Proceedings of ITC/ISHW2007

Formation of Sweet-Parker-like electron dissipation region in adriven open system
Bin Li1) , Ritoku Horiuchi1,2) and Hiroaki Ohtani1,2)
1)

The Graduate University for Advanced Studies,322-6 Oroshi-cho, Toki 509-5292, Japan
2)
National Institute for Fusion Science,322-6 Oroshi-cho, Toki 509-5292, Japan
October 4, 2007

Nonlinear development of collisionless driven reconnection is investigated by making use of the electromagnetic particle simulation code ”PASMO” developed for an open system which is subject to an external
driving source. The electric field at the reconnection point increases and approaches the external driving
field as time goes on. After the formation of x-shaped field structure around the reconnection point, thelength of the electron dissipation region continues to increase for a short time. Finally, it stops to grow andrelaxes to a steady state when the ratio of the width to length is a constant. Thus, Sweet-Parker-like electron
dissipation region is formed in a steady state, while the reconnection rate is controlled by the driving electric field.
Keywords: driven reconnection, collisionless, Sweet-Parker-like, steady state, electrons dissipation region

1 Introduction
Magnetic reconnection occurs in a wide variety of plasma
systems, such as collision-dominated plasmas in the solar
convection zone, weakly collisional plasmas in the solar
corona, and collisionless plasmas in the Earth’s magnetosphere, plasma experiments such as tokamak and reversed
field pinch[1, 2]. During magnetic reconnection, most ofmagnetic energy is effectively transformed into plasma energy.
The structure of dissipation region, where reversed
magnetic field lines are dissipated and reconnected, has
not been understood well for a long time. Recently experimental results [3] of Magnetic Reconnection Experiment (MRX) and observations from WIND satellite [4]shed new light on this research direction. Their results have
implied that the dissipation region has a Sweet-Parker likespatial geometry: a long and narrow current layer extends
along the downstream direction.
Unfortunately, Sweet-Parker model is simply based onclassical collisional resistivity assumption and cannot explain time scale of fast energy release in solar flare. Thus,
it is necessary to investigate what mechanism is responsible for breaking the frozen-in flux constraint in the dissipation region. In a series of previous two dimensional simulations, dissipation mechanisms are generally thought tobe provided by microscale collisionless kinetic effects, i.e.,
the inertia effect [5, 6] and the thermal effect [7, 8] based on
the non-gyrotropic meandering motion. It is implied thatthe evolution of collisionless reconnection is controlled by
the particle kinetic effects in two dimensions.
In this paper, we will pay attention to investigating
steady state of fast magnetic reconnection in a driven case
based on the three dimensional particle-in-cell (PIC) code
author’s e-mail: li.bin@nifs.ac.jp”PASMO” [9, 10]. It is found that in our simulation aroundthe X-line, an electron dissipation region is formed and extended to the downstream. In a steady state, it has a rectangular shape current layer, which is a typical prediction of
the Sweet-Parker model.
The outline of the paper is as following; first we
describe the simulation model, and second introduce thestructure of the electron dissipation region and finally discuss the simulation results.

2 Simulation model
The simulation is carried out by using three dimensional
PIC simulation code ”PASMO”, 2D version of which has
been successfully used in previous investigations[7, 11, 12,13, 14, 15].
Physical quantities at the boundary of z axis is (z =±zb ) are assumed to be periodic, upstream and downstream
boundaries are set at those of y axis (y = ±yb ) and x axis
(x = ±xb ), respectively.
At upstream boundary, plasmas satisfy the frozen-incondition. By adopting an external electric field Ez (x, t) at
y = ±yb along z-axis, ions and electrons are driven into thesimulation domain at the same drift velocity. The driving
electric field is relatively larger within the input window
size xd around x = 0 during the Alfvén time τA = yb /VA ,where VA is the initial average Alfvén velocity. Therefore, plasmas have larger speed within the size of the input
window so as to make the convergent plasma flow into thecenter of simulation domain where magnetic reconnection
will occur first. The driving field will approach a uniform
profile with a constant value along the upstream boundary. Magnetic fields can change spatially and temporallyaccording as the driving electric field evolves.

