J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)614–617
000–000

Dynamical Evolution of Thin Current Sheets in aThree-Dimensional Open System
HORIUCHI Ritoku, OHTANI Hiroaki and ISHIZAWA Akihiro
National Institute for Fusion Science, Toki, 509-5292, Japan(Received: 9 December 2003 / Accepted: 2 February 2004)

Abstract
Dynamical behavior of thin current sheets under the influence of collisionless reconnection in an open system isinvestigated by using newly developed electromagnetic (EM) particle simulation codes. In a three-dimensional opensystem collisionless driven reconnection evolves dynamically under the influence of an external driving flow andthree different types of plasma instabilities excited in a thin current sheet. Driving electric field imposed at the boundary
penetrates into the current sheet in accordance with the propagation of the lower hybrid drift wave excited in theperiphery. When the electric field reaches the neutral sheet, collisionless reconnection is triggered. The current sheet
is split as a result of collisionless reconnection, and thus small islands appear in the downstream. The accumulation ofcurrent density inside the islands excites the kink instability leading to the destruction of the island structure. A low
frequency EM instability is triggered in the current sheet after the island structure disappears in the system.
Keywords:
collisionless reconnection, open system, thin current layer, plasma instabilities, particle kinetic effects

1. Introduction

window size, because the inflow velocity is locally enhanced
within this region. The distribution function of incoming
particles at the input boundary is assumed to be a shifted
Maxwellian with a constant temperature and the averagevelocity equal to the E × B drift velocity.
As an initial condition we adopt a one-dimensional
equilibrium with the Harris-type anti-parallel magnetic
configuration, in which physical quantities depends only on
the y coordinate and the magnetic field is parallel to the xaxis. The initial particle distribution is assumed to be a shifted
Maxwellian with spatially constant temperature and average
particle velocity, which is equal to the diamagnetic drift
velocity.

Collisionless magnetic reconnection plays a crucial rolein a number of interesting phenomena with fast magnetic
energy release, plasma acceleration and heating both in spaceplasmas [1] and in laboratory plasmas [2]. Magnetic
reconnection dynamically evolves in an open system in which
both plasma inflow and outflow exist through the boundary.
Steady reconnection is realized only when the flux input rate
into the system is balanced with the reconnection rate.
However, this balance condition is not always assured because
magnetic reconnection is controlled by two different processes
with different time scales, i.e., an external global process andan internal microscopic process.
In order to study the dynamical evolution of collisionlessreconnection in the presence of an external driving source wehave developed a new open boundary model, in which a freecondition is used at the downstream boundary (x = ±xb) andan input condition is used at the upstream boundary (y = ±yb)
[3,4,5]. The plasma inflows are symmetrically driven fromtwo upstream boundaries by the external electric field
imposed in the z direction. The amplitude of driving field
Ezd (x,t) is initialized to zero at t = 0, and increases with timewhile keeping a bell-shaped profile near the center and a flat
profile in the periphery for an initial short time. After then aconstant profile is kept with maximum flux input rate E0 . The
spatial size of initial bell-shaped profile xd is named input

2. Dynamical evolution of the current
sheet
2.1 Two types of dynamic evolution
We have examined the relationship between the
dynamical behavior of kinetic plasmas and the driving fieldby carrying out several two-dimensional particle simulations
with different values of the input window size xd and the flux
input rate E0 [4,5]. As a result, it is found that there are two
dynamic regimes in the temporal behavior of collisionlessreconnection, which is strongly dependent on the value of xd ,but insensitive to the value of E0 . The steady collisionlessreconnection is realized when the input window size is small,©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: hori@tcsc.nifs.ac.jp
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Horiuchi R. et al., Dynamical Evolution of Thin Current Sheets in a Three-Dimensional Open System

while an intermittent regime appears as the window size
increases.

2.2 Plasma instabilities in non-driven caseIn three-dimensional case the spatial structure of current
sheet is dynamically modified by plasma instabilities excitedthrough wave-particle interaction. In the absence of an
external driving source the lower hybrid drift instability(LHDI) [6,7] is observed to grow in the periphery of current
layer in an early period, while a drift-kink instability (DKI)[8,9] is triggered at the neutral sheet as a second instability
after the current sheet is modified through nonlinear evolutionof the LHDI and its width becomes less than ion Larmor
radius [3,10]. In this way, the LHDI is not a direct cause ofan anomalous resistivity at the neutral sheet, but collisionlessreconnection is triggered by the DKI in the non-driven case.

