J. Plasma Fusion Res. SERIES, Vol.2 (1999) 532-535

The MHD Simulations of 3D Magnetic Reconnection near
Null Point of Magnetic Gonfigurations
BULANOV Sergei V., ECHKINA Eugenia YuI, INOVENKoV Igor N.', psGoRARo Franchesco2and PICHUSHKIN Victor V.lInstitute of General Physics, Russian Academy of Sciences, Moscow, Russia

IMosco. State (Jniversity, Moscow, Russia
2Dipartimento di Fisica dell' Universit'a di Pisa and INFM, Italy(Received: 9 December 1998 / Accepted: 18 February 1999)

Abstract
We investigate 3D plasma flow in the vicinities of critical points of magnetic configurations. The
study is based on the analysis of exact self-similar solution of the MHD equations and 3D computersimulations. Both the analytical solution and 3D MHD simulations demonstrate appearance of singular
distribution of the electric current density near the magnetic field separatrix surfaces of the form of thecuffent and vortex sheets.

Keywords:
magnetic reconnection, current sheets, 3D MHD simulation

The studies of the magnetic reconnection arebi = 0, a null point of the magnetic field occurs at r, = 0,

relevant to a broad range of problems of space andfusion plasmas (see [,2] and the references in). In thewhere B; = Aij xj. Further we assume that A;1 is not zero.

By virtue of the condition div B = 0, the trace of thematrix Aii is zero (Att = 0) and the sum of theeigenvalues vanishes, \ + h + L3 - 0 (Lo a = l, 2, 3

limit when the nonlinearity effects dominate overdissipation, self-similar solutions for the MHD equations

can be used in order to describe the self-consistentevolution of the plasma flow and of the magnetic field.
The process when in the plasmas it takes a finite time toform the singularity in the electric current distribution
near critical points is called the magnetic collapse. The
magnetic collapse corresponds to the self-pinching of
the electric current carrying plasma in inhomogeneous
external magnetic field when the gradient of the external

are eigenvalues) of the matrix A;.

In the present paper the magnetic configuration
with three nonvanishing eigenvalues is considered. We
assume that the initial configuration of the magnetic
field is current free with the matrix A,, of the diagonal
form:

A,, = diag Ia, b, - (a+ b)].

(l)magnetic field is larger or of order of that of themagnetic field associated with the plasma electriclf a and b have the same sign, the separatrix surface is zcurrent. In order to investigate non-self-similar plasma
motions, we use 3D MHD computer simulations.
In the vicinity of .r = 0, the magnetic field is equalto B(r, t) = B(O,l) + (x.V)A(r, t) + .... We introduce thenotation bi = Bi(0, r) for the uniform component of themagnetic field, and A;1 = dB/dxj,,=e for the matrix ofthe magnetic field gradients. If the uniform part vanishes

In dimensionless form the MHD equations can be-0.
written as

d,p +Y(pv)=0,a)@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research532

Bulanov S.Y. et al., The MHD Simulations of 3D Magnetic Reconnection near Null Point of Magnetic Configurations

d,v+ (r,V)r, = - pv@Dl2p + j x Bl p,(3)

P,o,j
6

p1d,r+(vY)r)/Q- trlxlr.l)^ ,,,a(4)

4(s)

2

where the adiabatic index is equal to T= 513.
As usual F = SnponE, where ps is a pressure at rs.

0d,B=Y x(v xB)- itLB, (V.B)=0,

The inverse Lundquist number is i, = c2/4Eovos. k=

20100.01.

The MHD equations (2-5) have self-similar

Fig. 1 Density p-a, electric current density /:b andvorticity (o-c versus time for a2z = -0.025, and ar!,
= 0.025; APr = diag{0.5, 0.5, -1}.

solutions when ? = 0 [3]. The discussions of the plasmapressure effects are presented in [3,4]. In these solutionsp = p(t), vi(x, t) = w;iQ)xi, B;(x, t) = AiiG)xi.
Substituting these expressions into eqs. (2)-(5) weobtain a system of ordinary differential equations forp(r), matrices w,,(t) and AilQ)

p +w*P =0,w,,+w,owo,=- (Aft- Aki)Akj /P,4,, +w*Au + A,owor=w,,A*j.

P,o,(6)

2A)

1(8)

0

In the simplest spatially nonuniform magnetic

w,i=l wzr wzz 0\0o ,,,,A_^i.i_(o,, At, oA,' AnOIj3-1

configuration with a null point, the plasma velocity andmagnetic fields are described by matrices [3](*,, wtz 0trl-2
Fig. 2 Density p-a. electric current density ,,:b and(e)vorticity a)-c versus time for a\z = -0, and a\, =4.025; Ali= diag{2, -1, -1}.

