J. Plasma Fusion Res. SERIES, Vol.2 (1999) 169-172

Gurrent Loop Coalescence in Dust Plasmas
SAKAI Jun-ichi and CRAMER Neil F''
lnboratory for Plasma Astrophysics, Faculty of Engineering
Toyama University, 3190, Gofuku, Toyama 930-8555' Japan
tTheoretical Physics Department and Research Centre for Theoretical Astrophysics,
School of Physics, The University of Sydney, N'S'W' 2006' Australia(Received: 9 December 1998 / Accepted: 19 February 1999)

Abstract
The weakly ionized plasmas that occur in protostellar disks and in the cores of molecular clouds
generally have a dust component. We consider the effects of the presence of charged dust on magnetic
reconnection process. We investigate the dynamics of two current loops coalescence, for both cases;
partial and complete magnetic reconnection. It is shown that the field-free gas produced during the twoloops coalescence with complete magnetic reconnection would be able to initiate star formation.

Keywords:
dust plasma, magnetic reconnection, coalescence of two loops, star formation

1. Introduction
The weakly ionized plasmas that occur in

2. Basic Equations

protostellar disks and in the cores of molecular clouds
generally have a dust component. The ionization
fraction of molecular clouds is typically only = 1g-r, un4

atomic and molecular species, the ionized atomic andthe dust may contribute = lvo of the mass of the cloud.
It has been shown that the ion-neutral drift leading
to ambipolar diffusion can lead to the steepening of thewhich employs the fluid momentum equations for

We consider molecular clouds consisting of neutral
molecular species, the electrons, and negatively charged
dust grains. We start a 4-fluid model of the plasma'
plasma ions (singly charged), neutral molecules, charged
dust grains and electrons:

magnetic field profile and to the formation of\| 0,,p'|,;+v' .Vr, | = Y p,+n,e(E+v,x B)

singularities in the current density of hydromagnetic
fluctuations [1,2] and Alfvdn waves [3,4]. Recently
Bulanov and Sakai (1998) t5l investigated in datails
magnetic reconnection process in weakly ionizedI- pivin(vi-vn)- pivid(vi-vd),

plasmas.

In the present paper we consider the effects of thepresence of charged dust on magnetic reconnection
process. We investigate the dynamics of two current
loops coalescence, for both cases; partial and complete(l)t" (* *u"' vu") = -Y pn- pnvn'(vn- v,)- pnvnd (yn - yd),magnetic reconnection. It is shown that the field-free gasQ)

produced during the two loops coalescence withcomplete magnetic reconnection would be able toinitiate star formation.
Corresponding author's e-mail : sakaij un@ ec s.toyama-u.ac.i p169

@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchSakai J. et al., Current Loop Coalescence in Dust plasmas,. (*+ u. vu.

the magnetic field by a reference field _86, and define the= Zonoe(E + vax B)

Alfv6n speed based on the field 86 and the ion density:)

uo= Bol(qtopi). Eq. (l) may rhen be written, using (g)_ paVan(yo _ yn)_ paVa;(yo _ y,and Faraday's law neglecting the displacement current,)

(3)

tt2
:$
1V xB1x B- {)o0 =- fi."€(E+v"x B)- p"v"n(v"-vn)^e,- vo ) xB(4)+vin(vi_ yn)+ v,o(v, _ v6 ), €)where E is the wave electric field, ln. is the speciesfrequency, {)i = Boelmi. The presence of dust introduces- P "v"a(v"- va ) - p "v"i(v" - v,),

where O. = Oi(l - 6)/6 and O; is the ion cyclotron

mass, A is the species mass density and r" is the species

the first and third terms on the rhs of (9).

velocity. pi and pn are the ion thermal and neutral

For strongly coupled dust grains and neutral gas,such that vd = vn, the ion equation of motion gives thethermal pressures, and 4, is the collision frequency of aparticle of species s with the particles of species r. We

following expression for the ion-neutral relative drift

have neglected electron inertia in (4), momentum

velocity:transfer to ions from electrons in (l) and to dust grainsu1
vo=v;-y"=F*l(VxA)xB

from electrons in (3), and the dust thermal pressurev,ogradient in (3).

(10)
- (R/p)((v xB) xB) xBl,
where 4 = tn * yia, R = {)^pnlvips, 82 = 81 + Bl + 82,
and F = l/(l + R2B2lp2). For a small dust number

The parameter 6 = n"lq < I measures the charge
imbalance of the electrons and ions in the plasma, withthe remainder of the negative charge residing on thedust particles, so that the total system is charge neutral.

- ene+ eni - Zoeno- 0.

density, vta << %n, so we approximate vi = v,n. Summing
the momentum conservation equations (1,2,3) of theions, neutrals and dust, the equation of motion for theneutral velocity is obtained. We normalize the densityby ps, definin g p = ( p" + pil po, and the pressure by pe.
The velocity is normalized by the Alfv6n velocity basedon Po, Va = B6/(1t()po)rt2, and space and time arenormalized by Le and Tt = LolVa. The result is(5)A typical value of d for molecular clouds is 6 = I - lO-a.
The charge on each dust grain is assumed constant, andfor simplicity we also assume that 6 is constant, eventhough nn and thus n1 are variable. The neutral massdensity obeys the continuity equation

Pot *o .(p,v^)=o.

