J. Pfasma Fusion Res. SERIES, Yol.2 (199\ 524-527

A Magnetohydrodynamic Model of MRX Discharges. WATANABE Tomo-Hiko, HAYASHI Takaya and SATO Tetsuya
National Institute for Fusion Science, Toki 509-5292, Japan(Received: l8 December 1998 / Accepted: 23 June 1999)

Abstract
A magnetohydrodynamic (MHD) model of the Magnetic Reconnection Experiment (MRX) has been
established. It can reproduce both of the "pull" and "push" mode operations with co-helicity injection.
Time evolutions of the MRX configuration, such as magnetic island formation, can be explained byquasi-static transition of the Taylor state according to change of poloidal field coil current. MHD
simulations agree well not only with the theoretical model but also with experimental evidences.

Keywords:
magnetohydrodynamics, magnetic reconnection, magnetohydrodynamic relaxation

1. Introduction

same toroidal field is induced by the FC coils into theplasma region, while anti-parallel toroidal fields are

Magnetic reconnection found in fusion, space, andastrophysical plasmas [1-3] has been attracted many
researcher's attention in the last two decades. This isgiven in the counter-helicity one. Changing theoperation scheme (four modes in total), Yamada and hisco-workers have found a morphological feature of fieldlines during reconnection. During the "pull" operation, aY-shaped current sheet is formed in the counter-helicity
case, while a magnetic island is created in a diffusion
region of reconnection in the co-helicity injection [7,8].
Our previous work based on the Taylor's relaxation

because reconnection could play a crucial role inmagnetohydrodynamic (MHD) relaxation processes. In

relaxation of an MHD plasma, there are two
characteristic features, such as conversion of magnetic
energy and topology change of field lines [4], for whichmagnetic reconnection is responsible [5]. Magnetic
Reconnection Experiment (MRX) in Princeton Plasma
Physics Laboratory is the first laboratory experimentfocusing on reconnection physics under a controllable
condition [6]. Field configuration in MRX is varied intime by changing currents in a pair of flux-core (FC)coils which consist of poloidal (PF) and toroidal field

theory and MHD simulations revealed a physicalmechanism of the island formation during the "pull"mode [9]. A main purpose of the present paper is to
study time evolutions of the MRX configuration duringthe "push" operation with co-helicity injection and tocompare the results with the "pull" mode.

(TF) coils. Then, a plasma flow is induced, which drives

2. Theoretical Model

reconnection at a separatrix point between FC coils.
MRX has two types of operation modes, so-called "pull"and "push" modes. In the "pull" mode, PF coil currents
are reduced in time, while it is increased in the "push"Here, we consider the Taylor state in a rectangular
plasma container with a pair of FC coils, neglecting

toroidal effects in e direction. Since no center rod isused in the MRX, one of the side boundaries at x = *L,case. In addition, two types of toroidal field
configurations are available in MRX, that is, co- andcorresponds to the main (symmetry) axis in comparison

counter-helicity injections. In the co-helicity case, the@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research

C o rre sp ondin g autho r' s e -mai I : t omo @ nifs. ac.j p524

Watanabe T. et al., A Magnetohydrodynamic Model of MRX Dischargesof the model configuration with the MRX device. Themodel is located in the poloidal plane of -L" I x I L'and -Lr < ! < Ly (L" = | and Z, = 1.4).A pair of FC
coils with their radii of 0.2 are, respectively, placed at(x, y) = (0, 10.6). Fig.l schematically shows the modelarr@configuration.

In a system with symmetry in z direction, theTaylor state can be given by
Y2Y + lt2Y =O(1)

in terms of the poloidal flux Y. Without loss ofol',@generality one may set Y= 0 on the outer boundary andY = Yo = 1 on the inner (FC coil) boundary. Solving
Eq.(l) for given p and Ye, one finds a Taylor state inLthe model configuration. Here, it is meaningful tomention a definition of magnetic helicity in a multiplyconnected region. According to Taylor [10], themagnetic helicity in a torus is defined as K = J A .Bdv-[ ,C.at I A.d.r so that K is indeed invariant to a multivalued gauge potential 7 (B and A are magnetic fieldand its vector potential). Here, d/ and ds denote loop
integrals the long and short way around the toroidal
surface. In the model configuration of MRX, the gauge-Fig. 1 A schematic plot of MRX configuration showing apath of integration defined in Eq.(2). Shaded
regions indicate cross-sections of flux-core coils.

lpl=1.6

invariant helicity is written as follows:rc =[ a.Bdxdy -Z vof d, at,,

X

100a)

80

where dl; means the loop integral around the i-th FC coilsurface, while d*dy denotes the integral in the multiply-

connected region between the outer and inneraOrutrr 40

boundaries.

