J. Plasma Fusion Res. SERIES, Vol. 8 (2009)

Ray Trace and Fokker-Plank Analyses for Electron
Bernstein Wave Heating and Current Drive in QUEST
H. Idei, N. Takeuchi1) , S. Kubo2) , A. Fukuyama3) , H. Nuga3) , M. Sakaguchi4) , N. Nakamura,
K. Hanada, H. Zushi, K. N. Sato, M. Sakamoto, M. Hasegawa, Y. Takase5) , O. Mitarai6) ,T. Maekawa7) and Y. Kishimoto7)
Research Institute for Applied Mechanics, Kyushu University, Kasuga 816–8580, Japan1)
Ariake National College of Technology, Ohmuta 836–8585, Japan
2)
National Institute for Fusion Science, Toki 509–5292, Japan3)
Department of Nuclear Engineering, Kyoto University, Kyoto 606–8501, Japan
4)
Communication Engineering Dept. Broad Wireless Corporation, Yokohama 220–0073, Japan
5)
Department of Complexity Science and Engineering, University of Tokyo, Kashiwa 277–8561, Japan
6)
Institute of Industrial Science and Technical Research, Tokai University, Kumamoto 862–8652, Japan
7)
Graduate School of Energy Science, Kyoto University, Kyoto 606–8501, Japan(Received: 10 September 2008 / Accepted: 18 December 2008)

The Q-shu University Experiment with Steady State Spherical Tokamak (QUEST) was proposed
at Kyushu University to study plasma-wall interaction phenomena in the steady-state plasma. The
steady-state current drive experiments are planned with Electron Bernstein Wave Heating and Current
Drive (EBWH/CD) in the QUEST. In order to evaluate the plasma current by the EBWH/CD, thepower deposition was ﬁrst evaluated using a ray tracing code TASK/WR. The local electric ﬁelds
calculated in the ray trace were used to the Fokker Plank (TASK/FP) analysis. The driven-current wasestimated from the WR and FP analyses in the O-X-B mode conversion scenario at rather low density(∼ 0.2 × 1019 m−3 ) and temperature (∼ 100eV).

Keywords: Electron Bernstein Wave, Ray Trace Analysis, Fokker-Plank Analysis, RF Heating, Current
Driveincident O-mode wave is converted into an X-mode
wave at the O-cutoﬀ, and propagates as X-mode wave.
The X-mode wave reaches the UHR, and converts to
the B-mode wave. In order to conduct the experiments, the launching mode, i.e the launching polarization, and the launching angle should be controlled
to attain the high conversion eﬃciency. A phasedarray antenna system, which enables us to control thelaunching polarization and angle, was proposed andunder development for the EBWH/CD experimentsin the QUEST. The prototype antenna system wasdesigned[1] and tested at low and high power levels.
In order to study the wave propagation and absorption of the B-wave, the wave trajectory has been
calculated, including the mode conversion processes,with some ray tracing codes[2]. The wave trajectory of the incident beam was calculated using the
TASK/WR ray-tracing code[3] for the QUEST. The
local wave electric ﬁelds were evaluated in the ray
trace calculation, and used for the Fokker-Planck(TASK/FP)[3] analysis. The driven-current proﬁlewas estimated in the FP analysis.
This paper is organized as follows. Section 2 describes the experimental setup using the phased arrayantenna in the QUEST. In Sections 3 and 4, the results of ray trace and FP analyses are shown, and thesummary is ﬁnally given in Section 5.

