J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)477–480
000–000

Passive Control of n = 1 Mode Motion on
Field-Reversed Configuration Plasmas
OKADA Masanori, FUJIMOTO Kayoko, GOTA Hiroshi, TAKAHASHI Tsutomu and NOGI Yasuyuki
College of Science and Technology, Nihon University, Tokyo 101-8308, Japan(Received: 9 December 2003 / Accepted: 16 March 2004)

Abstract
An n = 1 mode motion of a field-reversed configuration plasma is investigated from the point of a magnetic
structure of confinement field. The investigation suggests that the plasma having high separatrix elongation is preferable
for reducing the amplitude of the n = 1 mode motion because of a small negative gradient of the confinement fieldaround the plasma. In order to produce highly elongating plasmas, auxiliary conductors are installed into a device.
The observed amplitudes of the n = 1 mode motion are controlled at about a half level of that without any auxiliary
conductors, which is explained well from the magnetic structure.
Keywords:
field-reversed configuration, theta pinch, n = 1 mode motion, wobble motion, passive control, magnetic structure

1. Introduction

has been attempted. We show in the present article that themotion can also be controlled by a passive method. For thispurpose, the source of the n = 1 mode motion is investigatedfrom the point of view of the magnetic structure of theconfinement field. It is found that the behavior of the n = 1
mode motion can be well explained from a negative gradientof the confinement field around the plasma. Then, we
demonstrate a reduction of the n = 1 mode motion byinstalling two kinds of auxiliary conductors into anexperimental device.

A field-reversed configuration (FRC) is an elongatedcompact toroid consisting of a poloidal field only [1]-[5]. It
can confine plasmas with a high β value, which is the ratioof plasma pressure p to confinement field pressure Be2/2µ0. It
is possible to attain an average β value of about 0.9 inside aseparatrix. Moreover, the magnetic configuration has natural
divertors to lead fusion products like α-particles and protons
to a direct energy converter.
From these outstanding features, it is recognized that theFRC has potential application in a future reactor burning anadvanced fuel as deuterium and helium 3. There is atranslation region between the formation and the burning
regions in the reactor [6]. The FRC plasma must be moved
from the formation region into the burning region through thetranslation region along the guide field. The deviation of theFRC from the guide field may cause particle loss due to themagnetic reconnection between both fields inside and outside
the separatrix. Moreover, a high-energy neutral beam forheating the plasma can damage a vacuum wall, when the FRCplasma deviates largely from its planned position in theburning region.
The deviation of the plasma is observed as an oscillating
motion around the equilibrium position. This motion is calleda wobble motion in theta-pinch experiments [7], [8], and aradial shift and a tilt motion in the FRC experiments [9]-[11].
It is also generalized by an n = 1 mode motion (n is a toroidal
mode number).
Control of the motion in the FRC experiment by active
methods using a multipole field [12] or a neutral beam [13]2. Magnetic structure
It is generally known that field lines in a mirror coil havea bad curvature in the central region although a good
curvature in the mirror region. In order to investigate theeffect of the curvature of the field lines outside the plasma on
the n = 1 mode motion, the following Grad-Shafranov
equation for a flux function ψ without pressure and toroidal
current is solved in the region between the separatrix and thecoil [14]:∂ 2ψ 1 ∂ψ ∂ 2ψ–+=0.
∂r 2 r ∂r ∂z 2

(1)

In these calculations, the theta-pinch coil (ψ = 1) has a half
length rc = 0.75 m and a radius rw = 0.17 m at the center. Amirror field is produced using end coils with 0.3 m widthand 0.15 m radius. The separatrix shape of a FRC plasma
introduced into the coil (ψ = 0) has an elliptical form as(r/rs)2 + (z/rs)2 = 1, where rs is a separatrix radius at theCorresponding author’s e-mail: nogi@pyxis.phys.cst.nihon-u.ac.jp1477©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchOkada M. et al., Passive Control of n = 1 Mode Motion on Field-Reversed Configuration Plasmas