Proceedings of ITC/ISHW2007

value difference of two position Ez

0.05

0.04

0.03

0.02

0.01

0.00

-0.01

-0.02

0300

600

900t

1200

1500ceFig. 1 Difference of electric fields at X-line and upstream
boundary as a function of time.
trace1

0.12

0.08

downstream Ez

At the downstream boundary x = ±xb , plasmas number density is controlled by both the charge neutrality condition and the condition of the net number flux, which is associated with the fluid velocity in the vicinity of the boundary [14]. Thus, the plasma can freely flow in or out, thetotal number of particles may vary with time in this open
system. The field quantities E x ,Ey and ∂ x Ez are continuous at downstream boundaries. The other components ofthe electromagnetic field can be obtained by solving theMaxwell equations at the boundary.
One-dimensional Harris sheet equilibrium is adopted
as an initial condition, where magnetic field and plasmapressure
are given by Bx (y) = B0 tanh(y/yh ) and P(y) =.
B20 8π· sech2 (y/yh ) with the scale height yh . As a consequence of initially reversed magnetic field configuration,a neutral sheet appears at the center of y axis (y = 0).
The Lorentz force pushes plasmas toward neutral line,
while plasma pressure makes plasmas move away from it.
Thus the initial equilibrium is maintained by two forces
counterbalance. The distribution of particles is a shifted
Maxwellian with a uniform temperature Ti0 = Te0 . We set
the particle mass ratio mi /me = 100, the ratio of plasma
fre.
quency to the electron cyclotron frequency ωpe0 ωce0 = 2,
the strength of the inflow velocity E0 /B0 = −0.04, andthe ratio of input window size to inflow direction length
xd = 1.5 in our numerical simulations.

0.04

0.00

-0.04

trace8

-0.08

-0.12
-40

3 Electron dissipation region
In our simulation model, reconnection rate equals to theelectric field at the X-line[7, 9, 10, 11, 12, 13, 14, 15].
When system relaxes to a steady state, both electric field Ez
in the system and reconnection rate should equal to the external driving electric filed. After reconnection process begins, the electric field at X-line increases and equals to thedriving field after tωce = 750. The difference between electric fields at the X-line and upstream boundary is shown in
Fig.1. However, the electric field along the downstream direction does not reach uniform profile until tωce ≈ 1110,asshown in Fig.2. After then, the electric field becomes equal
to the driving field everywhere in the system, and thus thesystem relaxes to the steady state.
Fig.3 shows spatial profile of electrons current density
at tωce = 900. The electron current density evolves gradually in a long and narrow rectangular region at the centerof current layer. Thus, electron current sheet is formed in anarrow and long shape, which is the typical result predictedin the Sweet-Parker model. The current density increases
from upstream boundary to the X-line in the inflow direction and magnetic field Bx gradient also increases. To understand the structure of the current sheet, let us considerthree spatial scales:. electrons meandering motion scale lme ,skin depth de = c ω pe , and half width of the sharp peakin the current density profile. The electron meandering
scale is defined by the distance which satisfies the condi--30-20-10010203040xFig. 2 Spatial profile of electric field Ez along the outflow direction at eight consecutive time periods, where trace 1denotes the profile at tωce = 662, trace 8 at tωce = 1148.
The reference level of each line is shifted vertically toavoid overlapping of lines.

tion ρe (y)/y = 1 [5, 6], where ρe (y) is the local electron
Larmor radius. The half width of the current density sharp
peak is defined by the half width at 80% maximum value ofthe current density. In Fig.4, time evolution of three spatial
scales defined above are plotted, where they are normalized by Debye length and are measured along the verticalline through the X-line. The half width of the sharp peakvalue is close to the electrons meandering motion scale andhas similar tendency in time evolution. This result impliesthat electrons dynamics control the current sheet formation
around the X-line and thus a Sweet-Parker-like electrons
dissipation region is generated.
In a kinetic regime non-ideal effects becomes significant through microscopic physical processes and macroscopic frozen-in condition is not satisfied [8, 12, 13, 14,15]. Ion spatial scale in which non-ideal effects becomessignificant for ions are larger than that for electrons. In
other words, wider ion dissipation region is formed in
the current sheet compared with electron dissipation re-Proceedings of ITC/ISHW2007Electrons frozen in condition

T0.06

=1782

Electrostatic field Ecey

0.3

5y0

-5

-30

-20

-1001020

0.2

0.02

0.1

0.00.00

EyElectrons frozen in condition

0.04

-0.1

-0.02

-0.2
-0.04

30

x

-0.3
-0.06
-8

-6-4-2

2.5

Electrons skin depthElectrons meandering scale
2.0

Scale length/d0Half width of current sharp peak

024

68y

Fig. 3 Spatial profile of electrons current density at tωce = 900.

Fig. 5 Spatial profiles of electron frozen-in condition(E + ve × B)z and electrostatic field Ez along theinflow direction. Positions of electron skin depth andmeandering scale are indicated by vertical dotted linesand dashed lines, respectively.

1.50.101.00.08

0.50.06/L
0300

600

t

900

1200

1500

ce

0.04

Fig. 4 Time evolution of three spatial scales; electron skindepth, electron meandering scale, and half width of current density peak.

