J. Plasma Fusion Res. SERIES, Vol.5 (2002) 234-237

n=l Mode Global Motion
on Field-Reversed-Gonfiguration Plasmas
GOTA Hiroshi. TACHIKAWA Eri, OHKUMA Yasunori',
FUJIMOTO Kavoko.
'TAKAHASHI Tsutomu and NOGI Yasuyuki
College of Science and Technology, Nihon University, Tolqo 101-8308, Japant Graduate School of Technology, Tolcyo IJniversity of Agriculture and Technology, Koganei 184-0012, Japan(Received: 11 December 2001 / Accepted: 15 May 2002)

Abstract
An n = I mode motion of a field-reversed-configuration plasma is investigated using a mirror
configuration and a cusp configuration of the bias field. The n = I mode motion reaches 20-40 7o of theplasma radius in the former configuration, although it is at a low level in the latter configuration. It isexperimentally confirmed that the violent n = I motion can be controlled by a multipole field. The
critical strength of the multipole field needed to push back the plasma to the equilibrium position istheoretically derived using a simple model which includes the conductor effect of the wall, and iscompared with an experimentally obtained value. The two values agree within the range of experimental

reproducibility.

Keywords:
field-reversed configuration, FRC, n = 1 mode, global motion, multipole field, theta pinch

1. lntroduction
A field-reversed-configuration (FRC) plasma isformation region.

formed by the negative-biased theta-pinch method [1].
Since the separatrix surface of the FRC with a radius r,
at the center is far from the coil wall r*, for instance r,/r* = 0.3-0.4, the wall stabilization effect on the plasma
motion is weak. Therefore, the plasma possesses much
freedom of movement in the confinement field. It is wellknown that the FRC plasma deviates slightly from theequilibrium position and moves slowly around it. This
motion is called the n = 1 mode motion.

plasma without a violent n = 1 mode motion is firstdescribed. Secondly, the effect of a multipole field on
the n = I mode motion is shown. Lastly, the criticalstrength of the multipole field required to control themotion is discussed and compared with the experimental

An experimental method to produce an FRC

results.

2. Experimental Setup
The FRC plasma is formed using a theta-pinch
device called NUCTE-III which has a one-turn coil with

When the FRC plasma is translated from aformation region into a quasi-static field in order to

1.5m length and 0.34 m bore at the center (z = 0). It is

possible to generate the two kinds of bias field
configurations shown in Fig. 1 by changing the coilconstruction. The strengths of the bias field and theconfine the plasma for a long time and to heat it using aneutral beam [2], the possibility that the deviation of theplasma becomes large during the translation arises, even
if it is small in the formation region. Therefore, the n =I mode motion must be controlled to a low level in theconfinement field at z = 0 are Bv = -3.2x10r T and 8" =@2OO2 by The Japan Society of Plasma

Corresponding author's e-mail: kayoko@ pyxis.phys.cst.nihon-u.ac.jp
234

Science and Nuclear Fusion Research

Fujimoto K. et al., n = I Mode Global Motion on Field-Reversed-confieuration plasrnas

0.5 T, respectively. The FRC plasma is generated using
deuterium gas at l0mtorr fill. Two kinds of multipolecoils are installed ot r, = 0.14 m between the thetapinch coil and the vacuum vessel made of a transparent
quartz tube. The coils produce a quadrupole field (m =2) and hexapole field (m = 3). Each field is separately
applied to the FRC plasma just after its formation. Other
constituents of the device are described elsewhere [1].

The n = I mode motion is observed using a

multichannel optical system that consists of an .r-array
and a y-array, as shown in Fig. 2. Each channel of theoptical system can detect the radiation integrated along
the optical path. The wavelength sensitivity of thesystem is limited to 550+5 nm by optical filters, whichz [m]Fig. 1 Bias field configurations: (a) a mirror field and (b)a cusp fieldis the range where the radiation is mainly
bremsstrahlung [3].
Typical examples of the line-integrated radiation
profiles can be seen in Fig. 3, where the time noted ineach frame was measured from the start of theconfinement field. The closed circles and the open
circles correspond to the r-array and the y-affay,
respectively. The profiles show much different shapes inthe third frame of (a) where an n = 2 mode rotational

instability occurs and (b) becomes triangular upon
application of the quadrupole field. The amplitude of then = I mode motion is measured from the center

