J. Plasma Fusion Res. SERIES, Vol. 9 (2010)

Three-Dimensional Analysis of ICRF Wave Excitation on the GAMMA 10
Central Cell
Y. YAMAGUCHI, M. ICHIMURA, A. FUKUYAMA1, H. HOJO, S. SATO, T. MURAKAMI,
Y. IMAI, T. YOKOYAMA, Y. MIYAKE, and T. IMAI
Plasma Research Center, University of Tsukuba, Tsukuba, Ibaraki 305-8577, Japan1
Department of Nuclear Engineering, Kyoto University, Kyoto 606-8501, Japan(Received: 5 November 2009 / Accepted: 14 January 2010)

A three-dimensional full-wave code is newly employed to investigate the plasma production by Ion-Cyclotron
Range of Frequency (ICRF) waves on GAMMA 10. The spatial structure of the excited wave fields are calculated
for two types of ICRF antennas (Double Half-Turn and Nagoya Type-III) in the central cell. The plasma production
by ICRF waves depends on the wave excitation in the plasma. Eigenmodes are strongly formed when the boundary
conditions are satisfied. In the experiment, the saturation of the plasma density is observed when Nagoya Type-III
antenna is used for the plasma production. On the other hand, remarkable increase in the plasma density has beenobtained by controlling the phase difference between Double Half-Turn and Nagoya Type-III antennas. The spatial
structure of the wave field is calculated, and the antenna-plasma coupling is evaluated. It is predicted that Double
Half-Turn antenna strongly couples to the eigenmode, and therefore produces the higher density plasmas in the central
cell.
Keywords: GAMMA 10, 3D-FEM, ICRF, Eigenmode, Double Half-Turn antenna, Nagoya Type-III antenna

frequency with controlling the phase difference(∆φ ∼ 0) between them. Achieved plasma density is ~5×1018 m-3, which is more than twice as large as theconventional value.
The wave propagation may change sensitively to thedensity profile and the boundary conditions. In the present
experimental configuration, the central cell plasma iscylindrically bounded, and the characteristic dimension isof the order of the wave length. When the boundary
condition is satisfied, the wave field is strongly excited as
the eigenmode. For the study of the eigenmodes, it isnecessary to take into account the spatial inhomogeneity
and the boundary conditions of the plasmas. In order toinvestigate the ICRF wave excitation, propagation, andabsorption, a three-dimensional full wave code is used. The
central cell is divided into the three-dimensional mesh, andthe rf current straps are arranged in the shape of the DHT
and Type-III antennas. The spatial structure of the wave
field is calculated, and the profile of the absorbed power isevaluated. The calculations are performed for two antennaconfigurations, which are only Type-III and Type-III +DHT (with the phase difference of ∆φ = 0). The waves are
excited with large amplitude and absorbed in the coreplasma region when the DHT + Type-III antenna is used.
On the other hand, the power absorption concentrates near
the antenna in the case of Type-III antenna. It is suggested
that the strong excitation of the waves in the core plasma
region with DHT antenna is effective for high density

1. Introduction

The radio-frequency (rf) power in the Ion-Cyclotron
Range of Frequency (ICRF) is used to provide the plasma
production, ion heating, and stabilization against themagneto-hydrodynamic (MHD) instability in the tandem
mirror experiments [1-3]. In GAMMA 10, strong
ion-cyclotron heating has achieved the highion-temperature above 10 keV [3]. In recent years, one of
the most active topics on GAMMA 10 has been a higher
density plasma production in the central cell [4, 5]. In thepresent condition, the density is relatively low (~ 2×1018
m-3), which restricts the operation window for the study ofthe potential confinement, effective Neutral Beam Injection(NBI), and so on. To improve the tandem mirror
performance, the high density plasma production is a
crucial issue.
In the experiment, Nagoya Type-III (Type-III)
antennas [6] and Double Half-Turn (DHT) antennas are
installed at both ends of the central cell. On the standard
discharge, Type-III antenna is coupled to the fast wave
with the frequency of ~ 10 MHz (ω/ωci ~ 1.6). The excited
fast wave plays the role of plasma production in the central
cell. DHT (6.36 MHz, ω/ωci = 1) is used for thefundamental ion-cyclotron heating. Several methods havebeen adopted to produce the high density plasmas [3-5].
Recently, remarkable increase in the density was observedwith a new antenna configuration, where the neighboring
Type-III and DHT antennas are driven at the samee-mail: yamaguti@prc.tsukuba.ac.jp

23©2010 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Y. Yamaguchi et al., Three-Dimensional Analysis of ICRF Wave Excitation on the GAMMA 10 Central Cell

short pulse (1 ms) gun-produced hydrogen plasma flows
from each end, and is sustained by applying rf power in
combination with hydrogen gas puffing in the central cell.

plasma production.
The manuscript is organized as follows, In Sec. 2 wewill describe the central cell configuration of the GAMMA
10 experiment. In Sec. 3 the experimental results ispresented for high density plasma production. In Sec. 4 wewill introduce the numerical calculation of the wave
excitation, and compare with the experimental results.

