Download Scaling of the inner electron diffusion region in collisionless

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, A06217, doi:10.1029/2011JA017464, 2012
Scaling of the inner electron diffusion region in collisionless
magnetic reconnection
A. Divin,1 G. Lapenta,1 S. Markidis,1,2 V. S. Semenov,3 N. V. Erkaev,4,5
D. B. Korovinskiy,6 and H. K. Biernat6,7
Received 20 December 2011; revised 22 April 2012; accepted 30 April 2012; published 12 June 2012.
[1] The Sweet-Parker analysis of the inner electron diffusion region of collisionless
magnetic reconnection is presented. The study includes charged particles motion near
the X-line and an appropriate approximation of the off-diagonal term for the electron
pressure tensor. The obtained scaling shows that the width of the inner electron
diffusion region is equal to the electron inertial length, and that electrons are accelerated
up to the electron Alfvén velocity in X-line direction. The estimated effective plasma
conductivity is based on the electron gyrofrequency rather than the binary collision
frequency, and gives the extreme (minimal) value of the plasma conductivity similar to
Bohm diffusion. The scaling properties are verified by means of Particle-in-Cell
simulations. An ad hoc parameter needs to be introduced to the scaling relations in order
to better match the theory and simulations.
Citation: Divin, A., G. Lapenta, S. Markidis, V. S. Semenov, N. V. Erkaev, D. B. Korovinskiy, and H. K. Biernat (2012),
Scaling of the inner electron diffusion region in collisionless magnetic reconnection, J. Geophys. Res., 117, A06217,
doi:10.1029/2011JA017464.
1. Introduction
[2] Magnetic reconnection is a powerful nature phenomenon which allows the magnetic energy to be transformed
rapidly into the kinetic and thermal energy of plasma [Priest
and Forbes, 2000]. Magnetic reconnection is generally
viewed as an important mechanism triggering magnetospheric, solar and astrophysical activity. The broadest
description of the process involves two important concepts:
1) the property of magnetic field lines being frozen into
plasma and 2) the finite conductivity of plasma which breaks
the froze-in constraint. The magnetic energy is released
when plasmas having different magnetic field come in contact by convection. Magnetic flux tubes are then detached
from plasma by means of some diffusive process, reconnect
and are eventually ejected. The plasma is demagnetized
within the so-called Diffusion Region (DR) and the rate of
1
Centrum voor Plasma Astrofysica, Departement Wiskunde, Katholieke
Universiteit Leuven, Leuven, Belgium.
2
PDC Center for High Performance Computing, KTH, Stockholm,
Sweden.
3
State University of St. Petersburg, St. Petersburg, Russia.
4
Institute of Computational Modeling, Siberian Branch, Russian
Academy of Sciences, Krasnoyarsk, Russia.
5
Polytechnical Institute, Siberian Federal University, Krasnoyarsk,
Russia.
6
Space Research Institute, Austrian Academy of Sciences, Graz,
Austria.
7
Institute of Physics, University of Graz, Graz, Austria.
Corresponding author: A. Divin, Centrum voor Plasma Astrofysica,
Departement Wiskunde, Katholieke Universiteit Leuven, Celestijnenlaan
200B, Leuven, BE-3001, Belgium. ([email protected])
©2012. American Geophysical Union. All Rights Reserved.
reconnection inside the DR controls the overall efficiency of
the whole process.
[3] The model developed by Sweet [1958] and Parker
[1963] was the first self-consistent steady state model of
magnetic reconnection. The scaling provides the general
framework to analyze the DR properties in different plasma
environments. Spitzer (collisional) resistivity [Spitzer, 1962]
is assumed as a source of plasma dissipation. As usually
noted, such dissipation cannot account for the reconnection
rates observed in collisionless plasmas where the mean free
path of a particle is much larger than the width of the DR. In
previous decades much of the efforts were put into the
investigation of dissipative mechanisms in the absence of
binary particle collisions. Ion and electron inertia effects,
chaotization of particle trajectories, turbulence and smallscale kinetic instabilities were found to enhance significantly
the effective resistivity of plasma (see an overview of dissipative collisionless mechanisms in, e.g., Biskamp [2000]
and Schindler [2006]).
[4] Detailed study of magnetic reconnection is virtually
impossible without numerical simulations. In MHD
approach with a uniform resistivity an extended diffusive
layer is formed, in good resemblance with the Sweet-Parker
model [Birn et al., 2001]. Localized resistivity shortens the
DR considerably and speeds up reconnection and repeats the
Petschek-like configuration [Petschek, 1964; Vasyliunas,
1975].
[5] Proton-electron collisionless plasmas are common for
space plasma environments. Collisionless kinetic effects and
multifluid physics are known to modify the process considerably. In particular, the presence of species of different
masses creates the multiscale reconnection pattern with the
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Figure 1. Multiscale structure of collisionless magnetic reconnection: plasma is magnetized in MHD
region; electron and ion motions are decoupled in HMHD (Hall MHD) and EMHD (electron MHD)
regions. Ion velocity is small compared to that of electrons in EMHD region. Electrons are not magnetized
inside EDR. Width le and length Le mark the EDR spatial extent.