midplane (z = 0) and rs is a half separatrix length. It should
be noted that an edge-layer plasma outside the separatrix isneglected in Eq. (1) because of its thin width. It has been
known that the half width of the layer is few ion-radii in theconfinement field corresponding to a value less than 10% of
rs [2], [15].
Calculated magnetic fields at the z = 0 plane are shownin Fig. 1, where xs = rs /rw and zs = rs /rc. It is found that theradial gradient of the field has a positive sign in vacuum andbecomes negative by introducing the FRC plasma. The
strength of the field normalized by the wall one changes thevalue by about 3% between the separatrix and the wall on a
fat FRC with xs = 0.3 and zs = 0.3, although it is almost
constant on a slender FRC with xs = 0.2 and zs = 0.5.
In order to know an axial dependence of the field
gradient, average fields are calculated along field lines fromz = 0 to z = rs asr

∫ Bdr
B=.
∫ dr

The average field in vacuum is calculated from z = 0 to z
= rc /2. These radial profiles are similar to the B/Bw profilesin Fig. 1 except a small variation of the gradient on thevacuum field and the fat FRC. The variation is that the radial
gradient of the average field becomes almost zero on thevacuum field and reduces to about one-third of the B/Bw
profile on the fat FRC.
It can be known from these calculations that theconfinement field around the plasma forms a hill structure inwhich the field gradient has a negative sign to the wall. Thefield gradient becomes large with decrease of the separatrix
elongation E (= rs /rs). This structure suggests that the plasma
is possible to move toward the wall until the magnetic hill
disappears due to the wall effect of the theta-pinch coil.
Therefore, it is anticipated that the plasma having a small E
will deviate largely from the equilibrium position.

s0rs

3. Experimental setup and plasma
parameters(2)

FRCs are produced with deuterium at 15 mTorr fillpressure in a 1.5-m-long theta-pinch coil. The inner radii ofthe coil are 0.17 m at the center and 0.15 m at the ends, which
are the same sizes as the calculations in Sec. 2. This coilgeometry produces a magnetic field profile with a mirror ratio
RM = 1.28 on the z-axis in vacuum. The strength of thenegative bias field is 32~48 mT at the initiation of theconfinement field (t = 0) which is 0.6 T at the peak and has a
120 µs decay time. Conducting rings consisting of three setsof single-turns copper wires with 6 mm diameter are attached
to both ends of the coil. They have a smaller radius rc.r. =0.13 m than the mirror coils. The nearest ring is apart fromthe coil end by 0.05 m. And a set of resistive metal liners canbe installed under the mirror coils. They have a radius rm.l. =0.14 m and a width wm.l. = 0.25 m.
The separatrix radius and length are obtained from asingle-turn loop and a magnetic probe array [16]. Electron
density at z = 0 is measured using a helium-neon
interferometer. The sum of electron and ion temperatures isestimated from the equilibrium equation: 〈β 〉 = 1 – xs2 / 2 [17].
Sixty-channels optical detectors are used to observe the n = 1
mode motion [18]. Each optical detector has a wavelength
sensitivity in the range λ = 550 ± 5 nm. It is confirmed using
a monochromator that radiation from the FRC plasma consists
mainly of bremsstrahlung in the wavelength range.
The collimators of the optical detectors are arranged inparallel to the x-axis (x-array) and the y-axis (y-array). Since
the vacuum vessel is made of a transparent quartz tube, theradiation can be observed at a given axial position. At thepresent measurement, three sets of the x-array and the y-array
are installed at z = 0 and ± 0.22 m.
Typical plasma parameters at the quiescent phase (t =15~25 µs) are rs ~– 0.05 m, rs ~– 0.35 m, an electron densityne ~– 3×1021 m–3, and plasma temperatures Te ~– Ti ~– 100 eV.
Other constituents of the device and plasma parameters aredescribed elsewhere [15].

0– –The radial profiles of B/B w are shown in Fig. 2, where theabscissa denotes radial positions of the field lines at z = 0–and Bw is an averaged value along a field line near the wall.

Fig. 1

Radial profiles of magnetic field outside FRCs at themidplane for a mirror coil.

Fig. 2

Radial profiles of averaged magnetic fields calculated
from z = 0 to z =rs.

