Submitted to Physical Review Letters
Two-scale structure of the electron dissipation region during collisionless magnetic
reconnection
M. A. Shay∗
Department of Physics & Astronomy, 217 Sharp Lab, University of Delaware, Newark, DE 19716
J. F. Drake, M. Swisdak
University of Maryland, College Park, MD, 20742
(Dated: November 1, 2018)
Particle in cell (PIC) simulations of collisionless magnetic reconnection are presented that demon-
strate that the electron dissipation region develops a distinct two-scale structure along the outflow
direction. The length of the electron current layer is found to decrease with decreasing electron
mass, approaching the ion inertial length for a proton-electron plasma. A surprise, however, is that
the electrons form a high-velocity outflow jet that remains decoupled from the magnetic field and
extends large distances downstream from the x-line. The rate of reconnection remains fast in very
large systems, independent of boundary conditions and the mass of electrons.
PACS numbers: Valid PACS appear here
Magnetic reconnection drives the release of magnetic
energy in explosive events such as disruptions in labo-
ratory experiments, magnetic substorms in the Earth’s
magnetosphere and flares in the solar corona. Recon-
nection in these events is typically collisionless because
reconnection electric fields exceed the Dreicer runaway
field. Since magnetic field lines reconnect in a boundary
layer, the “dissipation region”, whose structure may limit
the rate of release of energy, understanding the structure
of this boundary layer and its impact on reconnection
is critical to understanding the observations. Because
of their ability to carry large currents the dynamics of
electrons continues to be a topic of interest. Early sim-
ulations of reconnection suggested that the rate of re-
connection was not sensitive to electron dynamics [1, 2]
and this insensitivity was attributed to the coupling to
whistler dynamics at the small spatial scales of the dis-
sipation region [3, 4]. The results of more recent kinetic
PIC simulations have called into question these results
by suggesting that the electron current layer stretches
along the outflow direction and the rate of reconnection
drops[5, 6]. The fast rates of reconnection obtained from
earlier simulations[1, 3, 7] were attributed to the influ-
ence of periodicity[5].
We present particle-in-cell (PIC) simulations with var-
ious electron masses and computational domain sizes and
an analytic model that demonstrate that collisionless re-
connection remains fast even in very large collisionless
systems. The reconnection rate stabilizes before the pe-
riodicity of the boundary conditions can impact the dy-
namics. The electron current layer develops a distinct
two-scale structure along the outflow direction that had
not been identified in earlier simulations. The out-of-
plane electron current driven by the reconnection elec-
∗Electronic address: shay@udel.edu;
URL: http://www.physics.udel.edu/~shay
tric field has a length that decreases with the electron
mass, scaling as (me/mi)
3/8, which extrapolates to about
an ion inertial length di = c/ωpi for the electron-proton
mass ratio. The surprise is that a jet of outflowing elec-
trons with velocity close to the electron Alfven speed cAe
extends up to several 10’s of di from the x-line. Remark-
ably, the electrons are able to jet across the magnetic field
over such enormous distances because momentum trans-
port transverse to the jet effectively “blocks” the flow of
the out-of-plane current in this region. The momentum
transport causing this “current blocking” effect has the
same source (the off diagonal pressure tensor[1]), but is
much stronger than that which balances the reconnection
electric field at the x-line.
Our simulations are performed with the particle-in-cell
code p3d [8, 9]. The results are presented in normal-
ized units: the magnetic field to the asymptotic value of
the reversed field, the density to the value at the cen-
ter 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 cy-
clotron frequency Ω−1ci , and temperatures to miv
A. We
consider a system periodic in the x− y plane where flow
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 cur-
rent sheets superimposed on a ambient population of uni-
form density. The reconnection magnetic field is given by
Bx = tanh[(y − Ly/4)/w0] − tanh[(y − 3Ly/4)/w0] − 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, Te = 1/12 and Ti = 5/12, are ini-
tially uniform. The initial density profile is the usual Har-
ris form plus a uniform background of 0.2. The simula-
tions 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 have explored the dependence of the rate of recon-
http://arxiv.org/abs/0704.0818v1
mailto:shay@udel.edu
http://www.physics.udel.edu/~shay
FIG. 1: (color online). Reconnection electric field versus time:
(a) 204.8 × 102.4, (b) 102.4 × 51.2, (c) 51.2 × 25.6. w0 is the
initial current sheet width.
nection on the system size in a series of simulations with
three different system sizes and three different mass ra-
tios. For mi/me = 25, the grid scale ∆ = 0.05 and the
speed of light c = 15. For mi/me = 100, ∆ = 0.025 and
c = 20. For mi/me = 400, ∆ = 0.0125 and c = 40. The
reconnection rate versus time is plotted for our simula-
tions in Fig. 1. The reconnection rate is determined by
taking the time derivative of the total magnetic flux be-
tween the x-line and the center of the magnetic island.
