J. Plasma Fusion Res. SERIES, Vol.2 (1999) 198-201

Tilt Stabilization by Energetic Beam lonsin a Field-Reversed Configuration
NISHIMURA Kazumi, HORIUCHI Ritoku and SATO Tetsuya
Theory and Computer Simulation Center,National Institute for Fusion Science, Toki 509-5292, Japan(Received: 9 December 1998 / Accepted: 19 February 1999)

Abstract
The stabilization of the tilt disruption in a field-reversed configuration by an ion beam isinvestigated by means of a three-dimensional particle simulation. The growth rate of tilt mode decreases
as the beam current increases. It is also found that the drift kink mode grows in the vicinitv of the fieldnull line and saturates in the early phase.

Keywords:
field-reversed configuration, tilt instability, drift kink instability

1. Introduction

decreases, while it becomes impossible to suppress theThe compact torus is attractive for a fusion reactor

instability by decreasing F"o. Then the other stabilizing
method is needed for future devices. The main purposeof this paper is to investigate the properties of the tilt
instability in the FRC plasmas including the beam ionsby means of three-dimensional particle simulation.

because it has some superior features compared with therepresentative nominated reactor, e.g., tokamak. A field-reversed configuration (FRC) is grouped into thecompact tori. Since the pressure of plasmas confinedwithin the separatrix reaches its maximum value in thevicinity of the O-point (the field-null line) and thetoroidal magnetic field does not exist ideally, theaveraged plasma beta (B) tends to be high in FRC
plasmas ((B) - l). In future, this configuration isespecially expected for a D-3He fusion reactor [,2].
The tilt mode (n = l) is known as one of the most

2. Simulation Model
We consider the FRC plasma confined by auniform external magnetic field within the cylindricalconducting vessel. The height ofthis vessel Zp is always
fixed to 6 times the vessel radius Rp in this paper. The
time development of all particles (thermal ion, thermal
electron, and beam ion) and field quantities are solved
with using the three-dimensional electromagnetic
particle simulation code [4j. It is assumed that thedangerous instabilities to disrupt the configurationcompletely, where z represents the toroidal mode
number. We focused on the two effects (ion finiteLarmor radius (FLR) effect [3,4] and profile controleffect [5,6]) and carried out the three-dimensional

physical quantities are periodic at the boundary of z axis
and the vessel wall is a rigid perfect conductor. The
particles are elastically reflected at the vessel wall.

particle simulation in the previous paper [7]. The

Two-dimensional MHD equilibrium solution isconclusion was that the growth rate of tilting instability
decreases as the plasma beta value at the magnetic
separatrix F.o increases. In order to apply the FRC
plasma for nuclear fusion device, not only stability but
also confinement is improved to be better. The particle
confinement is generally improved to be better as the B.oused as an initial condition. In the case where the beamions exists, we adopt the MHD equilibrium including@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Research198

Nishimura K. et al., Tilt Stabilization by Energetic Beam Ions in a Field-Reversed Configuration

beam components.

Table 1 Characteristics of beam ions used in particle

simulations, including beam velocity va/c;- vp + f,i oru =0,(l)number ratio N"/N,; current ratio Ql".

v xB =T CIo +,ru),a)

RUN ub/c

with the Cobb's pressure model [6], which can control
both the current profile and the beta value at theseparatrix easily. Ifthe profile ofbeam current density isdetermined, the set of these equations can be solved
numerically under the assumption that n6lni << l, where

R00

0.00

RA1
RA2
RA3
RA4

0.033 0.01 0.02
0.050 0.01 0.030.067 0.01 0.04
0.084 0.01 0.05

RB1RB2RB3RB4RB5RB6n6 dnd ni are the number density of beam ions and thatof thermal ions, respectively. The spatial distribution ofbeam current is assumed to be in a Gaussian form. The
velocity and the center position of beam distribution aredetermined so as to satisfy the force balance equation.
The total number of loaded thermal particles is fixed to106. The beam ions are the same kind of thermal ions,i.r., qo - 41. The number of beam ions can be changed
freely as one of the controlled parameters in thisNtlN; hlro
0.00 0.00

0.008

o.02 0.01

0.020
0.028
0.040
0.055
0.063

0.02 0.025
o.o2 0.035

RCl 0.029

0.005 0.009

0.02 0.05
0.02 0.07
o.02 0.08

RCz 0.029 0.01 0.018
RC3 0.029 0.015 0.027
RC4 0.029 0.02 0.037
RCs 0.029 0.025 0.046
RD1RD2
RD3

simulation.

3. Simulation Result
The typical parameters are as follows: the mass
ratio is tnilm" = 50, the ion cyclotron frequency definedby the vacuum magnetic field is @"i = 0.O2 arl{ (q:electron cyclotron frequency defined by the vacuum
magnetic field), the ion thermal velocity is v1 = 0.015 c(c: velocity of light), the temperature ratio is T;!7" = l,
the radius of the cylindrical vessel is Rp - 5.63 clay1,
and the toroidal current of thermal plasma is 1o = $.p
4Nah (qr..r: elementary charge). The Alfv6n transit time/a (= Rp/va where va is the Alfv6n velocity defined by
the ion density at the field-null line and the magnetic
field at the wall on the midplane) becomes about200 d. The profile control parameters are fixed as flo= 0.O2 and D = -0.6 for all cases, where D is thehollowness parameter which controls the current
distribution. The profile given by these parameters
becomes unstable against the tilt instability if the beamcomponent does not exist [7].
There are two important parameters which we may
change independently. One is the beam velocity v6lc,
and the other is the ratio of the number of beam ions to
the number of thermal ions N6/Ni. The parameters used
for the simulation are listed in Table I. The simulation
runs are classified into two types. One type is to vary
the velocity of the beam ions v6 with keeping the total
number of beam ions N5/N1 constant (RA and RB inTable I), and the other type is to vary the total number

0.059 0.005 0.017
0.059 0.015 0.054

0.059 0.01 0.035

of beam ions with keeping the velocity of the beam ionsvb constant (RC and RD in Table I).