\0 oAtWe assume that at the initial time the magnetic
field is current-free and is given by the diagonal matrix
given by eq. (1) which corresponds to the vicinity of thejust then the magnetic collapse occurs.
In Fig. 2 the time dependence of the same value asin Fig. 1 are shown for initial configuration described by
the matrices: w$ with w\r = 4.O25 and w0; = 0 for otherij: Alt = diael2, -1, -l ). In this case the the separatrix

magnetic null point.
The results of numerical solution of system of eqs.
(6)-(8) are presented in Figs. I and 2, where the timedependences of the density p, the vorticity (w21 - wp)/2
and the electric current density (A2t - An)/2 are shown.
In Fig. 1 the time dependence of the density p,surface is perpendicular to the plane z = 0, andexcitation of the electric current directed along theseparatrix surface. We see that the magnetic collapse
developes much faster than in the previous case: at t0 =1.9 the p, a,, j,tend to infinity.
We consider two initial confieurations of thevorticity a;. and electric current density j. is shown for
initial configuration described by the matrices: wou with
wlz = -0.025, wlz = 0.025 and wl1= 0. for other ij; Aoil
= diag{0.5, 0.5, -l }. In this case the separatrix surface
is in the plane z = 0, and excitation of the electric

magnetic field.
In the first case the magnetic field is described by
the vector potential:

current is perpendicular to the separatrix surface. We seethat at to= 35.25 the p, @2,j. tend to infinity. TheA(x,y, z)=-2€xze" +(l - e)xye,,density shows nonmonotonic dependence on time. At
the initial stage the density of the plasma decreases and(10)where t < I characterizes the space inhomogeneity. The533Bulanov S.Y. et al., The MHD Simulations of 3D Magnetic Reconnection near Null Point of Magnetic Configurations

matrix At given by eq. (l) has the components d = I + €excitation of the electric current. To described-l + €. The electric current is imposed from themagnetoacoustic waves we choose the vector potential
at the boundaries x = +1, y - +l of the form A(x, y, t) =and D =boundary to be parallel to the magnetic separatrix.
The second magnetic configuration is described byAo@, y) + f (t + lnr), where Ao@, y) is given by the (11).

Function/({) is equal to -E,(( - l)tl€ for ( > I and 0for { < 1. All the results of the computational
simulations presented below are obtained for i^ -expression

A(x,y, z)=-2txzey * exyez,(11)which corresponds to the separatrix perpendicular to theelectric current imposed from the boundary. The matrix
A; has the components a = b = s with € = 0.1.
The system of MHD equations (2)-(5) has beensolved numerically in cubic computational region: -1 <xls < l, -l < yls < l, -l 3 zls 31. We assume that, at /0
a plasma with a uniform density and pressure is at restin the current free magnetic field, described by thevector potential (10) or (l l).

0.006, F= O.0l2,E= 0.01, Er = 0.01 and € = 0.01 or e =0.5 in (l l).
The first series of the numerical calculations wasperformed for the vector potential given by eq. (10) withe = 0.01. The results of the simulations of theazimuthally symmetric magnetoacoustic perturbations

are shown in Fig. 3 where a slice at z = 0 of thedistribution of the electric current density, iso-surface ofthe electric current with iso-value equal 0.2 are shown.
We see the formation of the cunent sheet in the vicinitv

The boundary conditions correspond to the'-1A

Fig.3 Distributiona) andlevel surfacewithlevel valueequal 0.2of thecurrentdensityfor€=0.01.

Jz

Fig.4 Distributiona) andlevel surfacewithlevel valueequal 0.2of thecurrentdensityfore=0.5.
534

Bulanov s.y. et al., The MHD Simulations of 3D Magnetic Reconnection

neaf Null Point of Magnetic configurations

We have studied the process of the driven regimesof the separatrices.
The second series of the numerical calculations wasperformed for the vector potential given by Eq' (10)of the magnetic reconnection in the 3D magnetic
configurations. In the first case' when the electricwith e = 0.5. In the Fig. 4 the distributions of the samefunctions as in Fig. 3 are presented. We see theformation of the current sheet and vortex sheets in thevicinity of the separatrices. The explicit dependence ofnumerical simulations demonstrate the formation of theelectric current sheet and vortex sheets in the vicinity ofthe structure of the electric current sheet on z-axis isobtained.

In the case when the magnetic configuration isdescribed by the expression eq. (ll) we have notcurrent excited at the boundary of computational region
is mainly parallel to the magnetic separatrix, thethe separatrices. This qualitatively corresponds toanalytical solutions describing the magnetic collapse' In
second case, when the electric current excited at theboundary of computational region is orthogonal to themagnetic separatrix, the formation of the vortex andelectric current sheets have not been observed in theobserved formation of the current and vortex sheets' The
results of the computational simulation is shown in Fig'

computer simulations. Self-similar regimes of themagnetic configuration evolution show that for themagnetic collapse to appear in this case it takes a time
much longer than in the previous case.

References

tll D. Biskamp, Noninear Magnetohydrodynamics(Cambridge Univ. Press, London' 1993).

t2l V.S. Berezinskii, S.V. Bulanov, V.L. Ginzburg'V.A. Dogiel and V.S. Ptuskin, Astrophysics ofCosmic Rays (North-Holland, Elsevier Sci. Publ',
B.V. Amsterdam. 1990).
t3l S.V. Bulanov and M.A. O1'shanetskij, Phys' Lett'
A 100, 136 (1984); Sov. J' Plasma Phys. ll,'127(1985).

-1-rA
Fig. 5 Distribution of the electric current density fort4l S.V. Bulanov and J.-I. Sakai, J. Phys. Soc. Jpn 66'
3477 (1997).

(11)'

s35