Plasma and Fusion Research: Regular Articles

Volume 14, 2402003 (2019)

A Full Wave Simulation on the Density Dependence of a Slow Wave
Excitation in the GAMMA 10/PDX Central Cell withTASK/WF3D∗)
Ryuya IKEZOE, Yushi KUBOTA1) , Makoto ICHIMURA1) , Mafumi HIRATA1) , Shuhei SUMIDA1) ,Seowon JANG1) , Koki IZUMI1) , Atsuto TANAKA1) , Ryo SEKINE1) , Hiroki KAYANO1) ,Yoriko SHIMA1) , Junko KOHAGURA1) , Masayuki YOSHIKAWA1) , Mizuki SAKAMOTO1) ,Yousuke NAKASHIMA1) and Atsushi FUKUYAMA2)
Research Institute for Applied Mechanics, Kyushu University, Kasuga 816-8580, Japan1)
Plasma Research Center, University of Tsukuba, Tsukuba 305-8577, Japan2)
Department of Nuclear Engineering, Kyoto University, Kyoto 615-8540, Japan(Received 30 September 2018 / Accepted 15 November 2018)
Beach heating using a slow Alfvén wave in ion cyclotron range of frequencies would be the first candidate
for ion heating in a DEMO-relevant divertor testing linear plasma device if it is applicable to a high-density
regime. To clarify its availability, the density dependence of a slow wave excitation is investigated using a fullwave simulation with TASK/WF3D code in the GAMMA 10/PDX central cell configuration, where there is anextensive track record of a beach heating. A shielding eﬀect is successfully demonstrated and well understoodunder a three-dimensional configuration in the limit of cold plasma approximation. As the density increases,
excitable left-handed electric field, which contributes to ion cyclotron heating, degrades more and more from acore region, and resultantly the ion absorption region goes outwards with reducing its power. For core densities
above 1020 m−3 , the wave field exists only at a very edge, and ion heating becomes negligible unless the wave
frequency is much increased with a correspondent magnetic field enhancement.
c 2019 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: slow wave, wave excitation, shielding eﬀect, full wave calculation, mirror plasma
DOI: 10.1585/pfr.14.2402003

1. Introduction
Handling of huge particle and heat fluxes to plasmafacing components and PWI raises concerns to the availability of a fusion reactor. Long time scales and highcycle numbers expected in DEMO put a huge step [1, 2].
However, current confinement experiments do not cover
the operational conditions expected in a future fusion reactor. Thus, in order to meet the testing requirements,
various kinds of linear plasma devices have been developed and in use [3, 4]. While the required conditions havebeen partly attained on several devices, no device can oﬀer
all the requirements at the same time. To develop a conceptual design for an ultimate linear device which meetsa real DEMO-compatible environment, one of the mostdiﬃcult tasks would be the achievement of ion temperature of about 100 eV with a significant density of over
1020 m−3 . Existing steady-state high-density linear plasma
devices use a plasma source such as a DC arc discharge,
an ECR discharge and a helicon plasma, all of which use ahigh-pressure fuel gas and do not have an active ion heating mechanism. Therefore, increase of ion temperature isauthor’s e-mail: ikezoe@triam.kyushu-u.ac.jp∗) This article is based on the presentation at the 12th International Conference on Open Magnetic Systems for Plasma Confinement (OS2018).

of great diﬃculty.
Ion cyclotron range of frequencies (ICRF) heating using a slow Alfvén wave excited from a high-field side,which is called as beach heating, would be the first candidate if it is applicable to a high-density linear plasma
because a beach heating provides an eﬃcient ion heating
and is essentially appropriate for a steady operation. In
the GAMMA 10/PDX tandem mirror, hydrogen plasma is
produced and heated solely by ICRF waves [5, 6]. The ion
temperature is several keV in the perpendicular direction
to a magnetic field line and 100 - 400 eV in the parallel
direction for standard operations, where a 6.36 MHz slow
wave is used under a magnetic field strength of about 0.4 T
and electron density of about 2 × 1018 m−3 . Recently, abeach heating of a deuterium helicon plasma with a relatively high density of a few 1019 m−3 has been tried inproto-MPEX [7], and the increase of ion temperature wasobserved although it occurred only at the plasma periphery [8].
In order to assess the applicability of a beach heating
using a slow wave for ion heating of a future high-density
linear plasma device, we have been promoting three studies in GAMMA 10/PDX; they are (i) production of a high-2402003-1