The rate increases with time, undergoes a modest over-
shoot that is more pronounced in the smaller domains,
and approaches a quasi-steady rate of around 0.14, in-
dependent of the domain size. Earlier suggestions [5]
that reconnection rates would plunge until elongated cur-
rent layers spawned secondary magnetic islands are not
borne out in these simulations. The rates of reconnec-
tion approach constant values even in the absence of sec-
ondary islands, which for anti-parallel reconnection typ-
ically only occur transiently due to initial conditions[10].
Even these transient islands can be largely eliminated by
a suitable choice of the initial current layer width w0 (a
larger value of w0 is required for the larger domains).
A critical issue is whether the periodicity in the x direc-
tion can influence the rate of reconnection [5]. In each of
the simulations we have identified the time at which the
ion outflows from the x-line meet at the center of the mag-
netic island. This occurs at t ≈ 155 for the largest simu-
lation shown in Fig. 1a. The plasma at the x-line can not
be affected by the downstream conditions until t ≈ 255,
when a pressure perturbation can propagate back up-
stream to the x-line at the magnetosonic speed. This is
well after the end of the simulation. The electrons are
ejected from the x-line at a velocity of around cAe ≫ cA
and therefore might be able to follow field lines back to
the x-line. During the traversal time δt = Lx/cAe, the
amount of reconnected flux is vinB0L/cAe, where vin is
the inflow velocity into the x-line. Using the conservation
of the canonical momentum in the z-direction, the condi-
tion that an electron with a velocity cAe can not cross this
flux to access the x-line reduces to L > di(cA/vin) ∼ 7di,
which is easily satisfied for the simulations in Fig. 1. The
fact that the reconnection rates for all of the simulation
domains in Fig. 1 are essentially identical further sup-
ports this conclusion.
Also shown in Fig. 1 in the dashed lines are the rates of
reconnection for mi/me = 100 in (b) and mi/me = 400
in (c). Consistent with simulations in smaller domains [1,
2], the rate of reconnection is insensitive to the electron
mass.
We now proceed to explore the structure of the electron
current layer. Shown in Fig. 2 is a blow-up around the x-
line of the out-of-plane electron velocity for mi/me = 25
and two simulation domains, 204.8 × 102.4 in (a) and
51.2 × 25.6 in (b), and for mi/me = 400 in a simula-
tion domain of 51.2 × 25.6 in (c). All of the data is
taken in the phase where the reconnection rate and the
lengths of the region of intense out-of-plane current are
stationary. Reconnection forms intense current layers
that have a well-defined length (half widths of around
7di and independent of the size of computational domain
formi/me = 25) and then open up forming the open out-
flow jet that characterizes Hall reconnection [3, 7]. The
current layer in the case of mi/me = 400 in Fig. 2c is dis-
tinctly shorter than the smaller mass ratio current layers
in Fig. 2a,b, suggesting that the length of the electron
current layer depends on the electron mass and would be
shorter for realistic proton-electron mass ratios.
Shown in Fig. 3a is a blow-up around the x-line of
the electron outflow velocity vex for the mi/me = 25,
204.8 × 102.4 run corresponding to Fig. 2a. In contrast
with the out-of-plane current the electrons form an out-
flow jet that extends a very large distance downstream
from the x-line. This outflow jet continued to grow in
length until the end of the simulation. This simulation,
along with others at differing mass ratios, reveals that the
peak outflow velocity is very close to the electron Alfven
speed [8, 11]. One might expect that because of the colli-
mation of the outflow jet and its length, the reconnection
rate would drop. However, this is not the case. While
there is an intense jet in the core of the reconnection
exhaust, the exhaust as a whole quickly begins to open
up downstream of the current layer (Jz). The jet itself
therefore does not act as a nozzle to limit the rate of
FIG. 2: (color online). Blowups around the x-line of the out-
of-plane electron velocity for: (a)mi/me = 25, simulation size
204.8× 102.4, (b) mi/me = 25, 51.2× 25.6, and (c) mi/me =
400, 51.2 × 25.6.