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Okada M. et al., Passive Control of n = 1 Mode Motion on Field-Reversed Configuration Plasmas

4. n = 1 mode motion

The radial implosion of the plasma becomes uniform along
the coil similar to that in the straight coil. Consequently, aFRC having a long separatrix length is formed in spite ofusing the mirror coil. When the conducting rings are installed,
almost all the lines of force of the negative bias field arecurved and pulled out from the narrow regions between theends of the coil and the conducting rings. The negative bias
field connects quickly with the confinement field as soon as
it is applied. Since closed lines occupy the full length of thecoil, a FRC having a long separatrix length will be formed.
On the other hand, the separatrix radius may remainunchanged at the equilibrium phase because of the constraintof a radial pressure balance.
The deviations of the FRCs which are controlled by theconducting rings and the resistive liners are shown by open
circles and triangles, respectively, in Fig. 4. It is seen that the––elongation E = 5.5~7.0 is increased to E = 6.0~8.0 by theeffect of the auxiliary conductors and the deviation is reducedto ξm ≤ 0.01 m from ξm = 0.01~0.03 m.

The n = 1 mode motion is estimated from the radial shiftof the radiation profiles formed by the optical detector arrays.
For instance, the x-array and the y-array give the y-componentof the shift ξy and ξ x, respectively. The motion is shown inFig. 3, where the radial shifts are simultaneously measured atthree axial positions. The FRC plasma starts to move at t = 5µs toward the wall of the vacuum vessel and comes back near
the z-axis after reaching ξm ~– 0.011 m at t ~– 20 µs, where ξmis the maximum of ξ = ξx2 + ξy2 . The motion can beobserved during t = 5~30 µs until the onset of the n = 2
rotational instability [19] and [20]. It is also found that theFRC moves almost in phase along the z-axis. This denotesthat the n = 1 mode motion is not a tilt motion, but a radial
shift in parallel to the z-axis.
The shift motion emerges in all FRCs in the present
experiment although the direction and the amplitude of thetrajectories change on a shot-by-shot basis. However, it isfound that there is a clear dependence of ξm on the separatrix
elongation. In order to form FRCs with variable elongation,
strength of the negative bias field is changed. The
experimental results are marked by closed circles in Fig. 4,–– –where E = rs /r–s (rs and r–s are, respectively, a time averaged
value of rs and rs from t = 5 µs to the time reaching ξm). The–deviation ξm increases with decreasing E and reaches ξm ~––~
0.03 m at E – 6.0, which corresponds to about 60% of rs ~–0.05 m.

6. Summary
The magnetic field structure outside the separatrix is
calculated relating to the n = 1 mode motion of FRC plasmas.
It is found that the field profile has a negative gradient to thewall at the midplane of the separatrix. The averaged valuesof the field along lines of force show also the negative
gradient although it is smaller than that at the midplane. The
gradient becomes large with decreasing the separatrixelongation. These calculations suggest that the n = 1 mode
motion is always possible to occur in the confinement field
and becomes violent on the FRC with small elongation.
In order to reduce the n = 1 mode motion, it is neededthat the FRCs with large elongation are formed. For thispurpose, two kinds of auxiliary conductors, which are theconducting rings and the resistive liners, are installed into thedevice. It is observed that the amplitude of the n = 1 mode
motion can be controlled at about a half level of that without
any auxiliary conductors.

5. Passive control
The calculations in Sec. 2 suggest that the amplitude of
the n = 1 mode motion can be reduced if FRCs having a large
E are formed. For this purpose, two types of auxiliary
conductors are installed into the device. The first is a set ofresistive metal liners under the mirror coils, which have ashort time constant τ ~– 4 µs. The second is a set of conducting
rings, which have a long time constant τ ~– 6.5 ms.
The role of the resistive metal liners is expected to retard
the pinch time in the mirror regions at the formation phase.

Fig. 3

Trajectories of a plasma column on the x-y plane atthree axial positions, where Arabic numerals denotethe time in µs from a start of the confinement field.

Fig. 43479

Elongation dependence of n = 1 mode motions of FRCs
with the conducting rings (open circles), the resistiveliners (triangles), and without those (closed circles).

Okada M. et al., Passive Control of n = 1 Mode Motion on Field-Reversed Configuration Plasmas

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