J. Plasma Fusion Res. SERIES, Vol. 2 (1999) 58-61

Anomalous Resistivity and Particle Kinetic Effect inGollisionless Recon nection
HORIUCHI Ritoku and SATO TetsuYa
National Institute for Fusion Science, Toki 509-5292, Japan(Received: 5 January 1999 / Accepted:26 June 1999)

Abstract
Roles of anomalous resistivity and particle kinetic effect in collisionless reconnection areinvestigated by means of a three-dimensional particle simulation. For no external driving source, thelower hybrid drift (LHD) instability is observed to grow in the periphery of current layer, but it cannot
penetrate into a high beta region near the neutral sheet. Instead, a low requency electromagnetic(EM)instability grows to generate the electric field at the neutral sheet after the LHD instability is saturated.
When an external driving flow exists, collisionless reconnection is triggered by the particle kinetic effectbefore the low frequency EM instability is excited.

Keywords:
anomalous resistivity, particle kinetic effect, three-dimensional particle simulation, collisionlessreconnection

between charged particles carrying the equilibrium
current and waves propagating along the equilibrium
current by means of two-dimensional particle simulationand have discussed the anomalous resistivity in thecurrent layer. The lower hybrid drift (LHD) instability[1] grows in the periphery of current layer, but it cannot
penetrate into a high beta region in the vicinity of the

1. lntroduction
Magnetic reconnection in a collisionless plasma,"collisionless reconnection", is an interesting andimportant process in considering energetically active
phenomena observed in a high temperature, rarefied
plasma such as the solar corona, the geomagnetic tail,laboratory plasmas, fusion plasmas, and so on.
However, non-ideal effect which breaks the frozen-in
condition of magnetic field is needed for the excitationof magnetic reconnection in a collisionless plasma. A
number of theoretical and simulation studies haveneutral sheet. Instead, a low frequency electromagnetic(EM) instability [2] grows to generate the electric field

disclosed that there exist two types of triggering

nonlinear development of collisionless reconnection in
the presence of an external driving flow and clarified thein the vicinity of the neutral sheet.

Horiuchi and Sato [4,5] have examined themechanisms which break the frozen-in condition andlead to magnetic reconnection in a collisionless plasma.

role of particle kinetic effect in collisionlessreconnection. Collisionless driven reconnection develops

One is due to the wave-particle interaction which is a
cause of anomalous resistivity in the current sheet [1-3].

in two steps in accordance with the formation of twocurrent layers, i.e., ion current layer in which the ion

The other is due to the particle kinetic effect whichbecomes significant in a particle scale such as thekinetic effect is dominant, and electron current layer inwhich the electron kinetic effect is dominant. However,collisionless skin depth and the Larmor radius [4,5].

Ozaki et al 13) have examined the interaction
C o r re spo ndin g autho r' s e - mail : ho ri @ t c s c. nifs. ac@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Researchip58

Horiuchi R. et al., Anomalous Resistivity and Particle Kinetic Effect in Collisionless Reconnection

the whole development of the system or the growth timeof collisionless reconnection is controlled by thedynamics of the ion current layer but not by that of theelectron current layer.

Which of two triggering mechanisms plays aleading role in the real situation? Because the above twosimulations were based on the different models, we can
not compare the results directly. In other words, threedimensional treatment is needed to answer this question.
We carry out the three-dimensional particle simulationin the presence of an external driving flow [6]. The
simulation results will be discussed in Sec. 3 and Sec. 4.

2. Simulation Model
Let us consider three-dimensional open system
which evolves dynamically due to the convergent
plasma flow supplied through the boundary. The particle

simulation is carried out by using three-dimensional
electromagnetic codes [7]. As an initial condition weadopt a one-dimensional equilibrium with amagnetically neutral sheet at }l = 0 as

B(y)= [80 tanh (y I L),0,0],(1)

P (9 = n'oa)

8n sech2 1y / L's,

Fig. lThe spatiotemporal structure of the Fourier

where Bs is a constant and Z is the scale height along theamplitude Ej*t, and (b) that ol Bt*lt in the (y,f)

y-axis. The equilibrium current flows in the negative zdirection. The particle distribution is assumed to be ashifted Maxwellian with a spatially constanttemperature. The ratio of ion to electron temperature isplane, respectively, where n stands for the Fouriermode along the z-axis.

Figure I shows (a) the spatiotemporal structure of theFourier amplitude E!=3) and, (b) that of Btu=t) in the (y,r)
plane, respectively, where n stands for the Fourier modealong the e-axis. One can see in Fig. 1-(a) that the LHD
instability, g(a=3), grows in the periphery of the current
layer in the intermediate temporal phase, but it can not
penetrate into the central high beta region (y = 0). This