3.1 Drift kink instability in the early phase
Figure I shows the time development of themagnetic field on the midplane for the case RA4. Theleft panels are the contour plots of the z-component of
the magnetic field B. and the right panels are those of

the e-component of the perturbed magnetic field AB.
defined by B,(t) - (8.(t)) (the bracket stands for theaverage in the toroidal direction) in the 0- r plane. The
solid lines and the dashed lines represent the positive

values and the negative values, respectively. The
maximum and minimum values are plotted on the top

part of the panels. The strong deformation of themagnetic profile occurs in the vicinity of the field nullline, where the current density including the beamcomponent steepens. The toroidal mode number of thisinstability is equal to 4 and the real frequency ar, isequal to 0.1 ar"i.This instability is saturated in the earlyphase of the simulation (about t/to = 2) for all cases. In

Fig. 2, the maximum amplitudes of perturbed magnetic

field are shown as a function of the current ratio I5lIr.
The maximum amplitude or the growth rate decreases
with the beam current lbllD, and any significant
199

Nishimura K. et al., Tilt Stabilization by Energetic Beam Ions in a Field-Reversed Configurationt /te=lmax-3.89 min=-4.28 max=0.40 min=-0.53
>.

a;v

0.8

{ f "[,.

0.6TI0.4(,t ftP2RARB
RC

aooo1

0.2

max=4.09 min=-4.17 max=1 .08 min=-0.65

0iii:jiiii;.riiiii iiiiiiiiiiii

00.02 0.04 0.06 0.08 0.1
I iltpmax=4.12 min=-4.21

Fig. 3 Tilt growth rates as a function of the current ratio
loll, lor the cases where the beam velocity ischanged (RA and RB), and the cases where thebeam number is changed (RC and RD).

t /te3max=o.69 min=-o.73

5.64

3.2 Tilt stabilization by the ion beam
The development of tilt instability is defined byusing the Fourier amplitude of the z component of thefluid velocity V!\(t) after the drift kink instability
saturates. We follow the time development of thisquantity and obtain the growth rate. The relation
between the tilt growth rate and the current ratio ispresented in Fig. 3. From this figure, it is found that theFig. 1 Time sequential plots of (left) the magnetic field
contours and (right) the perturbed magnetic fieldcontours.

65growth rate remains almost the same as that for no beamcase (R00) for 0 < 16llo < R"(= 0.03). If the current ratio
exceeds this critical value R", the tilt growth rate
becomes small as the current ratio increases. There is no
significant difference between the fixed number cases(RA and RB) and the fixed velocity cases (RC and RD).

1O4u

J3
ai
32

In other words, the influence of beam ions on the tiltoo
10growth rate can be evaluated by the current ratio I'o/1,
regardless of whether the beam current is controlled by
the beam velocity or by the number of beam ions.
The values of the FLR parameter J roughly stands
for the ratio of the plasma radius to the ion Larmor
radius. It is found that 5 tends to become larger as thecurrent ratio increases. As the beam current is stronger,
the inclination of magnetic field at the field null issteeper and thus the magnetic field becomes stronger inthe vicinity of the field null. In other words, the thermal

rj?'
0 0.02 0.04 0.06 0.08 0.1
IulIp

Fig. 2 Dependence of the saturation level of n = 4 drifl

kink instability on the current ratio /o//o for allcases.

ions becomes more strongly magnetized and the averageion gyroradius or the inverse of J becomes smaller withthe beam current. Figure 4 shows the relation between
the tilt growth rate and the parameter 5. It is important tonote that the tilt mode is less destabilized due to theexistence of the ion beam even in the high J region.

modification by this instability cannot be observed in
the case without beam current. Thus, it is concluded that
this mode is a kind of the drift kink instability driven by
the beam current [8].

200Nishimura K. et aI., Tilt Stabilization by Energetic Beam Ions in a Field-Reversed Configuration

0.8
()
!Y0.6

F

0.4

remained almost unchanged until the current ratio
reached the critical value R" - 0.03, and then it
gradually decreased as the ratio exceeded R". (iii)'{r'trAlthough the kinetic parameter J which evaluates the ion
FLR effect became large as the current ratio I,olI,
increased, the tilt mode can be stabilized due to the ionbeam effect.

fiolrIl.
-ec

0.2

ReferenceslllUsFig. 4 Tilt growth rates as a function of the kinetic
parameter 5 for the same cases as Fig. 3.

H.Momota, A.Ishida, Y. Kohzaki, G.H. Miley, S.
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4. Summary

Seyler, Phys. Fluids 29,2616 (1986).

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By carrying out the simulation runs with different
values of 15/10 and N5/N1, the following new results werefl990).

[5] L.C. Steinhauer and A. Ishida, Phys. Fluids B 4,

obtained:(i) The localized instability due to the ion beam
current in the FRC plasmas was generated along thefield-null line and saturated in the early phase (0 < tlt^<
2). The deviation of the profile mainly developed in theperpendicular direction of the magnetic field on themidplane. (ii) It was effective against the tilt instability
to increase the current ratio l,ollo. The growth rate

645 (1992\.

[6] J.W. Cobb, T. Tajima and D.C. Barnes, Phys.
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201