J. Plasma
287–290
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)
000–000

Mass Ratio Dependence of Current Layer Structure inCollisionless Driven Reconnection
ISHIZAWA Akihiro, HORIUCHI Ritoku and OHTANI Hiroaki
National Institute for Fusion Science, Toki 509-5292, Japan(Received: 9 December 2003 / Accepted: 14 April 2004)

Abstract
A steady two-scale structure of current layer is demonstrated in the collisionless driven reconnection without aguide field by means of two-dimensional full-particle simulations in an open system. The current density profile
along the inflow direction consists of two parts. One is a low shoulder controlled by the ion-meandering motion,
which is a bouncing motion in a field reversal region. The other is a sharp peak caused mainly by the electronmeandering motion. The separation of the shoulder from the sharp peak is revealed by virtue of a large mass ratio
calculation mi /me = 200 because the ratio of the ion-meandering orbit amplitude to the electron-meandering orbit
amplitude is proportional to (mi /me)1/4.
Keywords:
driven reconnection, collisionless, particle simulation, current layer

1. Introduction

systems [6-12]. In the time dependent magnetic reconnection,
two structures related to the current layer are found [5,6]. One
is out-of-plane electron flow characterized by the electron
skin depth, the other is out-of-plane ion flow characterized
by the ion skin depth [5,6].
On the other hand, steady states of the reconnection arefound in the long-time scale dynamics of the driven
reconnection in an open system [13,14]. In the steadyreconnection, the reconnection rate is controlled only by anexternal driving flow due to an applied electric field because
the Faraday’s law requires that the out-of-plane electric fieldshould be uniform in the steady state [11-14]. The current
layer profile adjusts itself to accommodate the uniform
electric field.
In the steady reconnection, it is found that the current
layer width is controlled only by ion dynamics [13,14],
although the current is dominated by the out-of-plane electronflow. It is claimed that the ion-meandering motion controls
the current layer which adjusts itself to sustain the uniform
out-of-plane component of the electric field. These studies are,however, limited by the small mass ratio of ion and electron
mi /me = 25. The significance of the ion dynamics should beclarified by examining a dependence on the mass ratio.
In this paper, we will investigate steady states of thecollisionless driven reconnection in an open system for several
cases with different mass ratio, namely: mi /me = 25, 50, 100
and 200. The large mass ratio leads to a clear separation of
the ion-meandering orbit amplitude from the electron-

The collisionless magnetic reconnection is a fundamental
mechanism of the rapid release of magnetic energy in thesolar corona, the high temperature tokamak discharge, themagnetospheric substorm, and reconnection experiments[1,2,3]. Recent computer simulations reveal that a small scalecurrent layer, where the frozen-in condition of the plasma isviolated, adjusts its structure so as to realize a large
reconnection rate demanded by a large scaleideal
magnetohydrodynamics (MHD) evolution [4-14]. The
simplest physical model of the violation necessary to generate
the large reconnection rate, i.e. the large reconnection electricfield, is the Hall-MHD model which includes the Hall-term
characterized by the ion skin depth [4-9].
A plasma is frozen in magnetic field lines in the ideal
MHD, and thus reconnection does not occur. This
circumstance is related to the fact that there is no typical scale
length that characterizes a small scale current layer profile.
When we introduce non-ideal effects leading to violation of
the frozen-in constraint, the width of the current layer isdetermined by typical scale lengths of non-ideal effects. The
plasma profile in the kinetic approach is characterized by thespatial scale lengths such as the electron skin depth due tothe electron inertia effect, the ion skin depth due to the ion
inertia effect, the electron Larmor radius, and the ion Larmor
radius.
The current layer structure is studied using hybrid
simulations and full particle simulations [4-14]. Most of thesesimulations present time dependent reconnections in closed©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: ishizawa@tcsc. nifs. ac. jp1287

Ishizawa A. et al., Mass Ratio Dependence of Current Layer Structure in Collisionless Driven Reconnection

3. Current layer structure

meandering orbit amplitude because the ratio of theseamplitudes is proportional to (mi /me)1/4. We reveal a two-scale
structure of the current layer, a sharp peak and low shoulders.
The sharp peak is mainly controlled by the electronmeandering motion. The half-width of the current layer isdetermined by this sharp peak. The ion dynamics forms low
shoulders of the current layer, and thereby resulting in a twoscale structure of the layer. The structure is mainly caused by
the meandering motion of ions and electrons because theystrongly violate the frozen-in constraint.