Electron Diffusion Region (EDR) located inside the Ion
Diffusion Region (IDR). The DR thickness scales as the
inertial length of the corresponding species, whereas disagreement exists about the lengths of EDR and IDR. The
length of the IDR is presumably > 10di. Here di = c/wpi is ion
inertial length, wpi = (4pnie2/mi)1/2 is ion plasma frequency,
ni denotes ion density, mi is the ion mass and e is the ion
charge. Similarly, the electron inertial length is defined as
de = c/wpe, where wpe = (4pnee2/me)1/2 and ne is the electron
density, me is the electron mass, e is the electron charge.
[6] The EDR is composed of the inner part, where electrons are demagnetized and provide for the necessary dissipation, and of the external part, or ‘reconnection ejecta’
[Daughton et al., 2006; Karimabadi et al., 2007; Shay et al.,
2007], stretched over the distance 10di and having highly
accelerated electron jet outrunning the convection of magnetic field lines. The inner EDR is the focus of our study.
Therefore, we simply refer to the inner EDR as ‘EDR’ for
short throughout the paper.
[7] The equation of motion of electrons provides the direct
way to analyze the frozen-in constraint. By noting that the
quasi-stationary reconnection requires the uniformity of the
reconnection electric field throughout the reconnection
region (inflow, DR and outflow) one immediately notices
that inside the EDR the convective term [ve B] is canceled
out and non-ideal terms (r Pe, (ve r)ve) balance the
reconnection electric field. The latter term is cast out as well
because of flow stagnation, thus plasma anisotropy supports
the reconnection in the collisionless regime.
[8] Electron pressure anisotropy within the EDR is commonly found in kinetic simulations of magnetic reconnection. The closures for r Pe are discussed in various
publications. In the present work we use the approach
developed by Fujimoto and Sydora [2009] and by Divin
et al. [2010] to construct the EDR scaling in collisionless
plasmas.
[9] The general EDR properties are investigated. Particlein-Cell (PIC) simulations using implicit PIC code iPIC3D
[Markidis et al., 2010] are performed to verify our estimate.
The results of simulations with different mass ratios (mi/
me = 1836, 256, 64) are compared in order to explore the
parameter range.
2. Scaling of the Inner Diffusion Region
[10] Geometry of the model is similar to the collisionless
magnetic reconnection configurations studied before
[Kuznetsova et al., 1998, 2000; Hesse et al., 1999]. The
main component of the B field is directed along the x
direction; the y-axis is directed along the X-line and the
z-axis is normal to the current sheet. The X-line is located at
x = 0, z = 0. The plasma is assumed to be collisionless and
laminar, hence the dissipation is provided by thermal electron inertia (r Pe) near the X-line [Birn et al., 2001;
Tsiklauri and Haruki, 2007; Shay et al., 2007; Karimabadi
et al., 2007; Tsiklauri and Haruki, 2008]. Steady state
reconnection is considered, so the electric field component
Ey is constant throughout the EDR and its nearest vicinity.
[11] The diffusion region of magnetic reconnection consists of large IDR and smaller EDR enclosing the X-line. At
the inflow boundary z ≳ di, both ions and electrons are
magnetized and particles experience E B drift toward the
X-line (MHD region in Figure 1). Within de ⪅ z ⪅ di the
Hall term-related physics is important, because the magnetic
field is frozen-in here into the electron component of plasma
only. Several fluid models [Uzdensky and Kulsrud, 2006;
Korovinskiy et al., 2008, 2011] describe this region in Hall
MHD or electron MHD approach.
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[12] The inner EDR is the tiniest part of the diffusion
region. A dissipative electric field dominates the convective
electric field there. The EDR area is approximated by a
rectangle and is typically elongated in the direction of outflow. The inner EDR half length is denoted as Le, and the
EDR half width is denoted as le. The magnetic field at the
inflow EDR edge is B0 = (B0, 0, 0), where plus and minus
refer to the upper and lower edges of the EDR.
[13] The following EDR pattern is adopted [Divin et al.,
2010]. Electrons enter the EDR through the inflow boundary at z = le with a velocity vez. They are still magnetized at
the upper boundary and hence magnetic field is frozen into
electron fluid. Within z < le electrons are demagnetized and
accelerated along the X-line by the reconnection electric
field Ey.
[14] The accelerated electrons are deflected by the reconnected magnetic field Bz and leave the diffusion region.
Respectively, characteristic velocity vex is obtained at the
outflow EDR boundary (x = Le). Simultaneously, the electrons moving along the X-line, produce an electric current
which is able to support the initial magnetic field B0. Electrons, moving along meandering trajectories inside the EDR,
balance (via r Pe) the reconnection electric field Ey. The
whole process is expected to be self-consistent. For the sake
of simplicity, guide field effects and By component generation are not considered in the present study. The rise of the
By component and its associated quadrupole structure is
important in the Hall region unlike the EDR.
[15] The half length Le of the inner EDR, as well as the
magnetic field intensity B0 at the inflow boundary of the
EDR, are supposed to be known, since the inner part of
the EDR is investigated only. The magnetic field B0 is
smaller than the lobe magnetic field far in the inflow region,
because the EDR is located deep within the IDR. In order to
determine the exact value of B0, one should know the
behavior of plasma inside the IDR. Coupling the EDR-IDR
dynamics is required. The task is of high analytical complexity and it will be considered in some future study.