1. Introduction
Electron Bernstein wave heating and current dive
(EBWH and EBWCD) is one of attractive candidates of heating and current drive method to sustain the steady-state plasma in the spherical tokamak(ST). The Q-shu University Experiment with SteadyState Spherical Tokamak (QUEST) was proposed at
Kyushu University, and the QUEST device has been
constructed. The establishment of steady-state current drive method is a key issue to study plasma-wallinteraction phenomena in the steady-state QUEST
plasma. The steady-state current drive experiments
were planned with the EBWH/CD in the QUEST.
In the ST devices, the plasma frequency may become larger than the electron cyclotron frequency inthe operating density range due to the rather low magnetic ﬁeld, and the electron cyclotron wave can not
penetrate into the plasma due to the cutoﬀ layers. In
the EBWH/CD, some mode conversions from the electron cyclotron (electromagnetic) wave to the electron
Bernstein (electrostatic) wave are required. In the XB mode conversion scenario, a perpendicularly incident X-mode wave meets the R-cutoﬀ, and converts to
the B-mode wave by tunneling of the evanescent layer
between the R-cutoﬀ and the upper hybrid resonance(UHR). In the O-X-B mode conversion, an obliquely
author’s e-mail: idei@triam.kyushu-u.ac.jp1104©2009 by The Japan Society of Plasma
Science and Nuclear Fusion Research

H. Idei et al., Ray Trace and Fokker-Plank Analyses for Electron Bernstein Wave Heating and Current Drive in QUEST

2. Experimental Setup
The density gradient at the conversion into the
B-mode wave is a key parameter to attain high conversion eﬃciency from the X/O-mode wave to the Bmode wave. Even mixture modes of the X/O-mode
are required, depending the density gradient. In addition to the launching mode, the launching angle is also
an essential parameter to achieve the high conversion
eﬃciency. In the X-B mode conversion, the beam is
injected in perpendicular to the magnetic ﬁeld of thetokamak, while the beam is obliquely injected to theplasma with an optimum angle in the O-X-B mode
conversion. The phased array antenna system should
ensure the controllability of the launching polarization
and angle for the EBWH/EBWCD experiments.
In order to control the launching polarization, an
orthomode transducer was designed to mix two orthogonal electric-ﬁeld components. The arbitrary polarized ﬁeld was expressed using the two orthogonalcomponents with a phase diﬀerence. The Lower Hybrid Current Drive (LHCD) system in the previous our
TRIAM-1M tokamak will be used to the EBWH/CDin the QUEST. The operating frequency was 8.2GHz.
In the system, the power of 200 kW was transmitted
to the LHCD antenna using 16 rectangular-waveguide(WR-137) lines. The output polarization state at theorthomode transducer was controlled by the phaseshifters and attenuators in the transmission lines. Figure 1(a) illustrates a conceptual design for the orthomode transducer part. Two ﬁeld components with
diﬀerent intensity and phase at the rectangular waveguide input were mixed at the orthomode transducer,
and were lead to the antenna with the arbitrary elliptical polarization state. The orthomode transducer was
designed to be optimum at the operating frequency of
8.2GHz for the open-end operation. The VSWR at the
open-end operation was less than 1.1 at the 8.2 GHz.
The output waveguide of the orthomode transducer
was square, not rectangular. The directional couplermeasured the intensity and phase of the forward component, also did the component reﬂected from the antenna and the plasma. Figure 1(b) shows the designed
phased-array antenna. The output aperture of the antenna was a 4 × 2 square-waveguide array. Using thephase diﬀerences among the 8 square-waveguides, thelaunching beam was steered to attain high conversion
eﬃciency from the X/O-mode wave to the B-mode
wave. In order to evaluate the radiation ﬁeld patternfrom the antenna, a simple Kirchhoﬀ integral code has
been developed. The Kirchhoﬀ integral at the propagating z position was expressed as follows,� a/2 � a/2
ikdx� dz �(1)Ex,z (x, y, z) =2πy −a/2 −a/2rFig. 1 Conceptual designs of (a): orthomode transducer
part and (b): phased array antenna.

Fig. 2 Beam intensity proﬁle evaluated using the Kirchhoﬀ integral and HFSS codes. The steering point
coordinates are illustrated.

where the coordinates of (x� , z � ) and (x, z) were at
the antenna aperture position(y=0) and at the radiated y position as shown in Fig.2, and a was a sideof the square-waveguide, respectively. Figure 2 shows
the beam intensity proﬁles from the antenna, whichwere evaluated using the Kirchhoﬀ integral code anda 3D full-wave electromagnetic ﬁeld simulation-code,
HFSS[4]. The Kirchhoﬀ integral code result coincided
well with that of the HFSS code. The 1/e-folding
beam radius in the intensity proﬁle was about 60 mm.
The beam with the small beam radius was steeredto the desired launching direction. There were threeside-lobes, but the power levels of the side-lobes were
−6 ∼ −11 dB below the main lobe.