Solving Eq.(l) by the second-order finite difference
and the conjugate-gradient method, we have plotted the20helicity and energy of the Taylor state in Fig.2, showingthat a magnetic island appears between the FC coils
when lpl exceeds a critical value (=1.$). As ltrrl increases
further (but less than the lowest eigenvalue of Eq.(l),=3.2), the island with a spheromak-like configuration
grows and finally covers the whole system. The result
gives us an important suggestion on the co-helicity
injection of the MRX discharge. Since the MRX plasma
is low temperature (say, lower than 20 eV), it may be

00.5

t522.533.544.5-lt

Fig. 2 Magnetic energy versus ,u of the Taylor state
given by Eq.(1). For lpl > 1.6, a magnetic island
appears.

approximate to the force-free state, that is, a solution ofEq.(l) for lttl < 1.6 before decreasing the PF coildecreased during the "push" operation. The "pull" andcurrent. As the PF coil current is reduced in a time scale
longer than the Alfvdn transit time, Y is "pulled" into
the FC coils, namely, Y6 becomes smaller. This meansthat more plasma current is induced in the system,
namely, lpl is increased by decreasing Y. On the otherhand, in the "push" mode, the PF current is increased
resulting reduction of the plasma current. Thus, lpl is"push" operation, respectively, correspond to injection
and reduction of the helicity normalized Ay U/$tt,.
Therefore, the magnetic island will appear when lpl >1.6 in the "pull" mode, while the configuration tends tothe vacuum state in the "push" case.

525

Watanabe T. et al., A Magnetohydrodynamic Model of MRX Discharges

3. Numerical Simulation
The conjecture derived in the previous section
would be valid for slow variation of PF coil current

respectively, y= I x l0-3and l=5l3.Furthermore,itis
considered that the resistivity 4 might be larger nearby
the FC coils than in the central region because of higher(ideally, infinitesimal induction electric field). However,

impurity density. Thus, we have employed anin real experiments, the PF coil currents are changed ina finite time (typically several tens of;rsec). Thus, bymeans of numerical simulation, it is necessary to checkthe validity of the scenario in a realistic time scale.

inhomogeneous 4 model such as

Using the same model described before, we have carried

where r and r" denote a distance from a center of thenearest FC coil and its radius. The previous simulation
for the "pull" case using the inhomogeneous 4 has given
a better agreement with the experiment than the constantn = n,lr+ c4 exp

out two-dimensional MHD simulations, where thefollowing equations are solved;+=-y.(pv),dt(3),.- ,--_L# ff

nv ' v + r1i2 +Time evolutions of the magnetic helicity and<D'iY -- r&energy obtained in the "pull" and "push" simulations areplotted in Fig.3 where solid line shows the Taylor state

given by Eq.(l). Starting from the Taylor state withF = l, in the "pull" case, both of the normalized energy(5)and helicity are increased in time. One can see the time

evolution follows the lowest energy branch of theTaylor state. After r = 20 Tt, when the energy increases(6)

to 4.4, a small island appears in the diffusion region, andthen, grows larger and larger. Comparing Figs.2 and 3 it
is found that the energy at the first appearance of theisland corresponds to F = 1.2 in the Taylor state.
Although the estimated p is smaller than the criticalvalue of 1.6, the difference may be attributed toinhomogeneity of p which is often observed in dynamic
simulations of plasma relaxation. On the other hand, in

aB_;tcrt -- v xE".