[16] The plasma is assumed to be incompressible and
electron density is ne = const. Then we have the following
six unknown parameters:
[17] 1. le - the EDR half width,
[18] 2. Ey - electric field along the X-line,
[19] 3. vez - inflow electron velocity,
[20] 4. vex - outflow electron velocity,
[21] 5. vey - velocity of electrons, accelerated along the
X-line,
[22] 6. Bz - magnetic field intensity at the outflow
boundary of the inner EDR.
[23] They have to be found from the following six equations, that determine the inner EDR physics discussed above.
vez Le ¼ vex le ;
ð1Þ
vez B0 ¼ cEy ;
ð2Þ
vex Bz ¼ cEy ;
ð3Þ
B0 4p
ene vey ;
¼
c
le
ð4Þ
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1=3
e m2e cLe
vey ¼ m
Ey2=3 ;
me e2 Bz
ð5Þ
me vey vez
¼ eEy :
le
ð6Þ
[24] Next, we comment on equations (1)–(6).
[25] Equation (1) expresses the mass conservation.
Equation (2) is the frozen-in condition for the electrons at the
EDR inflow boundary. But the next equality, equation (3), is
not the frozen-in condition at the outflow boundary, rather it
is just the definition of the inner EDR size Le. The convective electric field vexBz/c vanishes at the X-line. However, it
increases with x and becomes twice the electric field Ey
inside the so-called “external EDR” [Shay et al., 2007;
Karimabadi et al., 2007; Drake et al., 2008]. Here we define
the half length Le as the point where cEy = vexBz.
[26] Next equation (4) implies that electrons, accelerated
along the X-line, produce electric current which supports the
gradient of magnetic field B0 at the distance z le.
[27] With the exception of a factor m, equation (5) is the
well-known result on the neutral point particle acceleration
[Bulanov and Sasorov, 1976; Moses et al., 1993; Vekstein
and Priest, 1995; Priest and Forbes, 2000; Divin et al.,
2010]. Approximating the EDR B field by BxE = B0z/le,
BzE = Bzx/Le and taking the stationary and uniform reconnection electric field (Ey), the equation of motion
me
dve
e
¼ ve BE eE
c
dt
reads
me
me
dvex
e
¼ vey BzE ;
dt
c
dvey
e
¼ ðvez BxE vex BzE Þ eEy :
c
dt
[28] During the unmagnetized acceleration phase terms
vezBxE and vexBzE are considered to be small near the X-line.
Therefore, vey can be written explicitly as vey = eEyt/
m + vy0, where the initial electron velocity is determined by
the term vy0. In the cold plasma limit vy0 0 and the vex
equation is
dvex e2 Ey Bz
¼ 2
xt:
dt
me Le c
ð7Þ
[29] A characteristic timescale t a = (e2EyBz/m2e Lec)1/3 of
equation (7) is the time required for a particle to traverse the
EDR. Velocity |vey| = eEyt a/me is gained during the acceleration process, hence the equation (5). The approximations
(vy0 0, vezBxE vexBzE 0) used to derive the equation (5)
are rather crude. In order to introduce flexibility to our study,
we put a factor m in the equation (5), which is responsible for
the preacceleration of electrons outside the inner EDR. A
rationale for that step will be examined in section 3.2.
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[30] The last equation (6) is the approximation to the
electron pressure tensor component Pyz, presented in
Fujimoto and Sydora [2009] and Divin et al. [2010]. For the
case of cold plasma [Fujimoto and Sydora, 2009], the electron behavior in the EDR creates off-diagonal pressure tensor term in the Ohm’s law near the X-line. Divin et al.
[2010] considered the case of warm plasma, which is summarized next in three main points:
[31] 1. Particles inside the EDR fall into two broad categories: accelerated and inflowing.
[32] 2. These classes occupy different positions in the
velocity space. Bi-Maxwellian distribution function renders
the electron anisotropy.
[33] 3. Pressure divergence r Pe appears because the
relative density of these populations changes, with the density of accelerated electrons peaking at the EDR center.
[34] Because the scaling of inner EDR is considered, it is
convenient to introduce the electron reconnection
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirate
ɛe = Ey/EAe. Here EAe = B0VAe/c and VAe ¼ B0 = 4pne me are
the electron Alfvén electric field and the electron Alfvén
velocity, respectively. Then the system of equations (1)–(6)
for the inner EDR can be solved to give
vey ¼ VAe ;
ɛe ¼ kde =Le ;
vex ¼ kVAe
vez ¼ ɛe VAe ;
ð8Þ
le ¼ de ;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
4pnp mp (mp and np are proton mass and proton density,
respectively). The potential drop across the EDR
[Korovinskiy et al., 2008, 2011] is estimated as
1
B2
Df ¼ VAe B0 de ¼ 0 :
c
4pne
Ey ¼ ɛe EAe :
3/2
ð9Þ
ð10Þ
[35] We renamed the factor k = m
here for short. It can
be seen from (8)–(10) that electrons are accelerated up to
electron Alfvén velocity in y direction. The half width of the
EDR is equal to electron inertial length de. The magnetic
field Bz at the outflow EDR edge is weak ( ɛe), and electron reconnection rate is of the order of de/Le. It is noteworthy that these results are obtained only as a solution of
the system of equations (1)–(6), with the equation (5) for
accelerated particle velocity and electron pressure component Pyz closure (6) containing the essential EDR physics.