3. Experiment of high density plasma production
In the experiments, saturation in the density(~2×1018 m-3) has been observed when RF1 is used (Fig.
2) [3-5]. The cause of the density saturation is now
considered as due to the eigenmode formation of theexcited waves. As the optimum density exists for thewaves to satisfy the boundary condition, the density will
be clamped at the value on which the eigenmode isformed. For effective excitation of the eigenmodes, a newantenna configuration was introduced. The previous
theoretical work suggests that DHT antenna is more
effective for the fast wave excitation [5]. Shown in Fig. 3,
DHT antenna is driven by the frequency of neighboring

2. GAMMA 10 Device

Line density [ m -2 ]GAMMA 10 consists of five mirror cells, which are acentral cell, two quadruple minimum-B anchor cells placed
immediately outside the central cell, and the plug/barriercells at both ends (Fig. 1). The central cell has an
axisymmetric mirror configuration and is 5.6 m in lengthwith the field strength of 0.4 T at the midplane. The mirror
ratio of the central cell is 5. The anchor cell has a
non-axisymmetric minimum-B field configuration and is
1.6 m in length with the field strength of 0.6 T at the
midplane. Two ICRF sources (RF1 and RF2) are currently
used for plasma production, heating and stabilizing against
the Magneto-Hydro-Dynamic (MHD) modes. Two kinds
of ICRF antennas (DHT and Type-III) are installed nearboth mirror throats of the central cell. Type-III antennas are
driven by the RF1 system. The frequencies of RF1 (9.9 and10.3 MHz) are selected as the fundamental ion-cyclotron
resonance frequency of the hydrogen plasma near the
midplane of the anchor cells. The frequency of west TypeIII antenna is slightly higher than the frequency of the east
one in order to avoid the strong interference between two
antennas. RF1 plays two roles, that is, plasma productionin the central cell, and ion heating in the anchor cells forstabilizing against the MHD modes. Fast Alfvén waves
excited in the central cell propagate to the anchor cell. At
the flux tube with an elliptical cross section which islocated between the central and the anchor cells, a part of
the fast waves is converted to the slow waves and heats
ions in the anchor cell [7, 8]. The production of the highbeta plasma in the minimum-B anchors is necessary tosustain the plasma in the central cell [9]. DHT antennas are
driven by RF2 and are used for ion heating in the central
cell. The frequency of RF2 is adjusted to 6.36 MHz, whichis the ion-cyclotron resonance frequency near the midplaneof the central cell. A discharge is started by injecting a

17

4

0

0

50

100

200

250

300

Fig. 2.
RF input power dependence of the line
density. Saturation of the density has been observedwith RF1. The line-density of 4×1017 m-2 corresponds
to the density of 2×1018 m-3 at the coreRF1 (10.3 MHz)

Phase ShifterPic
Co kupil

NagoyaType-IIIA/D Converter

t

s
We

DHT
a

sm

Pla

Z
Anchor Cell -East(Minimum-B)

Anchor Cell -West(Minimum-B)

Central Cell
(Axisymmetric Mirror)

al

ntr

Ce

Type-III(RF1, 9.9 MHz)

1
Mirror Throat

DHT (RF2, 6.36 MHz)

-2

ac
ll V

s
Ve

Ce

Type-III(RF1, 10.3 MHz)

Fig. 3.
Experimental arrangement for controlling
the phase difference between west DHT and west
Type-III antenna in the central cell. The phasedifference is monitored by use of pickup coil locatedclose to the antennas.

Anchor Midplane

-4

l

se

m

uu

2

0
-6

150

RF Input power [ kW ]3

Magnetic Field [ T ]5 x 10

0
Z [m]

2

4

6

Fig. 1.
Axial profile of the magnetic field strength,
locations of the ICRF antennas.

