. Thus data from the Polar and
Cluster satellite supports the conclusion that the electron exhaust is driven by the Hall fields and extends nearly to the
magnetic separatrices.
We contrast the structure of the electron exhaust in the
case of reconnection with mi me with that in an equalmass, electron-positron plasma. In the equal mass case the
out-of-plane velocities of the electrons and positrons have
equal magnitudes but opposite directions in the vicinity of
the x-line and therefore pull the newly reconnected field lines
in opposite directions out of the plane. As a result there are
no traditional Hall electric and magnetic fields.24,25 Thus, we
expect the structure of the electron out-of-plane current layer
and the exhaust to be very different than the case with
mi me. Shown in Fig. 9 are plots of vez, the magnetic flux
and vex. The out-of-plane current layer is no longer localized
in the outflow direction25 but scales with the system size as
70
60
50
0
(c)
vex
90
1.0
0.5
80
y/di
042306-7
0.0
70
-0.5
60
-1.0
50
0
20
40
60
x/di
80
100
FIG. 9. 共Color online兲 Blowup around the x-line of 共a兲 the electron out-ofplane velocity vez, 共b兲 the in-plane magnetic field lines and 共c兲 the electron
outflow velocity vex from a simulation with mi = me. The white contour in 共c兲
is the separatrix.
in the Sweet–Parker model 共in very large systems the outflow exhaust becomes turbulent and the current layer again
becomes localized along the outflow direction兲.26 Because of
the length of the current layers in the equal mass case, secondary magnetic islands regularly form and complicate the
interpretation. In Fig. 9 there is an island near the x-line and
another that is being ejected near the right boundary. The
electron exhausts display no evidence of the central jet and
no evidence of the widening pedestal whose boundary maps
the magnetic field lines. The exhaust on the left, which is less
affected by the islands, is localized well within the magnetic
separatrix 共white contour line兲. These data further confirms
the key role played by the Hall fields in opening the electron
outflow exhaust downstream of the x-line in the case of reconnection with mi me.
V. THE TRANSITION REGION
We have shown that whistler dynamics around the x-line
causes the magnetic field just upstream of the electron current layer to bend into the out-of-plane direction producing
the out-of-plane magnetic field Bz. This magnetic field is
produced by the current associated with the electrons being
ejected at high velocity from the x-line. An important question is what dynamics causes the electron outflow jet to transition from a collimated beam close to the x-line to the expanding outflow exhaust that develops further downstream.
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042306-8
Drake, Shay, and Swisdak
Phys. Plasmas 15, 042306 共2008兲
At this location the Bx field has been dragged into the z
direction and as a result 兩Bz 兩 兩Bx兩 in a substantial region
upstream of the symmetry line. The reduction of Bx essentially shuts off the flow of electrons into the current layer
since the convective outflow velocity along x, vex = c共EyBz
− EzBy兲 / B2, now dominates. Thus, it is the location downstream of the x-line where the reduction of Bx by the whistler
enables the electron outflow exhaust to begin expanding that
marks the end of the inner electron diffusion region.
VI. SUMMARY AND DISCUSSION
FIG. 10. In 共a兲 a blowup around the x-line of the electron flow vectors from
the mi / me = 400 simulation of Fig. 2. In 共b兲 vertical cuts of vey through the
x-line at x = 37.4di 共solid兲 and x = 41.2di 共dotted兲 and in 共c兲 vertical cuts of
−Bz 共solid兲 and Bx 共dotted兲 at x = 41.2di.
We focus on the physics of this region by showing in Fig.
10共a兲 the electron flow pattern in the vicinity of the x-line
from the simulation of Fig. 2. This data has been averaged
over a time interval of 2⍀−1
i just prior to the data shown in
Fig. 2. The x-line at this time is at x ⯝ 37.5di. In a region
around the x-line spanning several di the electrons flow into
the current layer without being significantly deflected into
the outflow direction. This behavior can be seen more clearly
in the cut of vey through the x-line along the inflow direction
共solid line兲 in Fig. 10共b兲. The flow is given by the y component of the E ⫻ B drift, cEzBx / B2 until the particles become
demagnetized in the current layer.