No prior scale factors (e.g., de, VAe) are introduced in the
equations.
[36] The EDR width and characteristic velocities (8)–(10)
appear to have an order-of-magnitude agreement with
Particle-in-Cell simulations reported in literature [Kuznetsova
et al., 2000; Daughton et al., 2006; Drake et al., 2008;
Pritchett, 2010]. More precise theory will estimate the k
factor entering the particle acceleration equation (5). In the
present paper we use Particle-in-Cell simulations in order to
verify the scaling relations (8)–(10) and find the numerical
value of k.
[37] Some general implications of the scaling (8)–(10) are
discussed next.
[38] It is a common fact that ion diffusion region dynamics
is coupled to the EDR flow properties. The electrons have
high outflow velocity (vex kVAe) at x Le at the inner EDR
edge. The accumulated energy is then transferred to protons
by means of electrostatic electric field. Electrons can accelerate protons up to the proton Alfvén velocity VA ¼ B0 =
ð11Þ
It means that fast electrons moving along the neutral line
create a strong electric field Ez veyB0/c = EAe inside the
EDR (that is, of the order of electron Alfvén electric field),
which in turn is convected along the separatrices.
[39] As it was pointed out previously, the electron velocity
is directed parallel to the X-line first. Electrons are turned to
the outflow direction away from the EDR approximately
over a local Larmor radius rLe ≅ mevAec/(eBz). From the
fluid point of view, this implies that whistler or kinetic
Alfvén waves are launched in the EDR which transfer the
potential drop (11) from the EDR to separatrices.
[40] The results (8)–(10) can also be presented in SweetParker format, by introducing an effective conductivity and
the corresponding magnetic Reynolds number. Since the
electric field Ey and the current j = nevey are known from the
EDR scaling (8)–(10), we can formally calculate the conductivity s:
s¼
Bz ¼ de =Le ¼ ɛe B0 =k;
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j
1 nec nec 1
:
¼
¼
Ey ɛe B0
Bz k
ð12Þ
This expression is similar to the conductivity provided by a
Bohm-like diffusive process [e.g., see Miyamoto, 1980; also
Lyons and Speiser, 1985; Priest and Forbes, 2000]. This can
be understood by calculating the formal collision frequency
n = nee2/mes, which, in turn, provides for the equality
between effective collisional and Larmor frequencies
n = kBze/mec = ɛeB0e/mec, since s is given by equation (12).
The same argument was used for the derivation of Bohm
diffusion coefficient, and the latter is often considered to be
the upper limit for diffusion in plasma. Evidently, Bohm
conductivity is many orders of magnitude smaller then the
classical Spitzer conductivity. The effective conductivity
(12), which is localized near the X-line, can provide nearly
the upper limit of the reconnection rate in collisionless
plasma.
[41] The physical meaning of the effective collision frequency n can be established, if one calculates the acceleration
time t a using the values, provided by equations (8)–(10),
t a mec/kBze ≡ n 1. The collision frequency n is equal to
and electron gyroboth the inverse acceleration time t 1
a
frequency, based on magnetic field kBz. This notion explains
the decrease of plasma conductivity s: the interaction of a
single electron and EDR is considered to be a collision, and
the inverse of collision frequency is equal to the typical
electron trapping time t a inside EDR.
[42] The electron Reynolds number can be introduced,
taking the characteristic values from equation (12) and
scaling equalities (8)–(10),
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Re ¼
4psVx Le
¼ L2e =de2 ¼ k 2 =ɛ2e :
c2
ð13Þ
DIVIN ET AL.: ELECTRON DIFFUSION REGION SCALING
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Table 1. Parameters of Different Runs
Run
me/mi
nb/n0
Lx Lz
Nx Nz
1
2
3
4
1/1836
1/256
1/256
1/64
0.2
0.1
0.2
0.2
20di 10di
200di 30di
30di 15di
25.6di 12.8di
2048 1024
2560 384
1472 736
640 320
[43] Hence, the EDR scaling can be rewritten in the
Sweet-Parker-like form,
vex =k ¼ vey ¼ VAe ;
pffiffiffiffiffiffiffi
vez ¼ Vex = Ree ;
pffiffiffiffiffiffiffi
Bz ¼ B0 = Ree ;
pffiffiffiffiffiffiffi
Ey ¼ kEAe = Ree ;
ð14Þ
Temperature ratio is Ti/Te = 5 for all runs. Temperatures of
background and current sheet plasmas are equal at t = 0. The
thickness of the initial current sheet is L = 0.5di. A small
initial non-GEM perturbation [Lapenta et al., 2010] is added
to start reconnection:
Yðx; zÞ ¼ Y0 cos
ðxL =2Þ2 þðzL =2Þ2 x
z
2pðx Lx =2Þ
pz
s2
cos e
;
Lx
Lz
where B′ ¼ r Yðx; zÞ^y, which is added to the initial
magnetic field (17). The intensity Y 0 is 0.1, and range s is
1di. Perfect electric conductor (PEC) boundaries are set at
z = 0 and z = Lz. Periodic boundaries are set in x direction.