24

Y. Yamaguchi et al., Three-Dimensional Analysis of ICRF Wave Excitation on the GAMMA 10 Central Cell

400
# 201187-2

Line Density [cm ]

13

8 10

13

Type-III East
(9.9 MHz)

13

4 10

200

Type-III West(10.3 MHz)

13

2 10

0

300

Line Density

6 10

100

DHT West(10.3 MHz)

0

50

0
150

100

RF Input Power [kW]Type-III antenna. RF1 signal (10.3 MHz) is split in two,and a part is newly connected to the amplifier of DHT
antenna via a phase shifter. The experiment was
performed to control the phase difference between two
antennas. The input power is evaluated by use of thebi-directional coupler installed outlet of the rf power
amplifier. The loss due to a mismatch of the impedance istaken into account to derive the input power. In order tomeasure the antenna phase, rf current on the antennas is
detected with pickup coils, which are installed close to
the antenna. In Figure 4, temporal evolution of theline-integrated density is shown.
In the experiments, the phase difference was changed
for each discharge. Whole power input for plasma
production was fixed to about 240 kW. In Figure 5, theelectron density at the plasma core is plotted versus thephase difference. The core plasma density is estimated by
the interferometer, which has the multiple measurement
channels in the radial direction. As is clearly indicated in
the figure, an optimum phase difference exists for theplasma production in the present density range.
Sometimes plasmas can not be sustained when the phasedifferences largely away from the optimum one. Figure 6
gives the rf power dependence of the plasma density. The
phase difference between two antennas was set to the
optimum one. The density considerably increases withapplied rf power. Together with the rf power, gas fuellingrate is also increased. In this experiment, the density isenhanced up to ~5×1018 m-3 (corresponding line density is~8×1013 cm-2), which is more than twice as large as theconventional value. No saturation in the density has been
observed. And more increase in the density can be
expected with increasing the rf power.

Time [ms]Fig. 4 The temporal evolution of the line-integrated
density on typical discharge is shown together with theapplied rf power for each antenna.
12-3Electron Density [cm ]

4 10

12

RF1E (9.9 MHz): Type-III East
RF1W (10.3 MHz): Type-III West
RF2W (10.3 MHz) : DHT West

3 10

12

2 10

Phase Difference:(DHT West) - (Type-III West)
12

1 10

Time : 130 - 149 ms
Pin ~ 240 kW

0
-1

-0.5

#201173#201177#201178#201180#20118100.5

1Phase Difference / πFig. 5.
Antenna phase dependence of the density at
the core region is shown. The phase-shift of DHT from
Type-III is measured. The amount of power input was
fixed to about 240 kW. The density profile wasestimated from 130 to 149 ms at 5ms intervals. It isclearly seen that an optimum phase difference exists
for the plasma production.

4. Three-Dimensional Analysis of the Wave
Excitation

12

6 10

While the phase controlled antennas have complicatedrf current path around the inhomogeneous plasma, it isimportant to carry out detailed wave analysis in the precise
central cell geometry. In this work, the effect of the antenna
structure to the wave excitation is computationallyevaluated by use of a three-dimensional full wave code
(PAF/WF3: Plasma Analysis by FEM / Wave analysis by
FEM 3D). This code is developed by one of the authors;
Prof. A. Fukuyama in Kyoto University. This code solves
the Maxwell's equation for the wave electric-field as aboundary-value problem using the finite element method(FEM). In the model, it is assumed that the cold andinhomogeneous plasma surrounded by the conducting
walls. The power absorption with the collisional damping
is described by introducing effective collisions in the
dielectric tensor. Figure 7 shows the generated finite-3

Electron Density [cm ]

12

5 10

#201183 - #201192#209005 - #209021
Averaged : 130 - 150 ms

12Type-III
+
DHT

4 10

12

3 10

12

2 10

Type-III

12

1 10

0
0

100

200

300

400

500

Input Power [kW]Fig. 6.
RF power dependence of the plasma density.
The phase difference between two antennas was fixed
to the optimum one.

25

Y. Yamaguchi et al., Three-Dimensional Analysis of ICRF Wave Excitation on the GAMMA 10 Central Cell(a) DHT + Type-III Antenna
φ1

0.5m

0.5

φ2

z = -3m

Z =0m

z = 3mY [m]Z
φ2 − 180deg.