It is assumed that physical quantities are periodic at
both the boundary of the x-axis (-r = 1.r5) and that of thez-axis (a = *zu), and driving electric field exists at theboundary of the y-axis Cy = lyu) in order to supply theplasma with the E x B drift velocity into the simulation
domain. The amplitude of driving electric field E6s(,r,t)
is described by a constant bell-shaped profile with amaximum input rate of magnetic flux E6 at the center ofresult indicates that the LHD mode can not be a cause ofan anomalous resistivity leading to collisionlessthe input boundary (-r = 0, y = tyb). The typicalreconnection at the neutral sheet. On the other hands, a
low frequency EM mode, Bf=r), grows with a large ratein the vicinity of the neutral sheet in the late phase after
the LHD mode is saturated, as is seen in Fig. 1-(b).
These results are in good agreement with those obtained
from the two-dimensional particle simulation in the (y,z)

simulation parameters are as follows. The mass ratio M1!
M"= lO0, the ion Larmor radius p1 = l.l2L, the ratio of

the electron drift velocity to the electron thermalvelocity 1)r"11)r = 0.16, and 2x,o/L = 2,5,3. Generation of Anomalous Resistivity
In this section we discuss how two plasma
instabilitie s, which were observed in the twodimensional particle simulation [3], evolve in threeplane [3]. The detailed analysis [6] confirms that the low

frequency EM instability is essentially two-dimensional.
An electric field is generated at the neutral sheet
through the plasma instability. Temporal evolution of
the DC component of the electric field Q at the neutral

dimensions in the absence of no external drivine flow.
59Horiuchi R. et al., Anomalous Resistivity and Particle Kinetic Effect in Collisionless Reconnection;:(a),-()

10"EoEz,n=o,c()i:{:.::::*'---'*-;;..;5:;..1.,i()()

tr0b l0-'F(.) -€ll0-4

02OO,100oL" tHt.

'F,.. i'-..,Fig. 3 Temporal evolutions of the Fourier modes for thecase of Eo = 0.024 where the solid, dashed,dotted, and dot-dashed lines represent the n = 1
mode of 8,(0,0,2l., the n = 0 mode of E,(O,O,zl, andz)aiP)

the m = 1 mode ol Br(x,O,Ol, respectively. Here, mstands for the Fourier mode number along the x(c)Yreaxis.

Vzi ---""--othe ion kinetic scale by the convergent plasma flow
[4,5]. The Poynting flux moves towards the neutral
sheet while creating the electron current layer inside theion current layer. As soon as the electric field reachesot)

9o500(t" t1000

the neutral sheet, magnetic reconnection is triggered andFig. 2 Temporal evolutions of (a) the n = 0 mode of
E,10,0,2l., (b) the n = 1 mode of 8,(0,0,21 (solid) andthe n = 3 mode of E,lo,yol2,z) (dashed), and (c) theaverage electron velocity (solid) and the averageion velocity (dashed) at the midpoint for the samecase as Fig. 1.

the topology of magnetic field lines is changed. The
temporal evolutions of the Fourier modes are shown in
Fig. 3 for the case of Ee = 0.0280. Because most of ionsinside the current layer is unmagnetized in the present
case, the Poynting flux supplied at the boundary reaches

the neutral sheet in the electron drift time ( t6" = ypdcEs
sheet ("reconnection electric field" ) is shown inFig.2,= 100 @*t).Thus, magnetic reconnection is triggered bytogether with those of the Fourier modes of B, at theneutral sheet and E, at the periphery (y = yol2), andthose of the average electron velocity and the averageion velocity along the z-axis at the midpoint (x = 0, y 0). After the LHD mode is saturated, the reconnection

the convective electric field earlier than the excitation ofelectric field starts to grow at the neutral sheet inof the current layer by the convergent plasma flow
results in the increase of the growth rate of the low
frequency EM mode, because the growth rate strongly
depends on the width of current layer [2]. Consequently,
the EM mode is excited earlier and grows with a largerrate compared with that for no driving flow.

the low frequency EM mode (solid line). The ycomponent of magnetic field with the m = I spatialstructure along the x-direction is generated as a result ofmagnetic reconnection (dot-dashed line). The thinningaccordance with the excitation of the low frequency EMmode (solid line in Fig. 2-(b)). The generated electricfield accelerates the electrons in the vicinity of theneutral sheet along the equilibrium current tocompensate the momentum loss through the waveparticle interaction, as is seen in Fig. 2-(c).

5. Summary
4. A Particle Kinetic EffectWe have investigated the generation process ofLet us consider the case when the system evolves

anomalous resistivity and the dynamical development ofdynamically due to an external driving flow. The
convective electric field can penetrate into the ion

collisionless reconnection in three dimensions by meansof a full particle simulation. When there is no external
driving source, the LHD instability grows in thecurrent layer if the current layer is compressed as thin as60

Horiuchi R. et al., Anomalous Resistivity and Particle Kinetic Effect in Collisionless Reconnection

periphery of current layer in the early phase while thelow frequency EM instability is excited in the vicinity of
the neutral sheet after the saturation of the LHD
instability. It is found that the interaction between thisEM wave and charged particles carrying the equilibrium
current can be a cause of an anomalous resistivity
leading to collisionless reconnection in the neutral sheet.

t2l K. Yamanaka, Phys. Scr.17, 15 (1985).
t3l M. Ozaki, T. Sato and R. Horiuchi, Phys. Plasmas

If an external driving flow exists, collisionless

t6l R. Horiuchi and T. Sato, Plasma Phys. Control.

reconnection is triggered by the particle kinetic effectearlier than the excitation of the EM instability.

Fusion 41, A477 (1999).
t7l C.K. Birdsall and A.B. Langdon, Plasma Physics

3,2265 (1996).

t4l R. Horiuchi and T. Sato, Phys. Plasmas 1, 3587
/L994J.

t5l R. Horiuchi and T. Sato, Phys. Plasmas 4, 277(1997).

via Computer Simulation (McGraw-Hill, Newtll

York. 1985).

References
N.A. Krall and P.C. Liewer, Phys. Rev. 4,2094(r97 r).

6l