Run parameters are summarized next in Table 1.
ð15Þ
ð16Þ
i.e., tangential values (vex, vey) are of the order of O(1)pwhile
ffiffiffiffiffiffiffi
the normal values (vez, Bz) are of the order 1= Ree .
Physically, this means that collisionless magnetic reconnection for the electron fluid can be interpreted as the SweetParker-like process locally inside the inner EDR, having
effective magnetic Reynolds number Ree. Bohm-like conductivity near the X-line reduces Ree significantly, whereas
the rest of electron fluid (far from the X-line) remains
magnetized and can be considered ideal. This configuration
resembles strongly fast Petschek-type reconnection, which
occurs if resistivity is enhanced in a region much smaller
than global scales. The present version of the scaling (8)–(10)
does not present any limitations on the inner EDR length Le.
However, some kinetic instabilities (e.g., tearing instability)
are known to effectively limit Le extent and preclude the
EDR elongation up to macroscopic scales, keeping the
reconnection rate high.
3. Comparison to PIC Simulation Results
[44] Particle-in-Cell simulations of antiparallel magnetic
reconnection are presented next to verify the derived scaling.
The code description and simulation setup are followed by
the EDR parameters study for different mass ratios and
derivation of improved scaling.
3.1. PIC Simulations Setup
[45] The full-particle implicit code iPIC3D [Markidis
et al., 2010; Lapenta et al., 2010] is utilized. The implicit
moment PIC method [Brackbill and Forslund, 1982], used
in the code, reduces the required computational resources
dramatically and good resolution is feasible for mi/me = 1836
runs. Here ions are considered to be protons. Code units are
normalized to conventional Alfvén units [e.g., Birn et al.,
2001; Zeiler et al., 2002].
[46] The following parameters are common for all runs.
The initial condition is represented by Harris [Harris, 1962]
current sheet, having the asymptotic field BH, peak current
sheet density n0 and background population nb:
z
Bx ðzÞ ¼ BH tanh ;
L
z
ns ðzÞ ¼ n0 cosh2 þ nb ; s ¼ e; i
L
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ð17Þ
3.2. Results
[47] Two runs (Run 1 and Run 2, mass ratios mi/me = 1836
and mi/me = 256, respectively) are reported in detail. Other
runs (Run 3 and Run 4, mass ratios mi/me = 256 and
mi/me = 64) are discussed briefly to show the consistency of
the scaling.
[48] Typical reconnection configuration is shown in
Figure 2 for Run 1 (Wci0t = 15.2). The nearest vicinity of the
EDR is almost stationary. The initial current sheet is reconnected by this moment (compressed remnants of the initial
current sheet are visible in Figure 2a, x > 14di and x < 6di).
The reconnection electric field Ey is constant around
8di < x < 12di and 4di < z < 6di (see Figures 2b and 2e). Ey
accelerates electrons near the X-line in y direction
(Figures 2c and 2f). The jet is turned in X-Y plane,
corresponding to vex component rise.
[49] Ohm’s law terms are plotted in Figure 2b for z = Lz/2
and panel (e) for x = X(), where X() denotes the current
X-point location. The electron pressure anisotropy term
(∂ Pyz/∂ z) breaks magnetic field lines, similar to simulations
performed in Ref. [Divin et al., 2010]. The approximation
nemevezvey is plotted in Figure 2d. This approximation
agrees well with the pressure component and was verified
for a range of me/mi mass ratios (not shown here).
[50] The scaling derivation (equations (8)–(10)) is done
for quasi-stationary process. However, PIC simulations are
inherently non-stationary. Therefore, comparison of the
typical EDR parameters (B, v, le, Le) and the scaling relations (8)–(10) should be performed at multiple times t; the
EDR position must be known, correspondingly.
[51] The X-line drifts constantly in the x direction; the
displacement in the z direction can be neglected. Therefore,
a special algorithm, that estimates the EDR parameters for a
given time step, is constructed.
[52] The algorithm finds the current X-line position in a
two dimensional simulation. All the points having Bz = 0 on
z = Lz/2 plane are either X- or O-lines in the present twodimensional geometry. Fluctuations and tearing instability
usually lead to formation of multiple small-scale neutral
lines far from the major reconnection site. After Wci0t > 5,
the X-line closest to the (Lx/2, Lz/2) point is the major one.
The Ohm’s law terms (ve B, r Pe, (ve r)ve) are
computed locally to mark the EDR extent. The EDR width le
is calculated as the z distance between X-line and a point
having vezBx = Ey (see equation (2)). Similarly, Le is calculated as the x distance between X-line and a point, where
vexBz = Ey (equation (3)). Here, the Ey is sampled from the
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Figure 2. General configuration of magnetic reconnection region, Run 1 (mi/me = 1836), Wci0t = 15.2:
(a) out-of-plane electron current enevey; (b) Ohm’s law terms along z = Lz/2 line; (c) ion and electron
velocities along z = Lz/2 line; (d) electron pressure component Pyz and approximation, nemeveyvez across
the X-line (x = X()); (e) Ohm’s law terms along x = X() profile; and (f) electron and ion velocities along
x = X() profile. Here Wci0 is ion gyrofrequency in ambient magnetic field BH.