× Ex,z (x� , z � )[exp(−ikr)/r],�(x − x� )2 + y 2 + (z − z � )2 ,=1105

H. Idei et al., Ray Trace and Fokker-Plank Analyses for Electron Bernstein Wave Heating and Current Drive in QUEST

3. Ray Trace Analysis
At ﬁrst the QUEST project will take the focus to
study how to sustain the plasma current in steadystate by the EBWH/CD. The steady-state current
drive in the ST plasma is most important to study theplasma-wall interaction in the steady-state operation.
In the ﬁrst step, it was planned to sustain the rather
low plasma current of 20kA in the lower density region,but in the steady state. In this scenario, the plasma
current will start up by only the RF-wave without
the induction electric ﬁeld[5]. In the lower density
and/or its lower gradient case, the O-X-B scenario
was suitable for the EBWH/CD with a high conversion eﬃciency to the B-wave. The incident O-mode
launched from the low -ﬁeld/density side can be converted to the X-mode if the beam was injected with anappropriate angle oblique to the magnetic ﬁeld. The
optimum refractive index in parallel to the magnetic
opt, at the O-cutoﬀ layer, was given byﬁeld N//
�
Y
opt
N// =
,
(2)
1+Y

Fig. 3 Wave trajectory at the toroidal cross section (z=0)
in the O-X-B conversion scenario. The (x, y, z) coordinates, and the parallel refractive index excitedat the antenna, N//0 , are explained. The origin of
the coordinates is a center of the QUEST device.

where Y was the raio of the electron cyclotron frequency ωce to the wave frequency ω. The wave propagation and absorption of the B-wave after the O-X-B
mode conversion was evaluated using the TASK/WR
ray tracing code. The wave trajectory was calculated
from the following diﬀerential equations,
∂D dk
∂D dt
∂D
dr
=
,
=−
,
=−
,
ds
∂k
ds
∂r
ds
∂ω

(3)

where r(s), k(s) and t(s) were the position/wave vectors and the time of the wave trajectory, respectively.
The wave propagated to satisfy the local dispersion relation, D(ω, k, r) = 0. The dispersion relations D for
O/X/B-modes were described with non-relativistic
hot plasma expressions without cold and electrostatic
approximations. The geometrical coordinates weretaken as a simple tokamak conﬁguration with circular poloidal cross-sections. The major and minor radii
were 0.64m and 0.36m in this study. The proﬁles ofelectron density and temperature, and plasma current
were assumed to be parabolic. The central electron
density and temperature were ne0 = 0.2 × 1019 m−3
and Te0 = 100eV, respectively. The total plasma current was 20kA. The O-cutoﬀ was at the plasma radiusopt
was 0.605. Figr=0.28, and the corresponding N//
ure 3 shows the wave trajectory at the toroidal crosssection, indicating the O-X-B mode conversion processes. The evolutions of the refractive indices N⊥ andN// in perpendicular and parallel to the magnetic ﬁeld
were shown in Fig.4 along the propagation. The inciopt
was converted todent O-mode with the optimum N//
the X-mode at the O-cutoﬀ layer. The X-mode wasconverted to B-mode at UHR again, and propagatedto the high-density side (i.e. to the plasma center).

Fig. 4 Evolutions of refractive indices N⊥ and N// in perpendicular and parallel to the magnetic ﬁeld along
the propagation in the O-X-B conversion scenario.

The absorption of the wave was evaluated from theoptical thickness τ , which was obtained by integrating the partial derivative of the imaginary part of D
along the propagation as follows,�∂ [Im(D)]
ds.
(4)
τ =2
∂s

Figure 5 shows the power deposition proﬁle calculated
from Eq.(4) in this O-X-B conversion scenario. The
wave was perfectly absorbed before it reached to thecold resonance layer by the Doppler-shifted resonancecondition of the B-wave.