Further downstream, however, the electron flow toward
the current layer is deflected into the downstream direction.
The dotted line in Fig. 10共b兲 is a similar cut of vey in the
downstream region at x = 41.2di. At this location downstream
the electrons flowing toward the current layer are deflected
well upstream of the current layer. It is evident from Fig. 10
that, consistent with the expanding exhaust shown in Fig. 5,
the y location of the deflection of the electron motion expands upstream with increasing distance downstream of the
x-line.
The important question is how far downstream from the
x-line do the electrons flowing toward the current layer begin
to be deflected prior to reaching the unmagnetized region.
This location marks the end of the collimated electron outflow channel, which, depending on its elongation, might constrain the rate of reconnection. To understand the physics
behind the deflection of the electrons, we show in Fig. 10共c兲
vertical cuts of −Bz 共solid兲 and Bx at x = 41.2di, the location
where the electrons show significant deflection in Fig. 10共b兲.
We have explored the role of the Hall fields in the dynamics and structure of the electron outflow exhaust during
reconnection in collisionless plasma. Just downstream of the
x-line the newly reconnected magnetic fields expand outwards as a convecting whistler wave: the magnetic field Bx,in
just upstream of the electron current layer rotates in a righthanded sense producing an out-of-plane magnetic field Bz,out
with a magnitude nearly equal to that of Bx,in. The electron
outflow velocity is equal to the electron Alfvén speed based
on Bx,in.
Downstream of the electron out-of-plane current layer
the electron outflow exhaust consists of a narrow superAlfvénic jet, which remains collimated far downstream from
the x-line,13–15 flanked by a pedestal whose width increases
monotonically with increasing distance downstream. Thus,
the super-Alvénic jet, while very interesting, does not appear
to act to constrain the rate of reconnection.
The broader outflow exhaust is driven by the Hall electric and magnetic fields at a velocity vex ⬃ cEyBz / B2, which
is comparable to the local field line velocity cEz / By. The
Hall fields are generated by the out-of-plane motion of the
electrons near the x-line17 and propagate with high velocity
nearly parallel to B downstream as a kinetic Alfvén disturbance. The rapid parallel propagation of the Hall fields in
comparison to the reconnection inflows causes the Hall fields
to closely map the magnetic field lines. In particular, the Hall
fields are strongest just downstream of the magnetic separatrix. Since the electron outflow exhaust is driven by the Hall
fields, the boundaries of the electron exhaust also closely
map the magnetic separatrix. The structure of this exhaust
differs greatly from that of the MHD model, where the outflow exhaust is driven by a slow shock that is well inside of
the magnetic separatrix 共the angle that the slow shock makes
with the symmetry axis is approximately half of that of the
magnetic separatrix in the Petschek model兲. Thus, the opening of the electron exhaust, which prevents the electrons
from throttling reconnection, is controlled by the Hall fields.
The dominance of the Hall fields in driving the electron
outflow as well as the conclusion that the boundaries of electron outflow exhaust closely map the magnetic separatrices
are supported by Polar observations of antiparallel reconnection at the magnetopause.22 The two-scale structure of the
electron exhaust, including the localized super-Alfvénic jet
and broader Alfvénic pedestal and associated Hall fields, was
seen in a recent magnetosheath reconnection event.27 In
these observations only 4% of the outward flux of electrons
was carried by the super-Alfvénic jet, the remaining 96%
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042306-9
The Hall fields and fast magnetic reconnection
being carried by the broader outflow driven dominantly by
the E ⫻ B motion in the Hall fields. Thus, while the superAlfvénic jet is certainly an interesting feature of reconnection, it appears to play only a minimal role in convecting
electrons away from the x-line.