Here we examine a spatial profile of out-of plane current
density when the system is relaxed into a steady state.
We show current density profiles along the vertical line
passing the X-point in Fig. 1 for mi /me = 25, 50, 100, and200. For mi /me = 200 we observe clear shoulders at y ≈ ±35λd
and a sharp peak that has a size nearly equal to 10λd in width.
On the other hand, the profile consists of one wide peak formi /me = 25 because ion-scale and electron-scale are not well
separated. We, therefore, conclude that the current density
profile has a two-scale structure consists of a sharp peak andlow shoulders for a large mass ratio.
Next, we consider the two-scale structure of the current
layer more quantitatively, and then reveal that the meandering
motions of ions and electrons play crucial roles in theformation of the structure. We introduce several spatial scale
lengths describing the layer structure, and consider the time
evolution of the spatial scale lengths, namely: the half-widthof the distance of shoulders, the half-width of a sharp peak,
the half-width of a current layer, the ion-meandering orbit
amplitude lmi, the ion skin depth di = c/ωci , the electronmeandering orbit amplitude lme, and electron skin depth de =c/ωce. These scales are normalized by the Debye length andare evaluated from the spatial profile of plasma along thevertical line passing the X-point. The meandering orbit
amplitude of species s is defined by the distance y which
satisfies the condition ρs( y)/y = 1 [7,14], where ρi( y), andρe( y) are the local ion Larmor radius, and the local electron
Larmor radius, respectively. The half-widths of the shoulder
and of the sharp peak are defined by the half-widths at 20 %and 80 % of maximum value of the current density,respectively. Figure 2 shows that the scales are relaxed into
the following values: lmi ≈ 35λd , lme ≈ 5λd , di ≈ 80λd , and de ≈7λd . The width of the shoulder closes to the ion-meandering
orbit amplitude, while the width of the sharp peak closes tothe electron-meandering orbit amplitude in the steady state.
The half-width of the current layer is dominated by the sharp
peak, and correspondingly the half-width of the layer relaxes
into the electron scale ≤ 10λd. Figure 3 shows the mass ratio
dependence of current layer width in a steady state. The halfwidth of current layer closes to the electron meandering orbit

2. Simulation model
We consider a square open region in xy-plane, the sizeof which is 2yb in height and 2xb in width. We use a two-anda-half-dimensional explicit electromagnetic particle simulation
code developed in the previous work [11-14].
Boundary conditions are as follows. At upstream
boundary, ions and electrons are frozen in magnetic fieldlines, and thus plasma inflow is driven by E × B drift due toan external electric field Ezd(x,t) applied in z direction at y =± yb. The condition for the incoming particle distribution is a
shifted Maxwellian with the averaged velocity given by E ×B drift. The boundary conditions for remaining field quantities
are following: Ex = 0 and ∂y Ey = 0 at y = ± yb. The external
field Ezd(x,t) evolves from zero, so as to induce a steadyreconnection. To excite magnetic reconnection at the centerof simulation domain, the external field is assumed to bestrong within the input window size xd around x = 0 at earlyphase, then the field profile becomes uniform E0 after one
Alfvén time τA = yb /VA, where VA is the initial average Alfvén
velocity. This uniform field E0 plays a role to maintaindeformed magnetic field lines within the input window at theinflow boundary, and correspondingly the system relaxes to asteady state [13,14].
At the downstream boundary x = ± xb, the plasma canfreely flow in or out. The boundary condition for particles isdetermined by both the charge neutrality condition and thecondition of the net number flux, which is associated withthe fluid velocity in the vicinity of the boundary [14]. Thus,
the total number of particles varies with time in this opensystem. The field quantities Ex, Ey and ∂x Ez are continuous atthe downstream boundary. These conditions enable magnetic
islands to go through the boundary. The remaining
components of the field quantities are given by solving theMaxwell equations at the boundary.
The initial condition is a one-dimensional Harris typeequilibrium as Bx ( y) = B0 tanh( y/yh), P( y) = B02 /8π sech2( y/yh), that has a neutral sheet at y = 0, where yh is the scale
hight. The distribution of particles is a shifted Maxwellian
with a uniform temperature Ti 0 = Te0.
We set the ratio of the plasma frequency to the electroncyclotron frequency ωpe0 /ωce0 = 3. 5, the strength of theinflow velocity E0 /B0 = –0. 04, andthe input window size xd= 0. 42 xb in our numerical simulations.