X-line, vx and Bz are computed at the outflow EDR edge and
vz, B0 = Bx are computed at the inflow EDR edge. The set of
scaling quantities is completed by taking vy and ne from the
EDR center. The vertical lines in Figure 2 visualize the Le, le
estimation. Vertical lines corresponds to Le extent (length) at
the left part of the image and the EDR width le at the right
part of the image.
[53] Simulated EDR parameters are shown in Figures 3
and 4 (Run 1 and 2, respectively) for different times t. The
simulation results are grouped into four panels (a–d),
representing such combinations of EDR parameters that
should be identical according to the scaling. Figures 3a and
4a display magnetic field estimate (from equations (8), (10),
Bz = B0ɛe/k = B0Ey/kEAe = B0de/Le). Figures 3b and 4b show
the reconnection rate estimates (ɛe = Ey/EAe = kde/Le = kvez/
vex = kBz/B0). Figures 3c and 4c show the computed EDR
width lecompared to the local electron inertial length (see
equation (9)). Figures 3d and 4d display the velocities
(according to equation (8), vex/k = vey = VAe). The k factor
is k = 0.36 (Run 1) and k = 0.3 (Run 2). The value of k is
estimated by minimizing the differences kVAe kvex||,
||ɛe kde/Le||, ||Bz ɛeB0/k|| during quasi-stationary phase.
[54] The following basic behavior is clearly visible in both
Figures 3 and 4:
[55] 1. Electrons are accelerated along the X-line (y direction) up to the local electron Alfvén velocity starting with
Wci0t 5 (Run 1, see Figure 3d) and Wci0t 10 (Run 2, see
Figure 4d). Notably, quasi-stationary regime is reached at
later times (Wci0t 10 for Run 1 and Wci0t 15 for Run 2),
when the reconnection rate estimate Ey/EAe reaches a constant value, see Figures 3b and Figure 4b. Also, as seen in
Figure 2c, the point with vex peak is outside the inner EDR.
The peak values of vex and vey are usually equal or comparable (Runs 2–4, not shown).
[56] 2. The electron inertial length de, calculated using
local electron density, agrees well with the EDR width. The
data spread is rather large, especially at the later stage
(Figure 4c, t ≥ 20), mainly because of the small EDR size
and the statistical PIC noise.
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Figure 3. Time evolution of EDR parameters, Run 1 (mi/me = 1836). Bx and vez are computed at the
inflow EDR edge (z Lz/2 = le, x = X()), Bz and vex are computed at the outflow EDR edge (x X() = Le,
z = Lz/2), Ey, vey, ne are taken from the point (x, z) = (X(), Lz/2). Electron Alfvén velocity is estimated for
inflow plasma parameters VAe = Bx/(4pneme)1/2.
[57] 3. The behavior of Bz = B0de/Le component of the
magnetic field (see Figures 3a and 4a) is consistent with the
model for Wci0t > 10 (Run 1), Wci0t > 15 (Run 2).
[58] 4. The k factor is needed to match the outflow magnetic field estimate Bz B0de/Le ɛeB0/k (see Figures 3a
and 4a) and reconnection rate estimates ɛe/k de/Le vez/
vex Bz/B0 (see Figures 3b and 4b) and electron velocity
estimate vex kVAe (see Figures 3d and 4d).
[59] An interpretation of the k factor is presented next.
[60] In Figure 5, a profile of Ez(z) and a convective term
vey(z)Bx(z) across the X-line are plotted for Run 1. It is clear
that a strong Hall electric field is established near the X-line,
peaking at the EDR edges. When a particle traverses the
enhanced Ez region and then gets unmagnetized inside the
EDR, a velocity ‘kick’ vy0 of the order of (E B)y drift
velocity is given. The estimate for the Hall E field is
expressed as Ez = veyB0, where vey and B0 are the characteristic EDR current velocity and upstream magnetic field,
respectively. At the time Wci0t 15 the EDR inflow edge B0
magnitude is B0 0.11 (see Figures 3a and 3b), and the
electron vey velocity is 15 (in the reference Alfvén units).
Hence, the peak Ez B0vey 1.6, in correspondence with
Figure 5b.
[61] We assume that the total velocity gained by a particle
during X-point acceleration is a sum of conventional vey
scaling (see equation (5) with m = 1) [Bulanov and Sasorov,
1976; Moses et al., 1993; Vekstein and Priest, 1995] and an
initial velocity vy0:
vey ¼
1=3
e m2e cLe
Ey2=3 þ vy0 :
me e2 Bz
ð18Þ
[62] The initial velocity ‘kick’ is estimated as a fraction of
(E B)y Ez/B0 drift velocity, which we express as
vy0 Ez
¼ ð1 m1 Þvey ;
Bx
that is the EDR velocity estimate vey appears implicitly in
vy0. With these in mind, the equation (18) takes the form
1=3
1
e m2e cLe
Ey2=3
vey ¼
m
me e2 Bz
ð19Þ
identical to equation (5). More precise theory will estimate
the unknown factor k analytically.
[63] The EDR scaling relations for all runs are presented in
Figure 6.