The role of electrons in limiting the rate of reconnection
is ultimately determined by the length of the intense electron
out-of-plane current layer that develops in the vicinity of the
x-line 关see Fig. 2共a兲兴. This region was denoted as the “inner
electron diffusion region”15 to distinguish it from the superAlfvénic outflow jet that extends far downstream of the
x-line. In the present simulations the end of this current layer
and the simultaneous broadening of the electron outflow
channel are controlled by the whistler. The whistler rotates
the in-plane Bx field in the vicinity of the current layer into
the z direction, creating the out-of-plane field Bz. The resulting reduction of Bx self-consistently reduces the electron outof-plane current. Of greater significance is the resulting diversion of the electron flow toward the current layer into the
outflow direction 共Fig. 10兲 and thereby broadening the electron outflow channel. The rotation of the fields by the whistler forces Bz Bx in the current layer and just upstream. The
electron inflow vey = cEzBx / B2 is therefore locally suppressed
and the outflow velocity vex ⯝ cEyBz / B2 strongly enhanced.
The resultant broadening of the electron outflow channel
marks the end of the inner electron diffusion region.
In light of these results on the structure of the electron
current layer and outflow exhaust, a fundamental question is
whether the electrons play a significant role in limiting the
rate of reconnection. The insensitivity of the rate of reconnection to the electron mass in PIC simulations6,7,15 seems to
suggest that the electrons simply adjust their dynamics to
match the rate of reconnection determined by the ions. The
earlier explanations for the insensitivity of the rate of reconnection to electron dissipation5,6 are borne out by the simulations reported here. In Table I the data from simulations
with mi / me = 25, 100 and 400 indicate that the outflow velocity of the electrons from the x-line is given by the electron
Alfvén speed based on the upstream magnetic field with the
width ␦y of the outflow channel scaling with c / ␻ pe. The net
flux from the x-line ⌫ = vex␦y is therefore independent of me.
The length of the electron current layer ␦x along the outflow
direction 共based on the half-width, half-maximum of vez兲 is
also shown in Table I along with the measured electron inflow velocity just upstream of the x-line vey and the corresponding inflow velocity veyc based on the continuity of the
flow. The measured and expected inflow velocities match.
The continued increase of vey as me decreases seems to suggest that the current layer is not acting as a bottleneck for
reconnection.
Simulations of magnetic reconnection in electronpositron plasma, where, because of the equality of the
masses of the two species, there are no Hall fields parallel to
those discussed here, provide an interesting contrast24–26 to
the results presented here. Reconnection remains fast even in
the absence of the Hall fields. On this basis it was suggested
that the Hall fields are not important in electron-ion
reconnection.25 Since we have shown here that the Hall fields
in our simulations are responsible for fast, electron-proton
Phys. Plasmas 15, 042306 共2008兲
reconnection, what is producing fast reconnection in the
electron-positron simulations? For systems with scale sizes
Lx ⬍ 400di Swisdak et al. have shown that Sweet–Parker current layers form which lengthen with increasing system size,
as expected in the absence of the Hall fields. In this regime
the exhaust remains collimated well within the magnetic
separatrix as in the MHD model. However, for systems with
Lx ⬎ 400di the outflow exhaust develops a strong pressure
anisotropy and the resulting Weibel instability produces a
turbulent outflow jet that broadens with increasing distance
downstream. As a result, the rate of reconnection of the
electron-positron system remains fast even in very large
simulation domains.
In light of the critical role that the Hall fields play in
controlling the structure of the electron exhaust during reconnection, it is important that simulation domains used for
studying reconnection be chosen carefully to minimize the
influence of boundary conditions on the development of the
Hall fields. The challenge is that the Hall fields propagate
along the magnetic field as kinetic Alfvén structures at velocities that peak at the electron sound speed cse, which for
values of ␤ exceeding me / mi, exceed the Alfvén speed. The
Hall magnetic field Bz is produced as electrons near the
x-line drag the in-plane field in the out-of-plane direction. It
is critical that the magnetic field downstream is anchored by
the inertia of the downstream plasma since it is the relative
out-of-plane motion of the field line at the x-line versus
downstream that generates Bz. In the case of periodic simulations the effective boundary condition on Bz is at the
middle of the magnetic island where by symmetry Bz = 0. In
periodic systems it is possible that the amplitude of Bz is
higher than a real system where Bz asymptotes to zero further
downstream.
We have checked how the periodic boundaries affect the
development of Bz by carrying out a set of simulations with
mi / me = 25 in three different simulation domains: 204.8di
⫻ 102.4di, 102.4di ⫻ 51.2di, and 51.2di ⫻ 25.6di.15 The reconnection rates are indistinguishable in the three cases as
are the lengths of the out-of-plane current layers. In Fig.