2.3 Effect of an external driving flow in threedimensions
How does an external driving flow affect collisionlessreconnection in three dimensions? Figure 1 shows thetemporal evolution of the integrated energies in the case of
the driving field E0 = –0.04 B0 and the mass ratio mi /me =100, where the solid, dotted, dashed, and dot-dashed lines
stand for the magnetic field, electric field, ion, and electronenergies, respectively. There are three typical phases in theevolution of the energirs, i.e., (1) the initial ramp-up phase,(2) the intermediate phase in which the energies drop
suddenly, and (3) the late quasi-steady phase. We can see
three different plasma instabilities in these temporal phases.
Figure 2 illustrates the spatiotemporal structure of the n= 7 mode of the electric field Ez in the (t,y) plane, where n is
the Fourier mode number in the z-direction and the neutral
sheet is located at the mid-point of y-axis (vertical axis). The
LHDI is excited in the periphery at the relatively early period
in the same way as in the non-driven case. This mode has anelectromagnetic feature as well as an electrostatic feature [11].
The driving electric field imposed at the upstream boundary
carries the plasma towards the current sheet and compresses
it. The anomalous resistivity generated through the interaction
between particles and the LHD wave leads to the penetrationof the driving electric field into the current sheet. Figure 3
shows the temporal evolution of two Fourier modes of theelectric field Ez at the midpoint. Figures 2 and 3 indicate that
the LHD wave (n = 7 mode) itself propagates towards thecenter of the current sheet together with the driving field (n =0 mode) and triggers collisionless reconnection at the neutralsheet.
Magnetic reconnection generates the fast divergent flowwhich carries the magnetic flux towards the downstream
region, thus leading to the change in the spatial structure ofthe current sheet. Figure 4 illustrates the perspective view ofmagnetic field strength B2x + B2y at ωce t = 456 in threedimensional space, where the weak field region is plotted by
an isosurface. The weak field region is split into three pieces
as a result of magnetic reconnection, i.e., the central region
2615Fig. 1

Temporal evolution of volume-integral values ofmagnetic field, electric field, ion, and electron energies.

Fig. 2

Spatiotemporal structure of the n = 7 mode of theelectric field Ez in the (t,y) plane.

Fig. 3Temporal evolution of two Fourier modes the electricfield Ez at the midpoint, where the solid and dashedlines correspond to the n = 0 and n = 7 modes,respectively.

Horiuchi R. et al., Dynamical Evolution of Thin Current Sheets in a Three-Dimensional Open System

Fig. 4 Perspective view of the magnetic field Bx2 + By2 at ωcet = 456, where the weak field region is plotted by an isosurface.

cyclotron frequency which is much lower than the lower
hybrid frequency. The detailed examination leads to theconclusion that this mode is nothing but the drift kinkinstability and is a possible candidate for anomalous resistivity
in the neutral sheet.

3. SummaryFig. 5

Dynamical behavior of thin current sheets controlled bycollisionless reconnection has been examined by means of
three-dimensional EM particle simulation in an open system
which is subject to an external driving source. Collisionlessdriven reconnection evolves dynamically under the influenceof three different types of plasma instabilities excited in a thincurrent sheet. Driving electric field imposed at the boundary
penetrates into the current sheet in accordance with thepropagation of the lower hybrid drift wave excited in theperiphery. When the electric field reaches the neutral sheet,collisionless reconnection is triggered. Small islands, which
are generated in the downstream as a result of collisionlessreconnection, suffer from the kink instability leading to thedestruction of the island structure. A low frequency EM
instability is also triggered in the current sheet in the latephase when there is no island structure in the system.

Spatiotemporal structure of the (n = 1) mode of themagnetic field Bx in the (t,y) plane.

around a magnetic “x”-point, and two islands in thedownstream which include a magnetic “o”-point. The
reconnected magnetic flux accumulates inside the magneticislands and increases the current density there. When thecurrent density exceeds some critical value, a kink instability
is triggered resulting in the destruction of the island structures,
as is shown in Fig. 4.
When the magnetic islands move out though theboundary, the extra energy is suddenly expelled from thesystem together with the magnetic islands (see Fig. 1). After
this event, the system relaxes into a quasi-steady state.
However, it is also found that a low-frequency EM instability
is excited near the central region in this late phase.
Spatiotemporal structure of the n = 1 mode of the magnetic
field Bx is plotted in Fig. 5. This EM mode is excited in thelate phase and has a frequency comparable to the ion

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