FIG. 3: (color online). Blowups around the x-line for system
size 204.8× 102.4 with mi/me = 25. (a) The electron outflow
velocity vex. (b) Momentum flux vectors, Γ = pexzx̂ + peyzŷ
(vectors in box surrounding x-line are multiplied by 20), with
a background color plot of | (Ez + (ve ×B/c)z)/Ez | .
reconnection: the rate of reconnection remains constant
even as the length of the outflow jet varies in time.
To understand how the electrons can form such an ex-
tended outflow jet while the out-of-plane current layer
remains localized, we examine the out-of-plane compo-
nent of the fluid electron momentum equation along the
symmetry line of the outflow direction. In steady state
Ez = −
mevex
vexBy −
∇ · Γ, (1)
where ve is the electron bulk velocity, Γ = pexzx̂+peyzŷ
is the flux of z-directed electron momentum in the recon-
nection plane (not including convection of momentum)
with pe the electron pressure tensor. In Fig. 4a we plot
all of the terms in this equation along a cut though the
x-line along the outflow direction from a simulation with
mi/me = 100 and Lx × Ly = 102.4 × 51.2. The data
has been averaged between t = 116.2 and t = 117.0.
The electric field (black) is balanced by the sum (red)
of the electron inertia (dashed blue), the Lorentz force
(solid blue) and the divergence of the momentum flux
(green). The major contributions to momentum balance
come from the Lorentz force and the divergence of the
momentum flux. At the x-line the electric field drive
is balanced by the momentum transport [1, 12]. The
surprise is that the Lorentz force, rather than simply in-
creasing downstream from the x-line to balance the re-
connection electric field, instead strongly overshoots the
reconnection electric field far downstream of x-line. This
tendency was seen in earlier simulations [12] but there
was no clear separation of scales because of the small size
of these earlier simulations. Downsteam from the x-line
the electrons are streaming much faster than the mag-
netic field lines. Thus, in a reference frame of the moving
electrons the z-directed electric field has reversed direc-
tion compared with the x-line. This electric field tries to
drive a current opposite to that at the x-line. Evidence
for this reversed current appears downstream of the x-line
in Fig. 2c. In spite of the strength of the effective elec-
tric field, the reversed current carried by the electrons
is small. As at the x-line, the momentum transfer to
electrons in this extended outflow region is balanced by
momentum transport. The momentum flux around the
x-line is shown as a 2-D vector plot in Fig. 3 for the same
run as in (a). The momentum flux has been multiplied by
20 in the box surrounding the x-line. The data for this
figure has been averaged between t = 172.5 and 174.5.
The background color plot is of | (Ez+(ve×B/c)z)/Ez |,
which is & 1 where the electrons are not frozen-in. Evi-
dent is the outward flow of momentum around the x-line
and the much stronger outward flow of negative momen-
tum in an extended downstream region. The momentum
transport is so large that the out-of-plane current down-
stream is effectively “blocked”. The force associated with
this “blocking effect” drives the flow of the large-scale jet
of electrons downstream of the x-line.