Multipole coilpositions of each profile, (, and (".
Fig. 2 Arrangement of optical fibers and quadrupole

field coil-arr

0.3
(a)

€a,0.1
d;

0€d

20ps

o.2

*/*\r*

Hx 0.2

€

40ps

(t)

0.10+)

F{

o.20.1

E,nE,

6ops

*n*

20psG)

40ps

*""ru

60ps*"afryq--

Sff#oo4osoffif-10-505-10-50510
r,y-axis [ro2 m]x,y-axis [ro2 m]

Fig. 3 Line-integrated radiation profiles from the x-array (closed circles) and the y-array (open circles), (a) without amultipole field and (b) with the quadrupole field235

Fujimoto K. et al., n = I Mode Global Motion on Field-Reversed-Configuration Plasrnas

3. Experimental Results

trajectory from the origin becomes 26 Vo of r.. Although
the FRC plasma depicts a different trajectory shot byshot, the deviation is within 20-40 Vo.
'fhe n = I mode motion for the case of the cusp

The deviation of the FRC plasma in the case of themirror bias configuration is plotted in Figs. 4(a) and5(a). The traces are terminated at t = 4O ps because theradiation profiles include a significant error due to thegrowth of the large-amplitude n = 2 mode.It is seen that
the FRC plasma remains for 25 ps near the origin andbias configuration is shown in Fig. 4(b). The motion
becomes mild comparing to that in the mirror biasconfiguration. The result of the cusp bias experiment isplotted at I^- 0 in Fig.7 where 6G J€?+€l )then moves to the opposite y-direction. After reaching y= -0.01 m, it turns to the -r-direction.
The equilibrium FRC plasma has no azimuthal field
Be. However, such a field is generated when the open

three FRC plasmas are averaged for r = 2015 ps.

When the multipole field is applied to the FRC
plasma, the observation of the n = I mode motion
becomes possible for a longer time than that without themultipole field because the n = 2 mode is controlled to alow level [5]. When a weak multipole field is applied,the behavior of the r = I mode motion does not change
significantly. However, the motion disappears abruptly
upon increasing the field, except in the initial 20-30 ps,

field is distorted by the deviated plasma. The
relationship between the FRC plasma motion and the Be
field had been reported in reference 4. Following that
relationship, the Be measured at z = *0.2 m shown inFig. 6(a) indicates that the axis of the FRC plasma shifts
at0.7 - 1.0x10-2 m in parallel with the z-axis and tilts to0.9 - 1.0x10-2 radian for t = 30 ps.
Since the FRC plasma maintains r" = 0.05 m during

the trace in Fig. 5(a), the maximum deviation of the5.0
2.5(a)

F -2.5(a)- -5.002.5

1x10(b)_,l^/l/ t'./l^ ..n..

n1"t_0.2[m]...-....+0.2[m]r \fiVV,o,.fJ.

-2.5-5.0050

1

I [ps]

100Fig. 6 Time evolution of an azimuthal field at z = +0.2 m
on the vacuum vessel (a) without a multipole fieldand (b) with the quadrupole field010203040
r [us]
Fig. 4 Time evolution of deviation of an FRC along the x-axis (solid line) and the y-axis (dashed line): (a)mirror bias configuration and (b) cusp bias con-I w/o Multipoleo Quadrupoleo Hexapole
X Cusp bias

figu ration(a)lro ml

lrob)

zml

I-1

20
10

1.

