J. Plasma Fusion Res. SERIES, Vol.2 (1999) 512-515

Ghaotic Reconnection Due to Mixing of the Vortex-Current
Filaments
YATSUYANAGI Yuichi, HATORI Tadatsugur and KATO Tomokazu
Department of Applied Physics, Waseda Llniversity, Tokyo 169-8555, Japan
lNational Institute for Fusion Science, Toki 509-5292, Japan(Received: 7 December 1998 / Accepted: 16 June 1999)

Abstract
We propose a new reconnection mechanism "chaotic reconnection". The basic mechanism of thechaotic reconnection is examined by means of numerical simulations of collision between two vortexcurrent filaments. We conclude the chaotic reconnection is a faster process than the traditional resistive
non-chaotic reconnection.

Keywords:
solar flare, chaos, magnetic reconnection, MHD, vortex-current filamentmodel in our previous papers [3,4]. The vortex-current
filament consists of the electric curent and the vorticity
inside it. As basic equations, we use the ideal MHD

1. lntroduction
Nowadays many researchers share a common
aspect that the magnetic reconnection is a fundamental
process in the solar flare [1]. Now it is accepted in mostequations with gravity. Integrating the equation of
motion over the small volumeelement, we obtain amacroscopic force balance equation correct to OIp-2)
where p is a local radius of curvature of the filament.

cases that the electric resistivity plays a minor role andmost of the theoretical works on the reconnection todate deal with the time scale problem [2]. Here we shallpropose a new fast reconnection mechanism "chaoticreconnection" of the vortex-current filaments in theThe velocity of the filament dRl}t is given in thefollwing form with the cutoff Biot-Savart integral
correct to O(p-'): [3]three-dimensional space.
In Sec. 2, we give simulation results of collision(#), =-tr*{B)r+(ur)r

between two straight vortex-current filaments anddiagnosis via the Lyapunov exponents. The collisionmeans the strong interaction between two filaments.
In Sec. 3, we show the basic mechanism of the(R-x)xdr*ltoJ'z S |\312
41TK JllR -rl' + a2a2l\/

chaotic reconnection. The mechanism is revealed by theevaluation of the reconnection rates and we conclude thechaotic reconnection is a faster process than the-frt

traditional resistive non-chaotic reconnection which has
the simple configuration of the magnetic field.

*4,(R -x)x dr(rn -'t' + Pzoz)3''z
xs +O(p-2\(1)2. Simulation Results
We have introduced the vortex-current filament
C o rre sponding autho r' se-mail : yyanag i @ mn'was e da. ac j p512

@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchYatsuyanagi Y . et al., chaotic Reconnection Due to Mixins of the vortex-Current Filamenrs

Illustration of the initial conditions of the simulation isgiven in Fig. l. The most crucial parameter in oursimulations is symmetry of the initial configuration of
the filaments. The symmetry of the initial configuration
is determined by an initial angle 0 between the two
filaments projected on the r - z plane shown in Fig. l.

(c) is due to the collision where the two filaments fully
overlaps with each other. We consider that the otherpeaks at that time for types (a) and (b) are caused by thesame reason. After the collision, the exponents only fortypes (a) and (b) are positive. We, therefore, conclude
that the chaotic configuration is induced by the collision

The values of the initial angle are chosen as ttlz,of the two filaments with initiallv low-svmmetrical2 arctan (ll2) and zero which are called types (a), (b)configuration.

and (c), respectively. The most symmetrical case is 0 =0 and the most asymmetrical case is 0 = nl2.

3. Ghaotic Reconnection

We show typical results of simulations in Fig. 2.
For all the types, the filaments are attracted each otherand collide. Although the results for types (a) and (b)show complicated configuration near the collisionalregion, the result for type (c) shows non-complicatedconfiguration. This is because the initial configurationsof the types (a) and (b) are more asymmetrical than thatof the type (c).
The time evolution of the instantaneous Lyapunov
exponent is shown in Fig. 3. The largest peak for type

