J. Plasma Fusion Res. SERIES, Vol.2 (1999) 528-531

The Evolution of Structurally Unstable 2-D Magnetic
Configuration with Two Null Lines
BULANOV Sergei V., ECHKINA Eugenia yu', INOVENKOV Igor N.r, PEGORARO Franche2and PICHUSHKIN Victor. V.l
Institute of General Physics, RAS, Moscow, Russia
lMoscow State University, Moscow, Russia
2Pisa Llniversity and INFM, Pisa, Italy(Received: 9 December 1998 / Accepted: 23 March 1999)

Abstract
We consider analytically and with MHD simulations the regimes of driven magnetic field line
reconnection in two planar magnetic configurations with two null lines (X-lines), with boundary
conditions that correspond to the excitation of the finite amplitude magnetoacoustic and Alfv6n waves.
We demonstrate that the change of the magnetic field topology due to the magnetoacoustic type
perturbations is accompanied by the global redistribution of the electric current curried by the Alfv6nwaves.

Keywords:
noindent magnetic reconnection, structural stability, current sheet

The problem of magnetic field line reconnection,
which is of great importance for space and laboratory
plasmas [] is closely related to the problem of thestructural stability of vector fields [2,3]. As a result ofmagnetic reconnection the topology of the magnetic
field changes. It is natural to expect that structurally
unstable magnetic configurations must be transformed
into structurally stable ones. We have investigated theprocess of transformation from structurally unstable tostructurally stable configuration and distributions of theconfigurations described by the vector potential with e2+ 0, €t * 0 is structurally stable.

electric current density at the quasistationary stage.

We consider an initial planar two-dimensional
plasma configuration. The initial magnetic filed lies inthe x - y plane, is current-free and vanishes at null lines(along z). The magnetic field is described by the vector-101Fig. 1 Structurally unstable magnetic configuration withpotential ,46 = Ao(x, y)erwith

two X-lines.

o4@, y) = a (r' - 3xy2)+ ttx + e2y.

(1)

JIn the configuration described by vector potential withsz = 0 the separatrix connects two saddle X-lines. It is@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Researchstructurally unstable and is shown in Fig. l. The
528

Bulanov S.y. et al., The Evolution of Structurally Unstable 2-D Magnetic Configuration with Two Null Lines

fo (0=- i (e313 -The equilibrium of the configuration is violated
under the action of magnetoacoustic and Alfv6n waves.

It(-c--rr:u''*'1.
u(^+€t\The problem of the propagation of the smallr>
w-wo=-J-;n"' I /El

amplitude MHD waves in the planar potential magnetic
field that depends on two coordinates was analyzed in2,reGr_ rE ))(9)\r6+If lel << I at large distance from the origin, l(l >>[4] where it was shown that neglecting dissipation andthermal effects the magnetoacoustic wave is described/g ttr" magnetoacoustic wave can be cylindrical with w- wo = 1/(. Assuming the wave to be excited at thecircle with unite radius l(l= I, with the intensity 4s(r),bv-l f'" l't"=0,(8)where r = -tl r ie2. The conformal mapping (5) is
given by the function

The magnetoacoustic and Alfv6n waves are imposed at
the boundary of the computational region.

dd,o€(),Q)

we find that near the front of the wave converging
toward the origin, the wave amplitude behaves as

and for the Alfv6n wave we have(3)

aa,,b -1BorV;2b=0.

Here a(x, y, /) and b(x, y, t) are the perturbed z-

where r = ^F lcomponents of the vector potential in themagnetoacoustic wave and the magnetic field in theAlfv6n wave, respectively. The magnetic field
associated with the MHD perturbations is b(x, y, t) =f.We

see that the azimuthalcomponent of the magnetic field associated with themagnetoacoustic wave increases as r-3l2 when r -+ 0.
Close to the X-lines, where ( = +fe, we have

0rae, - d,ae, + be".The time is normalized to 14np)tt2l
gs2. In equation (2) f 6 = dfs/d(, where /6 = Fo - iAo is
the complex potential of the magnetic field that depends
on the complex variable ( = , + iy. The unperturbed
magnetic field, Be1, can be expressed via the scalar Fs

where s = l( + lEl. Here the perturbations of theazimuthal component of the magnetic field grow as s-land vector potential Ao as

whens-+0[5,6].

als,t)=Equation (3) has a solution b(x, y, t) that is constanton characteristics A6(.x, )) = Ao and

Bo*=-d,Fo=0rAo, Bor=-0r4--E,Ao. (4)This is equivalent to 86, - iBo, = -670166.
To solve equation (2) we perform following [4] theconformal mapping of the complex plane 6 = x + iy tothe plane w = u + iv given by

If e2 = 0 we have the case with the X-X-separatrix,

w_wo='["#r,which is determined by the condition Ao = 0. Expression(12) for the characteristic on the separatrix takes thedx

co=ttI

(s)

(r2)

form
where w6 and (s are constant. This mapping transforms
equation (2) to the wave equation in the CartesianC(r =7

coordinates u, v and t;(6)

dd,,a -dd,,a -dDuua=0,i.