We define the length ∆x of the inner dissipation re-
gion as the distance from the x-line to the point where
the Lorentz force vexBy/c crosses the reconnection elec-
tric field Ez. At this location the effective out-of-plane
electric field seen by the electrons reverses sign, causing
the electron current jez to be driven in reverse, which
allows the separatrices to open up. Thus, the inner dis-
sipation region defines the spatial extent of the magnetic
nozzle that develops during reconnection. Since the sim-
ulations presented in this paper use artificial values of
me, it is essential to understand the me scaling of ∆x so
that this important length can be calculated for a proton-
electron plasma. The momentum equation of electrons
in the outflow direction yields a steady state equation for
ex) =
vezBy, (2)
where vez ∼ cAe. Thus, the profile of By along the out-
flow direction and its dependence on me must be deter-
mined. This profile is shown formi/me = 25 (system size
102.4× 51.2), 100 (102.4× 51.2) and 400 (51.2× 25.6) in
Fig. 4b. Surprisingly, the profile of By is apparently in-
dependent of me. Our original expectation was because
of the continuity of the flow of magnetic flux into and
out of the x-line that By ∼ B0vin/cAe ∝ m
e , where the
outflow velocity eventually rises to cAe. However, since
the electrons are not frozen into the magnetic field until
far downstream, the expected scaling fails. To calculate
vex we approximate By by a linear ramp and integrate
Eq. (2). Setting the Lorentz force equation to the re-
connection electric field, we then obtain an equation for
)3/8 (
)1/2 (
diBy′
di. (3)
For the three simulations shown in Fig. 4b the simula-
tions yield 2.9di, 1.8di and 1.0di for mi/me = 25, 100
and 400, respectively, which is in reasonable accord with
the scaling. Extrapolating to a mass-ratio of 1836, we
predict ∆x ∼ 0.6di. In contrast the outer dissipation
region can extend to 10’s of di.
We have shown that the electron current layer that
forms during reconnection stabilizes at a finite length,
independent of the periodicity of the simulation domain,
and aside from transients from initial conditions remains
largely stable to secondary island formation. Reconnec-
tion remains fast with normalized reconnection rates of
around 0.14. The length of the electron current layer ∆x
scales as m
e . Since the width δ of the current layer
scales with the electron skin depth c/ωpe, the aspect-
ratio δ/∆x ∝ (me/mi)
1/8. Extrapolating from our
mi/me = 400 simulations to mi/me = 1836 should not
significantly change the aspect-ratio and we therefore ex-
pect the current layer to remain stable for real mass ra-
tios.
The structure of the current layer is important to
the design of NASA’s magnetospheric multiscale mission
(MMS), which will be the first mission with the time reso-
lution to measure the electron current layers that develop
during reconnection. The length of the out-of-plane elec-
tron current layer projects to around c/ωpi for a proton-
electron plasma while the the outflow jet, which supports
a strong Hall (out-of-plane) magnetic field, extends 10’s
of c/ωpi from the x-line.
FIG. 4: (color online). Results for simulation size 102.4×51.2
with mi/me = 25 and 100; and 51.2×25.6 with mi/me = 400.
(a) Cuts through the x-line of the contributions to Ohm’s law
for mi/me = 100. 1 → −me/eve · ∇vez, 2 → −ẑ · ve × B/e,
3 → −ẑ · (∇ ·Pe)/(nee), 4 → sum of 1,2,3. (b) Cuts through
x-line of By for the three different mi/me.
This work was supported in part by NSF, NASA and
Acknowledgments This work was supported in part
by NASA and the NSF. Computations were carried out
at the National Energy Research Scientific Computing
Center.
[1] M. Hesse et al., Phys. Plasmas 6, 1781 (1999).
[2] M. A. Shay and J. F. Drake, Geophys. Res. Lett. 25,
3759 (1998).
[3] J. Birn et al., J. Geophys. Res. 106, 3715 (2001).
[4] B. N. Rogers et al., Phys. Rev. Lett. 87, 195004 (2001).
[5] W. Daughton et al., Phys. Plasmas 13, 072101 (2006).
[6] K. Fujimoto, Phys. Plasmas 13, 072904 (2006).
[7] M. A. Shay et al., Geophys. Res. Lett. 26, 2163 (1999).
[8] M. A. Shay et al., J. Geophys. Res. 106, 3751 (2001).
[9] A. Zeiler et al., J. Geophys. Res. 107, 1230 (2002),
doi:10.1029/2001JA000287.
[10] J. F. Drake et al., Geophys. Res. Lett. 33, L13105 (2006),
doi:10.1029/2006GL025957.
[11] M. Hoshino et al., J. Geophys. Res. 106, 25979 (2001).
[12] P. L. Pritchett, J. Geophys. Res. 106, 3783 (2001).

  Particle in cell (PIC) simulations of collisionless magnetic reconnection are
presented that demonstrate that the electron dissipation region develops a
distinct two-scale structure along the outflow direction. The length of the
electron current layer is found to decrease with decreasing electron mass,
approaching the ion inertial length for a proton-electron plasma. A surprise,
however, is that the electrons form a high-velocity outflow jet that remains
decoupled from the magnetic field and extends large distances downstream from
the x-line. The rate of reconnection remains fast in very large systems,
independent of boundary conditions and the mass of electrons.