We consider when the filament I approaches thefilament 2 and the electric currents and the vorticities

inside the filaments are antiparallel to each otherrespectively, the net electric current and vorticity must
be nearly zero. This means the filaments should locallyannihilate each other in that region. Here we introduce athree-dimensional space averaging. The threedimensional space averaging is a kind of mixing. It is
well known that the mixing increases the entropy of thesystem and this implies that the mixing is an irreversible
process. Thus the space averaging is considered tointroduce a dissipation process into the system
FilamentIF

artificially. We calculate the three-dimensional spaceaveraged distribution of the electric currents from theresults shown in Fig. 2 numerically and trace theilsrr:r*nt 3

gx.e Drifi

trajectories by its distribution. The results are shown inFig.4.
For types (a) and (b) in Fig. 4, the macroscopic
filaments, which mean the reconstructed filaments
obtained by tracing the distribution of the electric
current, are reconnected with each other. The
reconnection is due to the chaotic configuration inducedby the collision of the two filaments in low-symmetry
system. Thus it is obvious that the reconnection is notobserved for type (c) because the initial configuration isdistence;5r
Fig. l lllustration of the initial conditions
simulation is shown.

symmetrical and the configuration does not evolve intoa chaotic one. We call this reconnection mechanismof the"chaotic reconnection" henceforth [5].

?0Type(a)Type {c)Fig.2 Snap shots of the configuration of the filaments atT=78 x 104Af are shown.
513

Yatsuyanagi Y. et al., Chaotic Reconnection Due to Mixing of the Vortex-Current Filaments

One can estimate the efficiency of the reconnectiondescribable by MHD but much less than the scale length

process by a reconnection rate. We assume theof the phenomena while the collisionless process ismicroscopic. We consider the large A(t, q yields thereconnection rate Rs(r) of the chaotic reconnection is
given by the following form:a)fast reconnection. Unnormalized overlapping volume iscalculated from the time-developed configuration of thefilaments numerically and is normalized by the non-where D(t) is a traditional non-chaotic reconnection rate
determined by the resistive dissipation process of thechaotic overlapping volume which is determined by theinitial configuration.

system. The notation AG, 0) represents a normalized
overlapping volume between the filaments per unit
volume. The value of D(t) is usually zero for the case ofideal MHD and positive for the case of non-ideal MHD.
It is well known that a kind of dissipation process isneeded for the mangetic reconnection. Thus a

Evolution of the normalized overlapping volumeAG, 0) is plotted in Fig. 5. For each type, there is a peakat T = 25 x 104 or so and these peaks are due to thecollision. In later time the values of the normalized

Rr(t)= D(I)AG, 0),

overlapping volume become large for types (a) and (b)while zero for type (c). The large value of thenormalized overlapping volume is rapidly achieved by
the chaotic dynamics of the filaments based on the ideal

dissipation process, space averaging, is introduced in thepresent work because D(r) should be finite.

Here we introduce another mesoscopic

MHD and the "diffusion regions" are formed anywherein the collisional region along the tangled filaments.

enhancement factor, normalized overlapping volume, toThen the reconnection rate R6(r) becomes sufficiently
large if the factor D(/) in Eq. (2) has a nonzero value.
Thus we conclude the chaotic reconnection is a fasterprocess than the traditional resistive reconnection owing

solve the time scale problem of the fast reconnection.

The term mesoscopic means the scale which isto the enhancement by the large normalized overlapping

volume. It is the main mechanism of the chaotic
reconnection in the three-dimensional space. If theoverlapping volume 9(t, 0) is small, the reconnection

Eroo,ir.stime scale is the same as that of the traditional resistive
non-chaotic reconnection which is determined only byF r'0.}: o.s

the factor D(r).

og
64. Conclusions
In this work we have proposed a new fast

0.0-0.50255075

reconnection mechanism, "chaotic reconnection" of thevortex-current filaments in the three-dimensional space.
Note that the basic dynamics are based on the nonlinear
ideal MHD in which the time scale is mainly determinedby the Alfv6n transit time, although we have introduced

100

Time [1o4at]

Fig.3 Evolution of the instantaneous

Lya pu n ov

exponents are shown.

Type (c)Fig. 4 Trajectory of the three-dimensional space-averaged distribution of the electric current is shown. Original data are
given in Fig. 2.
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Yatsuyanagi Y. et al., Chaotic Reconnection Due to Mixing of the Vortex-Current Filaments

a dissipation process, i.e. three-dimensional space
averaging. We must extend the present model to therealistic one and apply our model to the solar flare.
Further progress in understanding chaotic reconnection
dynamics requires the fully three-dimensional MHD! 800
ts nnnGsimulations in the high magnetic Reynolds number.

400

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