Q)

Here, j, E, B, aD and, e;i are, respectively, given byj = j,2 +j, =-iY2Y +VB,xi,(8)E =E,i +Eo =-v xB+ e j ,(e)B =B,i+YYxi,(10)@=2v(,, ,u -l ro ,l') ,,,,=Lf'

9.9l

2 \dxt o*, )(13)resistivity case [9].

e y; =-Yp+ i x B + v(Y2v* 1 v (v y)),(4)=-{- r, -, "r',,?}f,(1l)

l4
t2

(r2)

Taylor stateatpull mode
push mode=gog

The suffix p denotes a poloidal (.r, y) component of avector. Velocity y is set to be zero on the boundaries.
We have used a perfect conductor for the outerboundary, while E, = 0 and E = 10.01 on the innerboundaries that correspond to the FC coil surfaces. The
positive E. means the "pull" operation, while thenegative E is imposed on the FC coil surface in the"push" case. All of physical quantities are normalized
by the typical length L" = l, a characteristic poloidal
magnetic field 8o6 = YolL, = l, and the initial density p6= 1. The Alfv6n velocity Vas given by B,o and ps isequal to l. Time is measured by the Alfvdn transit time
Te= L,/Vao. The initial pressure is set to be p = 9.2.
Viscosity y and ratio of the specific heats f are,

50 (pull)

tl0C)

tu6
420

t=0
3'= 40 (push)

46l0Hclicity
Fig. 3 Evolution of the magnetic energy and helicity
found in MHD simulations for "pull" and "push"modes. Solid line shows the Taylor state. Marks
representing simulation results are plotted atevery 5 ?a.
526

Watanabe T. et al., A Magnetohydrodynamic Model of MRX Discharges

Ministry of Education, Science, Culture, and Sports inthe "push" case, the normalized energy and helicity aredecreased in time, while total values of them (notnormalized by %) are increased by the poloidal flux
injection. Thus, the global configuration in the "push"case approaches to the vacuum solution, although asharp peak of the plasma current is found at theseparatrix point. Therefore, no magnetic island isJapan (09780445, 10898016, 0987890, and 10044105).

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[4] J.B. Taylor, Phys. Rev. Lett. 33, ll39 (1974).
[5] T. Sato and K. Kusano, Phys. Rev. Lett. 54, 808(1985).

theoretical prediction, and are also qualitatively
consistent with more recent experiments where no island

is observed in the "push" operation I l].

4. Gonclusion

t6l M. Yamada, H. Ji, S. Hsu, T. Carter, R. Kulsrud,

In this study we have investigated transition of field
configuration in "pull" and "push" operations of theN. Bretz, F. Jobes, Y. Ono and F. Perkins, Phys.
Plasmas 4,1936 (1997).

MRX discharge. The Taylor state analysis shows thatt7l M. Yamada, H. Ji, T.A. Carter, S.C. Hsu, R.M.

the quasi-static change of the force free state can explain

Kulsrud, N.L. Bretz, F.C. Jones, Y. Ono, M.

both of the operations with the co-helicity injection. In
the "pull" case a magnetic island is formed, while theconfiguration tends to the vacuum solution in the "push"case. The simulation results under a realistic condition
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Katsurai, T.-H. Watanabe, T. Sato and T. Hayashi,

Proc. l6th Int. Conf. Fusion Energy, Montreal,
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t8l M. Yamada, H. Ji, S. Hsu, T. Carter, R. Kulsrud,
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also successfully explain the experiments.

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t9l T.-H. Watanabe, T. Hayashi, T. Sato, M. Yamada

Acknowledgment

and H. Ji, Phys. Plasmas 6, 1253 (1999); T.-H.
Watanabe, T. Hayashi and T. Sato, Bull. Am. Phys.
Soc.43, 1840 (1998).
tl0l J.B. Taylor, Rev. Mod. Phys. 58, 741 (1986).

One of the authors 0.-H.W.) would like to thank

M. Yamada. H. Ji. and Y. Ono for their fruitfulcomments and discussions. Numerical computations inthis study are performed on the NIFS MISSION System(Man-Machine Interactive System for Simulation). This

[1] M. Yamada, H. Ji and Y. Ono, private
communication (1998).

work is partially supported by the grants-in-aid of527