Submitted to Physical Review Letters
Two-scale structure of the electron dissipation region during collisionless magnetic
reconnection
M. A. Shay∗
Department of Physics & Astronomy, 217 Sharp Lab, University of Delaware, Newark, DE 19716
J. F. Drake, M. Swisdak
University of Maryland, College Park, MD, 20742
(Dated: November 1, 2018)
Particle in cell (PIC) simulations of collisionless magnetic reconnection are presented that demon-
strate that the electron dissipation region develops a distinct two-scale structure along the outflow
direction. The length of the electron current layer is found to decrease with decreasing electron
mass, approaching the ion inertial length for a proton-electron plasma. A surprise, however, is that
the electrons form a high-velocity outflow jet that remains decoupled from the magnetic field and
extends large distances downstream from the x-line. The rate of reconnection remains fast in very
large systems, independent of boundary conditions and the mass of electrons.
PACS numbers: Valid PACS appear here
Magnetic reconnection drives the release of magnetic
energy in explosive events such as disruptions in labo-
ratory experiments, magnetic substorms in the Earth’s
magnetosphere and flares in the solar corona. Recon-
nection in these events is typically collisionless because
reconnection electric fields exceed the Dreicer runaway
field. Since magnetic field lines reconnect in a boundary
layer, the “dissipation region”, whose structure may limit
the rate of release of energy, understanding the structure
of this boundary layer and its impact on reconnection
is critical to understanding the observations. Because
of their ability to carry large currents the dynamics of
electrons continues to be a topic of interest. Early sim-
ulations of reconnection suggested that the rate of re-
connection was not sensitive to electron dynamics [1, 2]
and this insensitivity was attributed to the coupling to
whistler dynamics at the small spatial scales of the dis-
sipation region [3, 4]. The results of more recent kinetic
PIC simulations have called into question these results
by suggesting that the electron current layer stretches
along the outflow direction and the rate of reconnection
drops[5, 6]. The fast rates of reconnection obtained from
earlier simulations[1, 3, 7] were attributed to the influ-
ence of periodicity[5].
We present particle-in-cell (PIC) simulations with var-
ious electron masses and computational domain sizes and
an analytic model that demonstrate that collisionless re-
connection remains fast even in very large collisionless
systems. The reconnection rate stabilizes before the pe-
riodicity of the boundary conditions can impact the dy-
namics. The electron current layer develops a distinct
two-scale structure along the outflow direction that had
not been identified in earlier simulations. The out-of-
plane electron current driven by the reconnection elec-
∗Electronic address: shay@udel.edu;
URL: http://www.physics.udel.edu/~shay
tric field has a length that decreases with the electron
mass, scaling as (me/mi)
3/8, which extrapolates to about
an ion inertial length di = c/ωpi for the electron-proton
mass ratio. The surprise is that a jet of outflowing elec-
trons with velocity close to the electron Alfven speed cAe
extends up to several 10’s of di from the x-line. Remark-
ably, the electrons are able to jet across the magnetic field
over such enormous distances because momentum trans-
port transverse to the jet effectively “blocks” the flow of
the out-of-plane current in this region. The momentum
transport causing this “current blocking” effect has the
same source (the off diagonal pressure tensor[1]), but is
much stronger than that which balances the reconnection
electric field at the x-line.
Our simulations are performed with the particle-in-cell
code p3d [8, 9]. The results are presented in normal-
ized units: the magnetic field to the asymptotic value of
the reversed field, the density to the value at the cen-
ter 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 cy-
clotron frequency Ω−1ci , and temperatures to miv
A. We
consider a system periodic in the x− y plane where flow
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 cur-
rent sheets superimposed on a ambient population of uni-
form density. The reconnection magnetic field is given by
Bx = tanh[(y − Ly/4)/w0] − tanh[(y − 3Ly/4)/w0] − 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, Te = 1/12 and Ti = 5/12, are ini-
tially uniform. The initial density profile is the usual Har-
ris form plus a uniform background of 0.2. The simula-
tions 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 have explored the dependence of the rate of recon-
http://arxiv.org/abs/0704.0818v1
mailto:shay@udel.edu
http://www.physics.udel.edu/~shay
FIG. 1: (color online). Reconnection electric field versus time:
(a) 204.8 × 102.4, (b) 102.4 × 51.2, (c) 51.2 × 25.6. w0 is the
initial current sheet width.