/.

\pl?*r.Vrl=fVp +(VxB)xB. (ll)'tdr(6)I

To complete the system of equations, Maxwell'sequations ignoring the displacement current are used,the collisional electron momentum transfer termswith the conduction cuffent density given byneglected, and neglecting the Hall term,

j=e(nivi-fl.v"-noZovo).

The magnetic induction equation gives, using (8) withO)

S =v x|xB)+Aovx

where equilibrium charge neutrality is expressed by (5).

[ ',]xB)xB tx B + (Rt p) B'z(v xB)xB
lf{,,ol,

Equations (4) and (7) lead to the following

I'generalized Ohm's law:lr, (l-t) I .- jxB- m.- v",(v.-v,)
E+li-+vnlxB
e cr'\'e\o-? ,.orr.-eL)n",Iya)- !

v",1v.- v,).

}

where B = [B] and Ae= pnlViT;pi.

(8)

3. Simulation Results
We use a 3-D simulation code of the above
equations in which the numerical scheme is theWe now use the strong coupling approximation(Suzuki and Sakai 1996) [3], whereby the ion inertia
term (the left hand side) and the ion thermal pressureterm are neglected in (1), leaving a balance between themodified 2-step Lax-Wendroff method. The systemsizes are 0 (x = y 3 6rcL,and 0 < z 3o.SnLin the x, yremaining terms. At this point it is useful to normalizeand e directions, respectively. The mesh points are N,t70-Sakai !. et al., Current Loop Coalescence in Dust Plasmas

appears strong wave emissions of fast magnetosonic

200, N) = 200 and N. = 10 in the x, y and z directions,respectively. We used free boundary conditions (firstderivatives of all physical quantities are continuous) forwaves from the reconnection region, which arepropagating upward the incoming flows. There occurs
strong plasma heating near the wave fronts as seen in

the -r and y directions, while periodic boundary
condition is used in the z direction. We take an initialFig.l (d).
Fig.2 shows the simulation results for complete
magnetic reconnection case at t = 6.3 ra. As seen in
Fig.2 (a) there occurs strong density accumulation with
about 13 times larger than the initial value near theregion where the magnetic fields are almost free, due tocomplete magnetic reconnection. The field-free gas
produced during the two loops coalescence witheach current loop, which is placed along the z direction
to satisfy a force-free condition. Other parameters usedhere are F = 0.01 and Ap - 0.5.

We show the simulation results of both cases forsymmetric two current loops collision, where partial orcomplete reconnection occurs. Fig. I shows foursnapshots at t = 8.1 ra: (a) density distribution
overlapped with velocity field (u, - uy), (b) magnetic
filed B. with vector field plots of (8" - 4), (c) velocity
u. with vector plots of (u, - ur,), and temperature
distribution. As seen in Figs.l(a) and (b), during twoloops coalescence the density enhancement is about
twice of the initial state and relatively weak. There+.25 C.tS 8"92 I.35 I.rt*-u^!?$ "{.0ds -9.01: t,0;scomplete magnetic reconnection could be able to initiate
star formation.

4. Conclusion
We investigated the effects of the presence ofcharged dust on magnetic reconnection process for bothA,**tl.045 i}.8'}5*-0t)- 3$-d&- 30.{l},1L,1

1.*Fig. 1 (a) Density distribution overlapped with velocity field (u"- u"), (b) magnetic filed flwith vector field plots ol (8,B"), (c) velocity u. with vector plots of (u, - u"), and (d) temperature distribution at t = 8.1 ro for partial magnetic
reconnection.

t't1

Saku J. et al., Current Loop Coalescence in Dust Plasmas-L.+-f.t..t,,t,2t.l:.i i_a1.,,.,.8 a,*::-71{,[Fig. 2 lal Density distribution overlapped with velocity field (u" - u,), (b) magnetic filed Q with vector field plots of (4 Br), (c) velocity u, with vector plots of (u,- url, and (d) temperature distribution at t = 6.3 ro for complete magneticreconnection.

complete and partial magnetic reconnection during two
current loops coalescence. We showed that the field-freegas produced during the two loops coalescence withReferencestll

complete magnetic reconnection could be able to initiate
star formation.

A. Brandenburg and E. Zweibe| Astrophys. J. 427,L9r (r99q.
A. Brandenburg and E. Zweibel, Astrophys. J. 448,734 (1995).

tzl M.M. Mac Low et c/., Astrophys. l. 442, 726

Acknowledgment(199s).

We thank the Cosel and Densoku company for thet3l M. Suzuki and J.I. Sakai, Astrophys. J. 465, 393(lee6).

support.

t4l M. Suzuki and J.I. Sakai, Astrophys. J. 487,921(1997).

tsl S.V. Bulanov and J.L Sakai, Astrophys. J. Suppl.

rr7. 599 0998).

r72