11共a兲 we show Bz in a blowup around the x-line from the run
in the 102.4di ⫻ 51.2di domain. In Fig. 11共b兲 we show the
profile of Bz 共dotted line兲 along the vertical lines in Fig.
11共a兲. Because of the symmetry across x = 0 the data along
the two cuts were averaged by subtracting the values along
the two cuts and dividing by two. The two cuts are just
downstream of the out-of-plane electron current layer, 8.5di
on either size of the x-line, which is where Bz begins to
broaden the electron exhaust. Also in Fig. 11共b兲 are cuts from
the two other simulation domains: the solid for the 51.2di
⫻ 25.6di domain and the dashed for the 204.8di ⫻ 102.4di
domain. All of the cuts were taken 8.5di downstream of the
x-line and when the rate of reconnection was steady. The
cuts reveal that the location of the peaks of the Hall magnetic
fields lie at the same distance from the symmetry line at y
= 0 for all three simulations. This is especially evident in the
two largest simulations. This indicates that the opening angle
of the electron exhaust is the same in all of the simulations,
which is consistent with the reconnection rate being the
same. The magnitude of Bz decreases modestly with increas-
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042306-10
y/di
(a) 6
0.4
Bz
4
2
0
-2
-4
-6
-0.2
-20
ACKNOWLEDGMENTS
0.2
0.0
-30
(b)
Phys. Plasmas 15, 042306 共2008兲
Drake, Shay, and Swisdak
-10
0
x/di
10
20
-0.4
0.4
0.2
J.F.D. acknowledges helpful discussions with Dr. J. Eastwood, Dr. T. D. Phan, and Dr. F. S. Mozer. This work has
been supported by NSF Grant Nos. ATM0613782 and
ATM0645271, NASA Grant Nos. NNG05GL74G and
NNG05GM98G, and DOE Grant Nos. ER54197 and
ER54784. Computations were carried out at the National Energy Research Scientific Computing Center.
1
0.0
-0.2
-0.4
-4
-2
0
y/di
2
4
FIG. 11. 共Color online兲 In 共a兲 the Hall magnetic field Bz from a simulation
with mi / me = 25 on a 102.4di ⫻ 51.2di grid. The vertical lines a distance of
8.5di on either side of the x-line show the location of the cuts of Bz displayed in 共b兲. Cuts of Bz a distance of 8.5di on either size of the x-line from
three mi / me = 25 simulations differing only in the size of their computational
domain: 204.8di ⫻ 102.4di 共dashed兲; 102.4di ⫻ 51.2di 共dotted兲; and 51.2di
⫻ 25.6di 共solid兲.
ing domain size by around 23% between the two smaller
simulations and around 16% between the two larger simulations. Ideally, there should be no change but we emphasize
that this is a very stringent test since the overall electron
outflow exhausts in the larger domains are much longer than
in the smaller domains yet the cuts are taken at a fixed distance downstream from the x-line.
In open boundary simulations the condition on Bz on the
outflow boundary is typically dBz / dx = 0.13,19,28,29 This allows Bz to take on any value at the boundary. However, if the
electron out-of-plane velocity was nearly constant along the
magnetic field 共i.e., nearly a function of ␺, the in-plane magnetic flux兲 the field line would effectively be free floating
rather than anchored and could move in and out of the reconnection plane without producing the Hall field Bz. Thus,
Bz could be weaker than in an infinite system, thereby inhibiting the opening of the electron outflow exhaust and causing
the electrons to throttle the rate of reconnection. Such behavior was seen in a recent open boundary models13 although
whether the proximity of the x-line to the boundary produced
this behavior is unclear. In any case the importance of the
Hall fields to the structure of the electron exhaust and the
rapid rate of propagation of the Hall fields downstream, suggests that care must be taken to ensure that the boundary
conditions in open boundary models are not influencing the
physics. A demonstration such as that in Fig. 11 that differing
simulation domains do not influence the structure of the Hall
fields is essential to be sure that physics results do not depend on boundary conditions.
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