[64] Figures 3 and 4 clearly indicate that reconnection
parameters vary strongly during the process. Reconnection
is intermittent before tWci ≈ 15, whereas closed inflow
boundaries (z = 0, z = Lz) lead to a gradual magnetic flux
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Figure 4. As in Figure 3, but for Run 2.
depletion at later times tWci > 20. Hence, some specific time
should be selected in order to study the scaling properties
dependence on the mass ratio mi/me. A comparison of different runs is performed at the beginning of quasi-stationary
phase (tWci ≈ 15), with a few time points selected before and
after to cover the PIC statistical noise.
[65] Similarly to Figures 3 and 4, the EDR parameters in
Figure 6 are grouped into panels which represent scaling
identities (8)–(10). Figure 6a shows the magnetic field estimate: Bz = B0de/Le = (1/k)B0Ey/EAe. Figure 6b displays the
reconnection rate estimates (Ey/EAe = kde/Le = kBz/B0). The
EDR width le compared to the electron inertial length is
plotted in Figure 6c. Figure 6d shows the characteristic EDR
velocities (vex/k = vey = VAe). Theoretically, Bz should be
equal to B0de/Le and (1/k)B0Ey/EAe (see Figure 6a). However, the parameters are smoothed out by extra physics not
included into derivation, or by PIC fluctuations.
[66] Runs 1–4 are coded by blue, green, magenta, red
colors, respectively. The scaling reveals good consistency
for Runs 1–3 (high mi/me mass ratios), that is, the data
spread is relatively small. Scattering is more pronounced for
Run 4 (mi/me = 64, seen in Figures 6a–6c), probably because
of more significant ion contribution to reconnection
dynamics for low mass ratios.
[67] The EDR width le, shown in Figure 6c, decreases
roughly as (mi/me)1/2 with increasing mi/me, if expressed in
units of di. It confirms the basic scaling property
le de di(mi/me)1/2. Run 2 has a larger EDR width
(shown in green in Figure 6c), owing to the lower inflow
plasma density.
[68] The velocity components (Figure 6d) are found to
satisfy the improved scaling in all runs. In addition, the
Figure 5. Electric field Ez in Run 1 at t = 15.2W1
ci0 .
(a) Ez(x, z) near the X-line and (b) Ez(z) (blue line) and the
convective contribution vey(z)Bx(z) (gray line) at x = X().
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Figure 6. (a–d) Variation of EDR parameters relations with mi/me, arranged as in Figures 3 and 4.
Runs 1–4 are marked by color: blue (Run 1), green (Run 2), magenta (Run 3), red (Run 4).
dependence on background density nb is clearly visible for
mi/me = 256 case (compare Run 2, green and Run 3,
magenta), because the lower nb Run 2 corresponds to higher
VAe given that magnetic field B0 at the inflow EDR edge are
almost identical (Figure 6a) for a fixed mi/me. The vey and
VAe dependence on mass ratio is a more difficult question,
since VAe depends on the inflow boundary magnetic field B0,
which changes with mi/me as well. The study of B0(mi/me)
requires the coupling between electron and ion dynamics
and, therefore, can only be roughly calculated. Remarkably,
electron reconnection rate estimates are relatively uniform
(≈0.1) for a wide mass ratio interval. Then, considering the
a ffiffiffiffiffiffiffiffiffiffiffiffi
mass ffidependent quantity, we
expression Ey/EAe not to bep
immediately obtain that Ey me =mi B20 is constant as well.
Since global reconnection rate Ey is widely believed to be a
constant [Birn et al., 2001], a simple relation B0 (me/mi)1/4
is obtained. Weak mass relation can indeed be seen in
Figure 6: the ratio of EDR outflow magnetic field Bz for
Runs 1 and 4 (red and blue in Figure 6a, respectively) is of
the order of (1836/64)1/4 ≈ 2.3; the same ratio of 2.3 is found
between EDR velocities vey in Runs 1 and 4 (Figure 6d, red
and blue).
[69] A mass dependence study, displayed in Figure 6,
outlines the scaling (8)–(10) properties and allows to conjecture a B0(mi/me) relation. The scaling derivation requires
the reconnection electric field to be a known parameter of
the process. However, the actual value of Ey in collisionless
magnetic reconnection is determined by both the external
configuration and inner physics. Even though other attributes of collisionless process being relatively well described,
the exact reconnection rate value remains a riddle that needs
a more rigorous theoretical research.
4. Summary and Conclusions
[70] Theoretical and numerical studies of electron diffusion region of collisionless magnetic reconnection are presented in this paper. The dissipative electric field, generated
by electron pressure gradient inside the EDR is considered to
be the main mechanism for breaking the magnetic field lines.
Starting from the closure for Pyz component, provided by
Divin et al. [2010], the scaling relations for the inflow and
outflow velocities, the current velocity, the EDR width and
exhaust magnetic field are derived. The following theoretical
estimates are established:
[71] 1. The electron current velocity vey (at the EDR center) is equal to electron Alfvén velocity VAe, and the outflow
velocity vex is equal to kVAe.
[72] 2. The EDR width is equal to the electron inertial
length de.