-1, 080pi

L\.PgID1,)K-1"{li{,Ll { { to,t102030

L [kA]

Fig.5 Trajectories of an n= 1 motion on the x- yplane(a) without a multipole field and (b) with the qua-Fig. 7 Multipole coil current dependence of normalized
n = 1 displacement

drupole field236

Fujimoto K. et al., n = I Mode Global Motion on Field-Reversed-Configuration Plasmas
as seen in Figs. 5(b) and 6(b), where current Iz= 48kA

that reduces the amplitude to 4^= feo (f, reduction factor) is obtained from the relation h63 = k^€?" asflows in the quadrupole coil. The quadrupole current
produces 8e = 0.098 T in vacuum at r = 0.05 m. The
field does not exceed 20 Vo of the confinement fieldT_tmc-even when the conducting effect of the plasma and thetheta-pinch coil is included.

nr*1f@ -\s'(4)The numerical values under the present experimental
conditions give 12" = 29 kA for m = 2 and 13" = 38 kAfor m = 3 at f = 0.3. These values explain well the experimental values in Fig. 7.

The effects of the quadrupole field and thehexapole field on the mirror-biased plasma aresummarized in Fig. 7. The abscissa is the current in themultipole coil. The ordinate is ( normalized by the5. Summary
The n = I mode motion was investigated using two

separatrix radius. The data are averaged for l0 ps at themiddle of the discharge. The n = I mode motion iskinds of bias field configurations. It was found that thecusp bias configuration is preferable for generating theFRC plasma without violent n = 1 mode motion.
However, in the mirror bias configuration, the amplitudeof motion reached 20-40 Vo of the separatrix radiusbecause of the asymmetrical reconnection at theformation phase of the FRC plasma. The motion can beclearly depressed by applying quadrupole currrent above
35 kA. The hex.apole field also shows a similar effect on
the motion although higher current is needed than thequadrupole current.

4. Discussion
When the FRC plasma deviates from theequilibrium position, a restoring force due to theconductor effoct of the theta-pinch coil arises. The

controlled by the multipole field. The field strength
needed to reduce the amplitude of the motion wasderived considering the wall effect of the theta-pinch
coil and was compared with the experimentally obtained

motion of the IrRC plasma can be revealed by a simple
analysis of the following equation of motion:,22ni B1
M(t.r. =_knr . kn=''.-'sle"dt2
Ilorlvalue. The two values agree within the range of(l)experimental reproducibility.

where r and M are the radial position of the center ofmass and the total mass of the FRC plasma, respec-References

tll Y. Ohkuma, M. Urano, M. Nakamura, Y.

tively. The k6 coefficient is derived from a model in

Narushima, T. Takahashi and Y. Nogi, Nucl.

which the FRC plasma is replaced by a magnetic dipole
moment between two parallel plate conductors with adistance 2r*.

Fusion 38, l50l (1998).

t2l H.Momota, A. Ishida, Y. Kohzaki, G. Miley, S.
Ohi, M. Ohnishi, K. Sato, L.C. Steinhauer, Y.
Tomita and M. Tuszewski, Fusion Technol. 21,

Another restoring force is added to the FRC plasma
when the multipole field is operated. The equation ofmotion becomes [6]z" l*:r) B?

2307 (1992).

[3] T. Takahashi, T. Asai, Y. Narushima, Y. Ohkumaand Y. Nogi, Proc. lst General Assembly of Asian
Plasma and Fusion Association Joint with 3rd Asia
Pasific Plasma Theory Conference, Beijing, 1998,

121(Chinese Physics Letters), 92 (2000).

The magnetic lield B. is the vacuum field strength produced by the rrtultipole coil at r = r"asp -Pory!^[D'= or,"'[r.i"\'-'

t4l S. Kumashiro, Ta. Takahashi, M. Ooi, T.
Takahashi, S. Shimamura and Y. Nogi, J. Phys.
Soc. Jpn. 62,1539 (1993).
t5l S. Ohi, T. Minato, Y. Kawakimi, M. Tanjyo, S.
Okada, M. Kako, S. Goto, T. Ishimura and H. Itoh,Phys. Rev. Leu. 51, 1042 (1983).
t6l K. Fujimoto, A. Hoshikawa, S. Ohmura, T.
Takahashi, Y. Nogi and Y. Ohkuma, Phys. Plasmas
9, r7r (2OO2).

(3)

where 2m, r- and gxrepresent the order of the multipole

field, the radia.l position of the multipole coil and thepermeability in vacuum, respectively. Therefore, the am-

plitude of motion without the multipole field fs can becontrolled by the coil current 1-. The critical current 1."

237