nection on the system size in a series of simulations with
three different system sizes and three different mass ra-
tios. For mi/me = 25, the grid scale ∆ = 0.05 and the
speed of light c = 15. For mi/me = 100, ∆ = 0.025 and
c = 20. For mi/me = 400, ∆ = 0.0125 and c = 40. The
reconnection rate versus time is plotted for our simula-
tions in Fig. 1. The reconnection rate is determined by
taking the time derivative of the total magnetic flux be-
tween the x-line and the center of the magnetic island.
The rate increases with time, undergoes a modest over-
shoot that is more pronounced in the smaller domains,
and approaches a quasi-steady rate of around 0.14, in-
dependent of the domain size. Earlier suggestions [5]
that reconnection rates would plunge until elongated cur-
rent layers spawned secondary magnetic islands are not
borne out in these simulations. The rates of reconnec-
tion approach constant values even in the absence of sec-
ondary islands, which for anti-parallel reconnection typ-
ically only occur transiently due to initial conditions[10].
Even these transient islands can be largely eliminated by
a suitable choice of the initial current layer width w0 (a
larger value of w0 is required for the larger domains).
A critical issue is whether the periodicity in the x direc-
tion can influence the rate of reconnection [5]. In each of
the simulations we have identified the time at which the
ion outflows from the x-line meet at the center of the mag-
netic island. This occurs at t ≈ 155 for the largest simu-
lation shown in Fig. 1a. The plasma at the x-line can not
be affected by the downstream conditions until t ≈ 255,
when a pressure perturbation can propagate back up-
stream to the x-line at the magnetosonic speed. This is
well after the end of the simulation. The electrons are
ejected from the x-line at a velocity of around cAe ≫ cA
and therefore might be able to follow field lines back to
the x-line. During the traversal time δt = Lx/cAe, the
amount of reconnected flux is vinB0L/cAe, where vin is
the inflow velocity into the x-line. Using the conservation
of the canonical momentum in the z-direction, the condi-
tion that an electron with a velocity cAe can not cross this
flux to access the x-line reduces to L > di(cA/vin) ∼ 7di,
which is easily satisfied for the simulations in Fig. 1. The
fact that the reconnection rates for all of the simulation
domains in Fig. 1 are essentially identical further sup-
ports this conclusion.
Also shown in Fig. 1 in the dashed lines are the rates of
reconnection for mi/me = 100 in (b) and mi/me = 400
in (c). Consistent with simulations in smaller domains [1,
2], the rate of reconnection is insensitive to the electron
mass.
We now proceed to explore the structure of the electron
current layer. Shown in Fig. 2 is a blow-up around the x-
line of the out-of-plane electron velocity for mi/me = 25
and two simulation domains, 204.8 × 102.4 in (a) and
51.2 × 25.6 in (b), and for mi/me = 400 in a simula-
tion domain of 51.2 × 25.6 in (c). All of the data is
taken in the phase where the reconnection rate and the
lengths of the region of intense out-of-plane current are
stationary. Reconnection forms intense current layers
that have a well-defined length (half widths of around
7di and independent of the size of computational domain
formi/me = 25) and then open up forming the open out-
flow jet that characterizes Hall reconnection [3, 7]. The
current layer in the case of mi/me = 400 in Fig. 2c is dis-
tinctly shorter than the smaller mass ratio current layers
in Fig. 2a,b, suggesting that the length of the electron
current layer depends on the electron mass and would be
shorter for realistic proton-electron mass ratios.