c 2019 The Japan Society of Plasma
Science and Nuclear Fusion Research

Plasma and Fusion Research: Regular Articles

Volume 14, 2402003 (2019)density target plasma in GAMMA 10/PDX and ion heating
experiments in a higher density regime, (ii) development ofwave measurements and (iii) evaluation of wave excitation
with a help of simulations. As for (i), production and sustainment of a double the typical electron density, of about
4 × 1018 m−3 , was achieved by a phase-control ICRF heating [9, 10]. Furthermore, an ECR plasma with the densityof about 1 × 1019 m−3 was produced by using a 28 GHz
gyrotron although it was still transient, and thus, foundto be diﬃcult for ion heating experiments at present. As
for (ii), a two-channel microwave reflectometer for wave
studies was developed, which provides an antenna switching for a multipoint detection in the axial direction and asimultaneous two-point measurement for wavenumber detection [11, 12]. This paper corresponds to (iii), describing the first investigation on the density dependence ofa slow wave excitation in the GAMMA 10/PDX central
cell configuration based on a full wave calculation using
TASK/WF3D code, which is developed in Kyoto University by one of the authors (A. Fukuyama) [13].

2. Simulation Setting for TASK/WF3D CodeTASK/WF3D is a three-dimensional full wave codesolving Maxwell’s equation as a boundary-value problem
with three-dimensional finite element method. It has beensuccessfully applied to the GAMMA 10/PDX configuration along with its two-dimensional version TASK/WF[14, 15]. Basic equations are shown in such as [15]. Absorption power is treated by adopting a collisional cold
plasma model, where the eﬀect of collisions is includedin the dielectric tensor. The absorption occurs near a cyclotron resonance layer by the collisional eﬀects.
The calculation conditions such as a magnetic field
configuration and ICRF antenna configuration, are set tothose of the GAMMA 10/PDX central cell. The magnetic
field strength is about 0.4 T and 2.0 T at the midplane andthe throat, respectively, consisting of an axisymmetric mirror field. The plasma diameter and the axial length between the mirror throats are 0.36 m and 5.60 m, respectively. The grids are separated by 1.5 cm and 2.0 cm inthe axial direction for |z| < 1.5 m and 1.5 m < |z| < 3.0 m,
respectively, where z axis is in the direction of the magnetic field line at the plasma center with the origin at the
midplane. In the radial direction, the grids are separated
by 2.2 cm and 7.0 cm for rcc < 27.0 cm and 27.0 < rcc <34.0 cm, respectively, where rcc is the radius at the midplane and varies along a magnetic field line.
The shape of the density profile used in this study isdetermined based on the typical GAMMA 10/PDX discharges and is fixed with varying a peak density from about
1 × 1017 m−3 to 1 × 1021 m−3 . As an example, the density
profile for 1 × 1018 m−3 is shown in Fig. 1 along with themagnetic field configuration. A tenuous plasma is locatedoutside of the confined plasma. The density of the tenuous

Fig. 1 Electron density profile with a peak density of 1 ×1018 m−3 (a) in x and z plane, and its cross sections at(b) x = 0, (c) z = 0 along with the magnetic field profile, and (d) a schematic of the double-half-turn RF line
current simulating the ICRF antenna.

plasma is uniform and is set to 1 × 1015 m−3 for all the density profiles used in this study. It is confirmed that its value
does not aﬀect much the wave structure inside the plasma.
At the outside of both mirror throat, a high-density plasma
is introduced to simulate a propagation boundary preventing a reflection by increasing a collision frequency.
To closely see a density dependence of wave excitation, single antenna is fed and the other antennas are not in
use in this study. The configuration of all the ICRF antennas is shown in such as [9]. The double-half-turn (DHT)
antenna in the west side of the central cell is simulated
by a DHT line RF current at z = 171.5 cm with diameter of 20.0 cm. The frequency of an induced RF current is
6.36 MHz and the amplitude is 620 A. It is confirmed thatthe excited wave intensity is proportional to the antenna
current in this calculation; no nonlinear plasma reaction is
included.