10-Jz(a.u.)

8
6

mi/me=200
mi/me=100
mi/me=50
mi/me=25

Sharp peakLow shoulder

420-100

-50

050

100

y/λd0Fig. 1

2288

The current density profile along the vertical line
passing through the X-point. The profile consists of a
sharp peak and shoulders for large mass ratio.

Ishizawa A. et al., Mass Ratio Dependence of Current Layer Structure in Collisionless Driven Reconnection

90807060504030201003020151050246810ωci t

1214160246810ωci t

121416

Scale length / λd0

The time evolutions of several scale lengths: (a) the half-width of the distance of current shoulders, the ion-meandering orbit
amplitude λmi, and the ion skin depth di = c/ωci, (b) the half-width of a sharp peak of current, the half-width of a current layer,
the electron-meandering orbit amplitude λme, and electron skin depth de = c/ωce.

50454035302520151050

dominantly violates the frozen-in constraint near the X-point.
The sharp peak structure dominates the current layer profile,and thus the half-width of current layer is controlled by theelectron dynamics.
The ion-meandering motion creates shoulders of thecurrent layer. The shoulder structure corresponds to thecurrent layer found in Refs. [13,14]. The shoulder structure
is controlled by the ion-meandering orbit amplitude because
the ion frozen-in constraint is strongly broken by themeandering motion. Although the ion inertia also violates thefrozen-in constraint at a scale below the ion skin depth c/ωpi ,the violation due to the ion inertia is weak compared to theviolation caused by the ion-meandering motion. The
mechanism of current layer formation and its relation to theviolation by the meandering motion will be discussed indetails elsewhere [15].

Current layer width
Ion orbit amplitude
Electron orbit amplitude

050100150

200

mi/meFig. 3

Half-width of current
Half-width of current sharp peakElectron meandering amplitude
Electron skin depth

25

Half-width of current shoulder
Ion meandering amplitude
Ion skin depth

0Fig. 2(b) mi/me=200

scale length/λd0

scale length/λd0(a) mi/me=200

The mass ratio dependence of scale lengths in thesteady state. The half-width of current layer closes tothe electron scale for a large mass ratio.

amplitude rather than the ion meandering orbit amplitude forthe large mass ratio. The current layer width is, therefore,
mainly controlled by the electron dynamics. The ion dynamicsgenerates low shoulders which are characterized by the ionmeandering orbit amplitude l mi ≈ 35 λ d. Therefore, themeandering motion of ions and electrons is the dominantmechanism of the two-scale structure formation of the out-of
plane current density in the steady state of the driven
reconnection.

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4. Summary
We have newly found a steady two-scale structure of thecurrent layer, a sharp peak and low shoulders, in thecollisionless driven reconnection. This two-scale structure issignificantly different from that characterized by the ion skindepth and the electron skin depth in previous simulations of
the time dependent reconnection [5,6].
The sharp peak is mainly controlled by the electronmeandering motionbecause the motion strongly violates theelectron frozen-in constraint. In spite of the fact that theelectron inertia also violates the frozen-in constraint at a scale
below the electron skin depth, the meandering motion
3289

Ishizawa A. et al., Mass Ratio Dependence of Current Layer Structure in Collisionless Driven Reconnection[11] R. Horiuchi and T. Sato, Phys. Plasmas 1, 3587 (1994).
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