Shown in Fig. 3a is a blow-up around the x-line of
the electron outflow velocity vex for the mi/me = 25,
204.8 × 102.4 run corresponding to Fig. 2a. In contrast
with the out-of-plane current the electrons form an out-
flow jet that extends a very large distance downstream
from the x-line. This outflow jet continued to grow in
length until the end of the simulation. This simulation,
along with others at differing mass ratios, reveals that the
peak outflow velocity is very close to the electron Alfven
speed [8, 11]. One might expect that because of the colli-
mation of the outflow jet and its length, the reconnection
rate would drop. However, this is not the case. While
there is an intense jet in the core of the reconnection
exhaust, the exhaust as a whole quickly begins to open
up downstream of the current layer (Jz). The jet itself
therefore does not act as a nozzle to limit the rate of
FIG. 2: (color online). Blowups around the x-line of the out-
of-plane electron velocity for: (a)mi/me = 25, simulation size
204.8× 102.4, (b) mi/me = 25, 51.2× 25.6, and (c) mi/me =
400, 51.2 × 25.6.
FIG. 3: (color online). Blowups around the x-line for system
size 204.8× 102.4 with mi/me = 25. (a) The electron outflow
velocity vex. (b) Momentum flux vectors, Γ = pexzx̂ + peyzŷ
(vectors in box surrounding x-line are multiplied by 20), with
a background color plot of | (Ez + (ve ×B/c)z)/Ez | .
reconnection: the rate of reconnection remains constant
even as the length of the outflow jet varies in time.
To understand how the electrons can form such an ex-
tended outflow jet while the out-of-plane current layer
remains localized, we examine the out-of-plane compo-
nent of the fluid electron momentum equation along the
symmetry line of the outflow direction. In steady state
Ez = −
mevex
vexBy −
∇ · Γ, (1)
where ve is the electron bulk velocity, Γ = pexzx̂+peyzŷ
is the flux of z-directed electron momentum in the recon-
nection plane (not including convection of momentum)
with pe the electron pressure tensor. In Fig. 4a we plot
all of the terms in this equation along a cut though the
x-line along the outflow direction from a simulation with
mi/me = 100 and Lx × Ly = 102.4 × 51.2. The data
has been averaged between t = 116.2 and t = 117.0.
The electric field (black) is balanced by the sum (red)
of the electron inertia (dashed blue), the Lorentz force
(solid blue) and the divergence of the momentum flux
(green). The major contributions to momentum balance
come from the Lorentz force and the divergence of the
momentum flux. At the x-line the electric field drive
is balanced by the momentum transport [1, 12]. The
surprise is that the Lorentz force, rather than simply in-
creasing downstream from the x-line to balance the re-
connection electric field, instead strongly overshoots the
reconnection electric field far downstream of x-line. This
tendency was seen in earlier simulations [12] but there
was no clear separation of scales because of the small size
of these earlier simulations. Downsteam from the x-line
the electrons are streaming much faster than the mag-
netic field lines. Thus, in a reference frame of the moving
electrons the z-directed electric field has reversed direc-
tion compared with the x-line. This electric field tries to
drive a current opposite to that at the x-line. Evidence
for this reversed current appears downstream of the x-line
in Fig. 2c. In spite of the strength of the effective elec-
tric field, the reversed current carried by the electrons
is small. As at the x-line, the momentum transfer to
electrons in this extended outflow region is balanced by
momentum transport. The momentum flux around the
x-line is shown as a 2-D vector plot in Fig. 3 for the same
run as in (a). The momentum flux has been multiplied by
20 in the box surrounding the x-line. The data for this
figure has been averaged between t = 172.5 and 174.5.
The background color plot is of | (Ez+(ve×B/c)z)/Ez |,
which is & 1 where the electrons are not frozen-in. Evi-
dent is the outward flow of momentum around the x-line
and the much stronger outward flow of negative momen-
tum in an extended downstream region. The momentum
transport is so large that the out-of-plane current down-
stream is effectively “blocked”. The force associated with
this “blocking effect” drives the flow of the large-scale jet
of electrons downstream of the x-line.