[73] 3. The outflow magnetic field Bz to inflow magnetic
field B0 ratio, the EDR width le to length Le ratio, the inflow
velocity vez to outflow velocity vex ratio are equal to the
parameter ɛ/k = Ey/kEAe,where EAe = VAeB0/c.
[74] 4. The conductivity estimate (calculated as s = j/Ey)
can be expressed in terms of Bohm diffusion nec/B0; therefore, the effective electron collision frequency is of the order
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of electron gyrofrequency, giving rise to the anomalous
laminar resistivity in purely collisionless environments.
[75] At last, the parameters VAe and B0 are calculated at the
inflow edge of the EDR. The ion diffusion region current
should reduce the B0 when compared to some reference
magnetic field far in the ambient region. However, this task
is of high analytical complexity and requires matching of
ion, electron, and large-scale MHD dynamics.
[76] The factor k appears as an unknown free parameter in
the scaling relations. If particle acceleration mechanism is
similar to that studied by Bulanov and Sasorov [1976],
Moses et al. [1993], and Vekstein and Priest [1995], then
k = 1, but le/Le and vex show a factor of 3 discrepancy
when compared to Particle-in-Cell simulations. The k factor
was interpreted as a preacceleration by strong Hall electric
field at the inflow EDR edge (see Figure 5). It produces a
‘kick’ to particles, which are at the initial stage of acceleration by reconnection electric field. The factor m = k2/3
in equation (5) turns out to be m 2 which means that the
electrons are preaccelerated to the half of VAe by the strong
Hall electric field and then they are accelerated by the
reconnection electric field to another half of VAe in the y
direction. Self-consistent analytical estimates would require
the matching of the inner EDR and the IDR solutions. As
an initial step, we found the factor k numerically by means
of PIC simulations.
[77] By using the implicit PIC code iPIC3D [Markidis
et al., 2010], the calculations were performed with mi/me
ratios equal to 64, 256 and 1836. Simulations revealed that
inner EDR parameters are in a good agreement with the
scaling relations (8)–(10).
[78] Noteworthy that similar expressions (vex VAe and
le de) were obtained in past works [Hesse et al., 1999;
Kuznetsova et al., 2000; Tsiklauri and Haruki, 2007;
Tsiklauri, 2008; Pritchett, 2010]. The present article uses the
closure for electron pressure anisotropic component Pyz of a
different form [Divin et al., 2010]. In addition, particle
acceleration mechanism is introduced in equation (5). The
equation for Pyz is valid for the case of “warm” plasma,
when electron thermal velocity is of the order of inflow
velocity vz. The thermal motion of electrons can influence
the EDR physics significantly. For hot enough plasma
(where electron thermal gyroradius rLe exceeds greatly the
electron inertial length), the parameter rLe should provide
for the better estimate for le.
[79] The scaling relations are developed for the antiparallel reconnection. Several studies [Hesse et al., 2004;
Pritchett and Coroniti, 2004] address the role of the out-ofplane (By) component in EDR dynamics. The guide field
magnetizes the electrons inside EDR and distorts the meandering trajectories, hence the equation (5) appears to be
inconclusive. The mechanism, proposed by Divin et al.
[2010], is no longer valid and some other closures for Pyz
should be considered.
[80] Two-dimensional PIC simulations cannot capture
the multitude of waves and instabilities that develop in the
X-line direction. Therefore, the condition ∂/∂ y = 0 stabilizes
the process significantly. The flows near the X-line are relatively quiet, hence the major non-ideal contribution in the
Ohm’s law is in the form of laminar electron collisionless
dissipation. Three-dimensional effects are the matter of
active research now [Pritchett and Coroniti, 2004; Yin et al.,
A06217
2008; Daughton et al., 2011; Divin et al., 2012; Markidis
et al., 2012]. Current-driven instabilities produce anomalous
drag that enhances the dissipation near X-line [Zeiler et al.,
2002; Drake et al., 2003; Che et al., 2011], and the interplay between the turbulent and r Pe-based mechanisms is
still poorly understood.
[81] Concluding, the scaling developed in the present
article provides for the order-of-magnitude estimates for the
EDR parameters in laminar regime. Including the effects of
guide field, thermal motion of electrons and electron pressure anisotropy in the external EDR would improve the
actual values, but impact the apparently scaling simplicity.
These effects will be a matter of further studies.
[82] Acknowledgments. The present work is supported partially by
the Onderzoekfonds KU Leuven (Research Fund KU Leuven), by the
NASA MMS grant NNX08AO84G and by the European Commission’s
Seventh Framework Programme (FP7/2007-2013) under the grant agreement 263340 (SWIFF project, www.swiff.eu) and 269198 – Geoplasmas
(Marie Curie International Research Staff Exchange Scheme). Additional
support is provided by RFBR grants 09-05-91000-ANF-a, 12-05-00152-a
and 12-05-00918-a, Austrian Science Fund project I193-N16 and by SPSU
grants 11.38.47.2011 and 11.38.84.2012. The simulations were conducted
on the resources of the Vlaams Supercomputer Centrum (VSC) at the
Katholieke Universiteit Leuven. The authors wish to thank the reviewers
for their comments that helped to improve the manuscript.
[83] Masaki Fujimoto thanks the reviewers for their assistance in evaluating this paper.
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