-0.5m

ν/ω = 0.01

Z =0m

ν/ω = 0.01

0.18

1500

0.0

0

-0.18
Double Half TurnXφ

1

Nagoya Type-III X

Z = 1.715 m-1500Electric Field Vector-0.5-0.5 -0.18 0.0 0.18 0.5
X [m]

φ2

∆φ = φ1 − φ2

Z = 2.317 m

ν/ω = 0.01

3000

Er [ V/m]

X

-3000
-0.5

-0.18 0.0 0.18 0.5
X [m](b) Type-III Antenna
Y

0.5

Z =0m

ν/ω = 0.01

Z =0m

ν/ω = 0.01

500

0.18

φ2 − 180deg.

0

0.0

-500

-0.18

Fig. 7. Finite element mesh of the central cell.
Configuration of DHT and Type-III antennas areshown.

1000

Er [ V/m]

Y-1000
Electric Field Vector-0.5-0.5 -0.18 0.0 0.18
X [m]

-1500
0.5

-0.5

-0.18 0.0 0.18 0.5
X [m]Magnetic Field Profile
Y [m ]

0.5

Fig. 9. The electric-field vector on each node, and the
profile of the radial component of the wave electric field.
It is clearly shown that DHT + Type-III (the phasedifference between two antennas is ∆φ = 0) excites m =+1 mode. In the case of Type-III antenna, m = +3 modeis dominantly excited at the core plasma region.

0.18
0.0
-0.18
-0.5
0.4 0.8 1.2 1.6 2.0 2.4 T

Plasma Density Profile
Y [m ]

0.5
0.18
0.0
-0.18
-0.5

-2.8

0.0

the phase of the rf current from 0 to 360 degrees withkeeping the phase difference between DHT and Type-III tobe ∆φ = 0, we remarked the time-variation of the field
profiles. Then, it is found that the electric-field vector
rotates right-handed (co-rotation with electron cyclotron
motion) at the core plasma region (r < 0.18m), and rotatesleft handed at the peripheral plasma region. In addition, we
confirmed that the global structure of the electric fields
rotates right-handed, which means the sign of theazimuthal mode number is positive [10]. The excitedwaves are suggested to be the fast wave. It is clearly seenthat the antenna strongly couples to m = +1 mode at thecore plasma region in case of DHT + Type-III (Fig. 9(a)).
On the other hand, the radial electric field excited by
Type-III antenna indicates the appearance of the m = +3
mode (Fig. 9(b)). The amplitude of the excitedelectric-field at the core plasma region is small ascompared with the case of DHT + Type-III antenna. Figure
10 gives the contour plots of radial electric-field andabsorbed power on the y-z plane for two cases. For thecase of DHT + Type-III antenna, which is shown in Fig
10(a), it seems that the antenna strongly couples to theaxial eigenmode and the power is dumped on the coreplasma region. In the case of Type-III (Fig.10(b)), however,
the amplitude of the coupled wave is small in the coreplasma, and the power absorption concentrates around theantenna. From the calculation results, it is suggested thatthe strong excitation of the waves in the core plasma regionis effective for high density plasma production.

2.8

Z [m]
0.5 1.5 2.5 3.5 4.5
x 10 18 m -3

Fig. 8. The parameters used in the calculations.
Contour plots of the external magnetic field and theplasma density on y-z space.
element mesh. The element size is optimized in order tohave sufficiently large numbers of spatial elements within
the core plasma region. The number of finite elements
within the system was 117270 in this calculation. DHT and
Type-III antennas are approximated by external rf current
straps. To take into account the width of the antennas, eachantenna is composed of four current straps. In the present
calculations, rf current on each strap is assumed to be same.
The external magnetic field and the plasma density can be
non uniform in all directions. Here we used the magnetic
field of the central cell and the measured profile of thedensity (Fig.8).
In order to identify the mode of the excited wave,
spatial structure of the electric-field is calculated. The peakdensity is set to 5×1018 m-3. The rf power radiated from theantenna is fixed to be 100 kW. For the case of DHT +Type-III antennas, the phase difference between two
antennas is set to ∆φ = 0. Figure 9 shows the electric-field
vector on each node of the finite-element mesh on the x-yplane at the mid-plane (z = 0). And the contour plot of theradial electric-field is given in same figure. By changing

26

Y. Yamaguchi et al., Three-Dimensional Analysis of ICRF Wave Excitation on the GAMMA 10 Central Cell

0
-4000

5

0.18
0.0
0.18

4000

0 0

0

en

D
ma

s

Pla

-3

18 m

]