4. Fokker-Planck Analysis
The local wave electric ﬁelds were calculated in
the ray trace calculation, and used for the FokkerPlanck (TASK/FP) analysis. In the TASK/FP code,
a quasi-linear RF operator in the velocity space1106

H. Idei et al., Ray Trace and Fokker-Plank Analyses for Electron Bernstein Wave Heating and Current Drive in QUEST

scenario at the lower density and temperature case,depending on the coupled power PCD that was eﬀective to drive the current. The window of the launchingangle to attain the high conversion eﬃciency to the Bmode was narrow. The N// spectrum, which were excited at the antenna system, should be taken into account to evaluate the O-X mode conversion eﬃciency.
The calculated driven-current proﬁle was not consistent with the assumed proﬁle at the ray trace analysis.
In the plasma start-up phase, the high energy(∼5keV)component was observed at the Compact PWI Device experiment[7]. The energetic electrons played an
important role on the formation of closed magnetic
surfaces in the start-up phase[8]. The power deposition and driven-current proﬁles at the EBWH/CD
became broader if there were energetic electrons. The
more consistent ray trace and Fokker-Planck analyses
were required in future, taking the mode conversion
eﬃciency and the high-energy component into consideration.

Fig. 5 Power deposition proﬁle in the O-X-B conversion
scenario at the QUEST.

5. Summay
The ray trace code TASK/WR and the Fokker
Plank code TASK/FP were applied to evaluate theplasma current by the EBWH/CD in the QUEST. In
the WR and FP analyses, the ratio of the total drivencurrent to the coupled power was 0.11A/W in the OX-B mode conversion scenario at the lower density(∼ 0.2×1019 m−3 ) and temperature(∼ 100eV). The dimensionless current drive eﬃciency η EBCD was 0.46.
If there were high energy components, the power deposition and driven-current proﬁles became broader.
The eﬀects of the high-energy component and the N//
spectrum, which were excited at the antenna system,
should be taken into account in the evaluation of thedeposition and driven-current proﬁles.

Fig. 6 Current proﬁle driven in the O-X-B conversion scenario at the QUEST.

6. Acknowledgment

was evaluated from the wave ﬁeld, and the bounceaveraged velocity distribution function was calculated to evaluate the plasma current driven by the
EBWH/CD. The nonlinear collision operator was usedwith the trapped electron eﬀect. Figure 6 shows
the current density proﬁle driven by the EBWH/CDin the O-X-B scenario. The ratio of the total
driven-current Ip to the power PCD was evaluatedas Ip /PCD =0.11 A/W. The dimensionless current
drive eﬃciency η EBCD (≡ 33 ne,20 Ip R/[PCD Te,keV ])was 0.46. Here, ne,20 and Te,keV were the electron density and temperature in units of [1020 m−3 ] and [keV],
respectively. The obtained eﬃciency η EBCD was comparable to the previous W-7AS result of 0.43[6].
In the ﬁrst stage of the QUEST project, the total
power of 400kW was available by 2 antenna systems
at the 8.2GHz operating frequency. The plasma current of 20kA may be generated even in this O-X-B

This work was realized in the industryacademic collaboration between Kyushu University and Broad Wireless Corporation, and wasdone in part, in the NIFS bi-directional collaboration (NIFS06KUTR016/NIFS08KUTR033), andthe RIAM Kyushu University collaboration programs(H18FP-11/H19FP-16).
[1] H. Idei et al.,Proc. of 32nd Int. Conf. on Infrared andmm-Waves, 789 (2007).
[2] T. Maekawa et al., J. of the Phy. Soc. of Japan,48 247(1980).
[3] A. Fukuyama, Fusion Eng. and Design,53, 71 (2001).
[4] http://www.ansoft.com/products/hf/hfss[5] T. Maekawa et al., Nuclear Fusion, 45 1439 (2005).
[6] H. P. Laqua et al., Phys. Rev. Lett.90 075003 (2003).
[7] T. Yoshinaga et al., in preparation to be submitted.
[8] K. Ishiguro et al., in this congress.

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