0.020.00
400

600

800

1000

1200

1400

1600

1800

t

cegion. Figure 5 demonstrates the spatial profiles of electron
frozen-in condition (E + ve × B)z and the electric field Ey .
Because electrons remains magnetized while ions are notmagnetized in the ion dissipation region, the driving field
pushes mainly electrons inward and creates electron richregion inside the electron dissipation region. The in-planeelectrostatic field is generated due to the charge separationin the kinetic regime and is largest at edge of electrons dissipation region, as shown in Fig. 5. The violation of electron frozen-in condition starts at electrons skin depth andbecomes significant below the electron meandering scale
In the steady state, spatial configuration of magnetic
field does not change with time and thus the width of electron dissipation region is expected to be a constant too,
which is defined by the electrons skin depth. After reconnection rate becomes a constant (tωce = 750), the skindepth also becomes constant. Based on the electrons massconversation let us define the length of the electrons dissipation region by the distance from the X-line to the position where electrons outflow speed is maximum. The ratio of these two scales, r = de /le , can be used to evaluate
the structure of electrons dissipation region. In Fig.6, time
variation of this ratio is plotted. Although the reconnection

Fig. 6 Time evolution of the ratio of electron dissipation region
width to its length.

rate, electrons skin depth and its meandering motion scalealready becomes constant(see in Figs.1 and 4), the curve
still gradually decreases after that. The length of electrons
dissipation region increases until tωce ≈ 1100 and then theratio becomes constant finally. This result corresponds thefact that the electric field in the downstream reaches a constant profile at tωce ≈ 1100 (see Fig.2).

4Summary

We have found that a Sweet-Parker-like electron dissipation region is formed during the process of magnetic reconnection. Its width is determined by electron skin depth
while its length is defined by the distance from X-line tothe position where electron outflow speed has maximum
value. The electron dissipation region is mainly controlled
by microscopic electron dynamics.
It should be mentioned that there is the delay in theProceedings of ITC/ISHW2007
period when inflow and outflow regions relax to the steadystate. The length of electrons dissipation region and downstream electric field Ez still evolve in a while after reconnection rate at the X-line already become equal to diving
electric field. The length of electron dissipation region isdetermined by using the position where electron outflow
speed has maximum value. This means that the electron
outflow speed starts to decrease as soon as they move into
the ion dissipation region, and it gradually approaches theion outflow speed. The simulation result also indicates thatthe ratio of the width of the electron dissipation region toits length is approximately given by 12 (me /mi )1/2 . Thus, therelaxation of electron outflow speed may be determined by
the cooperative action of both ion and electron dynamics.
The details about this will be discussed in a future paper.
[1] D. Biskamp, Energy Conversion and Particle Accelerationin the Solar Corona (Springer Berlin/Heidelberg, Germany,2003) p.109.
[2] E.Priest and T.Forbes, Magnetic reconnection (Cambridge
University Press, Cambridge, UK, 2000) p.322.
[3] Hantao Ji, Masaaki Yamada, Scott Hsu, and Russell Kulsrud, Phys. Rev. Lett. 80, 3256 (1998).
[4] M. Oieroset, T. D. Phan, M. Fujimoto, R. P. Lin and R. P.
Lepping, Nature. 412, 414 (2001).
[5] M. Ottaviani and F. Porcelli, Phys. Rev. Lett. 71, 3802
(1993).
[6] Biskamp, E. Schwarz, and J. F. Drake, Phys. Rev. Lett. 75,3850 (1995).
[7] R.Horiuchi and T.Sato, Phys. Plasmas. 1, 3587 (1994).
[8] M.Hesse,K.Schindler,J,Bim, and M.Kuznetsova, Phys.
Plasmas. 6, 1781 (1999).
[9] H. Ohtani, R. Horiuchi and A. Ishizawa, J. plasma phys.
72, 929 (2006).
[10] R. Horiuchi, H. Ohtani and A. Ishizawa, J. plasma phys.
72, 953 (2006).
[11] R.Horiuchi and T.Sato, Phys. Plasmas. 4, 277 (1997).
[12] A.Ishizawa, R. Horiuchi, and H. Ohtani, Phys. Plasmas. 11,3579 (2004).
[13] W. Pei, R. Horiuchi and T. Sato, Phys. Plasmas. 8, 3251
(2001).
[14] A. Ishizawa and R. Horiuchi, Phys. Rev. Lett. 95, 45003(2005).
[15] W. Pei, R. Horiuchi and T. Sato, Phys. Plasmas. 8, 3251
(2001).