0

x1
ty [

si

ν/ω = 0.01

0.5 1

1.5 2

2.5 3

(b) DHT Antenna
Antenna Loading
Resistance [Ω]
14
6
-2
360

10

4

90
0 0

0

2

2

The antenna-plasma coupling is computationallyevaluated to compare with the experimental result of theantenna phase control between DHT and Type-III
antennas. The calculation is performed by changing thephase difference (0 ~ 360deg.) and the plasma density(0.2×1018 ~ 10×1018 m-3). Figure 11 shows the loading
resistance of each antenna as the functions of the phasedifference and the plasma density. From the viewpoint of
the antenna-phase dependence, the loading resistances of
two antennas have different characteristics. As is clearlyseen in Fig. 11(a), the loading resistance of Type-III
antenna depends strongly on the phase difference. The
optimum phase difference is shown to be ∆φ ∼ −74 deg.,
which is near the experimental observation. On the other
hand, the loading resistance of DHT antenna has lessdependence on the phase difference as compared with
Type-III antenna. As for the density dependence, several
sharp peaks of the loading resistance can be seen on eachfigure. When the density is changed and the eigenmode isstrongly excited, the loading resistance indicates the peakvalue. In comparison to Type-III antenna, the loading
peaks of DHT antenna have much higher values. This
result indicates that DHT antenna couples to theeigenmodes stronger than Type-III antenna. Therefore, inthe phase controlled experiment with DHT + Type-III
antenna, it is suggested that DHT antenna takes main role

4

sity

en

D
ma

s

Pla

g.]

[de

ce

-2

-3
18 m

6

180

ren

Fig. 10. The contour plot of the radial electric field andabsorbed power on y-z plane for the cases of theType-III antenna and DHT + Type-III antenna (thephase difference between two antennas is ∆φ = 0).

8

270

iffe

eD

as

Ph

2.8

Er [ V/m ]

Y [m]
Y [m]

2

g.]

0.5
0.5

0.0
Z [m]

90

de
e[

4000
2500
1000
-500
8000

2.8

4

-2 -1.5 -1 -0.5 0

ν/ω = 0.01

0.18
0.0
0.18

0.5

0

6

180

nc

(b) Type-III Antenna
0.5

2.8

8

270

re
ffe

0.0
Z [m]

10

Di

2.8

2
0
-2
360

se

0.18
0.0
0.18

4

P [ W/m3 ]10 x10

a
Ph

Y [m]

0.5
0.5

Antenna Loading
Resistance [Ω]P [ W/m3 ]

Y [m]

4000

0.18
0.0
0.18

0.5

(a) Type-III Antennaν/ω = 0.01

Er [ V/m ](a) DHT + Type-III
0.5

6

]

0
[ x1

ν/ω = 0.01
8

10

12

14

Fig. 11. The antenna loading resistance as the functionof the phase difference and the plasma density.
for the density increase.
In the calculation of Fig. 11, we reduced the numberof finite elements from 117270 to 18000 because of thelimitation of the calculation time. Due to rather coarse
FEM grid, the field structure which has the small wave
length cannot be described. If the calculation is performed
with the large element-number of 117270, detailed
structure of the excited waves can be described as shown in
Fig. 10(a). In Figure 10(a), the excited wave has theaveraged wave length of ~2.4m on Z axis at the coreplasma density of 5×1018 m-3. However, in case of Fig. 11,
the excited wave has the averaged wave length of ~5.6m
on Z axis around the loading peak at ~4.4×1018 m-3. Thisdifference is caused by the reduction of spatial resolutionof the finite element mesh. If the number of finite elements
is large enough, the eigenmodes which have the small
wave length will appear. As the result, the number of theloading peaks may increase in the density range on Fig.11.

5. Summary
We have presented the first application of the27

Y. Yamaguchi et al., Three-Dimensional Analysis of ICRF Wave Excitation on the GAMMA 10 Central Cell

three-dimensional wave calculation code to the GAMMA
10 experiment. The remarkable increase in the plasma
density is experimentally observed by use of new antennaconfiguration, where the phase difference is controlled
between neighboring DHT and Type-III antennas. The
spatial structure of the wave field is calculated, and thedegree of antenna-plasma coupling is evaluated. It is
predicted that DHT antenna is effective to produce thehigher density plasmas in the central cell.

Acknowledgment

The authors acknowledge the support of GAMMA 10group. This work is partly supported by the bi-directional
collaborative research program of National Institute for
Fusion Science, Japan (NIFS09KUGM042).

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Y.Yamaguchi et al., Plasma Physics and Controlled Fusion,
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[4]

28