.

o("d, _v)

4l Et

fn

3tr + x2

(13)

3et + x2

The amplitude of the Alfv6n wave remains constant
while the wave approaches the X-X-separatrix x = 0 aswith the Green function+a+l*-fx * llt, but the electric current density associated withthe wave tends to infinity as x-2 for x -+ 0.
We consider non-planar two-dimensional
perturbations where all quantities depend on x, y and /only, but the perturbed magnetic and velocity vector
fields have three components, which allow bothmagnetoacoustic and Alfv6n-type perturbations, andI

ar|

tw=w0

Here 1(.x) = I forx > 0 and X@) = O forx < 0.
Equation (l) corresponds to the complex potentialof the masnetic field of the form

solve the system of the MHD equations numerically. We
use the MHD equations in dimensionless form [3]. To529

Bulanov S.Y . et al., The Evolution of Structurally Unstable 2-D Magnetic Configuration with Two Null Lines-1

Fig. 2 Distributions of the electric current density for magnetoacoustic (a) and Alfv6n (b) perturbation.

model magnetoacoustic waves we choose the vector
potential at the boundaries .r = *l and y = 11 of theform A(x, !, t) = Ao@, y) + r1t2 9Q - llr + l), where
Ao@, y) is given by (l), and 12 = x' + y', and g(e) =-E1E- 1)2le fot 6> 1,0 for { < l. Alfv€n waves aredescribed by the boundary condition for the zcomponent of the magnetic field B,(x = +1, y, t) =+Brmin{1, t/t"*}. All the results of the numericalsimulations presented are obtained for dimensionless

I-1

electric field Er = 0.06 and magnetic field By = 0.1. We
consider the boundary conditions related to perturbation

01Fig. 3 Structurally stable magnetic configuration with
two X-lines.

of the magnetoacoustic and Alfv6n type.
The first series of numerical calculations wasperformed for the structurally unstable magnetic
configuration. The results of the simulations are shownin Fig. 2.
Here and below we use the term "longitudinal"
current for z-component of the electric current (a) and"transverse" current for its poloidal component (j,2 +j'r)t'' (b). The magnetoacoustic wave changes the rightleft symmetry of the initial magnetic field. Two currents
sheets are formed by the magnetoacoustic perturbationnear the two null lines. The transverse current isThe main result shown in Figure 8 is in the symmetry

breaking due to the nonlinear interaction of themagnetoacoustic and of the Alfv6n perturbations on nulllines. The transverse electric current associated with

Alfv6n type perturbations is localized mainly in thevicinity of the null lines and of the separatrices at thetop of the computational region.
We have studied the transformation of structurally
unstable magnetic configurations into structurally stableconfigurations. We have observed a global redistributionof the electric current in the -r - y plane which is due tothe nonlinear interaction of the maenetioacoustic and of
the Alfv6n perturbations.

concentrated near magnetoacoustic shock waves locatedin the vicinity of the magnetic separatrices.
The second series of numerical calculations wasperformed for the initially structurally stable magnetic
configuration. The configuration of this magnetic field isshown in Fig. 3. The results of the simulations areshown in Fig. 4.
The nonlinear interaction of the magnetoacousticand Alfv6n waves leads to a situation where only one
null point dominates. The longitudinal and transverse
currents are concentrated near the dominant null line.

Referencestll D. Biskamp, Nonlinear Magnetiohydrodynamics(Cambridge Univ. Press. 1993).

[2] T. Poston and Y. Stewart, Catastrophe Theory andits Applicatiozrs, Pitman London, 1978.
S.V.
Bulanov, V.V. Pichushkin, and K. Schindler,[3]530Bulanov S.Y. et al., The Evolution of Structurally Unstable 2-D Magnetic Configuration with Two Null Lines

1050Fig. 4 Distributions of the electric current density for magnetoacoustic (a) and Alfu6n (b) perturbation.

[5] S.I. Syrovatskii, Sov. Phys. JETP 50, I 133 (1966).
[6] S.V. Bulanov and S.I. Syrovatskii, Sov. J. Plasma

Plasma Phys. Reports 22,979 (1996).

I4l

S.V. Bulanov and F. Pegoraro, Phys. Lett. A 180,275 (1993); Plasma Phys. Reports 19, ll20 (1993).

Phys.6, 1205 (1980).

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