We define the length ∆x of the inner dissipation re-
gion as the distance from the x-line to the point where
the Lorentz force vexBy/c crosses the reconnection elec-
tric field Ez. At this location the effective out-of-plane
electric field seen by the electrons reverses sign, causing
the electron current jez to be driven in reverse, which
allows the separatrices to open up. Thus, the inner dis-
sipation region defines the spatial extent of the magnetic
nozzle that develops during reconnection. Since the sim-
ulations presented in this paper use artificial values of
me, it is essential to understand the me scaling of ∆x so
that this important length can be calculated for a proton-
electron plasma. The momentum equation of electrons
in the outflow direction yields a steady state equation for
ex) =
vezBy, (2)
where vez ∼ cAe. Thus, the profile of By along the out-
flow direction and its dependence on me must be deter-
mined. This profile is shown formi/me = 25 (system size
102.4× 51.2), 100 (102.4× 51.2) and 400 (51.2× 25.6) in
Fig. 4b. Surprisingly, the profile of By is apparently in-
dependent of me. Our original expectation was because
of the continuity of the flow of magnetic flux into and
out of the x-line that By ∼ B0vin/cAe ∝ m
e , where the
outflow velocity eventually rises to cAe. However, since
the electrons are not frozen into the magnetic field until
far downstream, the expected scaling fails. To calculate
vex we approximate By by a linear ramp and integrate
Eq. (2). Setting the Lorentz force equation to the re-
connection electric field, we then obtain an equation for
)3/8 (
)1/2 (
diBy′
di. (3)
For the three simulations shown in Fig. 4b the simula-
tions yield 2.9di, 1.8di and 1.0di for mi/me = 25, 100
and 400, respectively, which is in reasonable accord with
the scaling. Extrapolating to a mass-ratio of 1836, we
predict ∆x ∼ 0.6di. In contrast the outer dissipation
region can extend to 10’s of di.
We have shown that the electron current layer that
forms during reconnection stabilizes at a finite length,
independent of the periodicity of the simulation domain,
and aside from transients from initial conditions remains
largely stable to secondary island formation. Reconnec-
tion remains fast with normalized reconnection rates of
around 0.14. The length of the electron current layer ∆x
scales as m
e . Since the width δ of the current layer
scales with the electron skin depth c/ωpe, the aspect-
ratio δ/∆x ∝ (me/mi)
1/8. Extrapolating from our
mi/me = 400 simulations to mi/me = 1836 should not
significantly change the aspect-ratio and we therefore ex-
pect the current layer to remain stable for real mass ra-
tios.
The structure of the current layer is important to
the design of NASA’s magnetospheric multiscale mission
(MMS), which will be the first mission with the time reso-
lution to measure the electron current layers that develop
during reconnection. The length of the out-of-plane elec-
tron current layer projects to around c/ωpi for a proton-
electron plasma while the the outflow jet, which supports
a strong Hall (out-of-plane) magnetic field, extends 10’s
of c/ωpi from the x-line.
FIG. 4: (color online). Results for simulation size 102.4×51.2
with mi/me = 25 and 100; and 51.2×25.6 with mi/me = 400.
(a) Cuts through the x-line of the contributions to Ohm’s law
for mi/me = 100. 1 → −me/eve · ∇vez, 2 → −ẑ · ve × B/e,
3 → −ẑ · (∇ ·Pe)/(nee), 4 → sum of 1,2,3. (b) Cuts through
x-line of By for the three different mi/me.
This work was supported in part by NSF, NASA and
Acknowledgments This work was supported in part
by NASA and the NSF. Computations were carried out
at the National Energy Research Scientific Computing
Center.
[1] M. Hesse et al., Phys. Plasmas 6, 1781 (1999).
[2] M. A. Shay and J. F. Drake, Geophys. Res. Lett. 25,
3759 (1998).
[3] J. Birn et al., J. Geophys. Res. 106, 3715 (2001).
[4] B. N. Rogers et al., Phys. Rev. Lett. 87, 195004 (2001).
[5] W. Daughton et al., Phys. Plasmas 13, 072101 (2006).
[6] K. Fujimoto, Phys. Plasmas 13, 072904 (2006).
[7] M. A. Shay et al., Geophys. Res. Lett. 26, 2163 (1999).
[8] M. A. Shay et al., J. Geophys. Res. 106, 3751 (2001).
[9] A. Zeiler et al., J. Geophys. Res. 107, 1230 (2002),
doi:10.1029/2001JA000287.
[10] J. F. Drake et al., Geophys. Res. Lett. 33, L13105 (2006),
doi:10.1029/2006GL025957.
[11] M. Hoshino et al., J. Geophys. Res. 106, 25979 (2001).
[12] P. L. Pritchett, J. Geophys. Res. 106, 3783 (2001).