3. Full Wave Simulation on the Density Dependence of a Slow Wave Excitation
Figure 2 shows the spatial distribution of calculated
ion absorption power for four diﬀerent densities. The dotted curves show the magnetic field lines at the edge and thehalf plasma radius. For the peak density of 3 × 1018 m−3 ,a significant absorption occurs near the ion cyclotron resonance layer in a core region while as the density increases,
the absorption region goes outwards in the radial direction
with weakening its power. In the middle of 1019 m−3 , absorption occurs only at a periphery. For 1 × 1020 m−3 , ion
absorption inside the plasma is greatly reduced and so it
does not contribute much to ion heating of the plasma of

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Fig. 2 Spatial distribution of ion absorption power for the density of (a) 3 × 1018 m−3 , (b) 1 × 1019 m−3 , (c) 3 × 1019 m−3
and (d) 1 × 1020 m−3 . The dotted curves show magnetic
field lines at rcc = 0.09 and 0.18 m, where rcc denotes theradius at the midplane.

ond one in the radial direction.
The detail of the shielding process is shown in Fig. 4
with the variation of the spatial structure of the left-handed
electric field against the increase of the density. Nine structures for the densities between 1×1017 m−3 and 1×1021 m−3
are shown in the order of the density, which is denoted byarrows. Focusing on the plasma region between the resonance around z = 1.1 m and the throat, it can be seen thatthe wave structure is a fundamental eigenmode in the radial
direction and has one node in the axial direction near the
resonance when the density is the lowest, 1 × 1017 m−3 . For
the middle in 1017 m−3 , a higher radial eigenmode emerges.
For the densities in 1018 m−3 , short wavelength structures
become visible near the resonance. It should be noted thatthe wavelength of a slow wave becomes shorter near the
resonance according to the dispersion relation. In addition,
the intensity near the core begins to drop in the middle of1018 m−3 . Inside the half plasma radius, the intensity is significantly reduced when the density reaches 1 × 1019 m−3 .
Then, as the density further increases, the shielding region
extends to the outside. As a result, a weak intensity exists at a very edge of the plasma in 1020 m−3 . For a further dense plasma, a significant shielding makes the electric field negligible except near the antenna.

4. Application of a Higher-Frequency
Wave

Fig. 3 Density dependence of the square of the averaged lefthanded electric field strength normalized by the antenna
current.

interest.
The reduction of the ion absorption power inside theplasma is caused by the variation of the field structure of
the slow wave due to shielding eﬀect. The squared intensity of the left-handed electric field that is averaged over
the region including the ion cyclotron resonance layer inside the half plasma radius is shown in Fig. 3 as a functionof the density. As clearly shown in Fig. 3, the squared intensity significantly decreases as the density increases; thesquared intensity for the density of 1 × 1020 m−3 is lessthan one thousandth of that for 1 × 1018 m−3 . The indentation seen at 3 × 1017 m−3 is caused by the emergence ofa node near core in association with the transition of the
wave structure from a fundamental eigenmode to the sec-

As indicated by the simulations shown above, ashielding eﬀect is essential when a wave is externally excited in a plasma by an RF current induced in an antennasurrounding the plasma whatever the configuration is. The
frequency of a slow wave is limited by ion cyclotron frequency, then of course, an increase in a magnetic fieldstrength reduces the shielding eﬀect. Figure 5 (a) shows
the left-handed electric field profile for 1×1020 m−3 densityand 63.6 MHz wave frequency. Comparing with the left
bottom panel in Fig. 4, the electric field strength at plasma
periphery clearly recovers. The ion absorption powershown in Fig. 5 (b) shows that as the frequency increases,
the absorption power increases almost linearly. About tenfold the power is available for 63.6 MHz although in thatcase the required field strength is quite high; 4 T and 20 T
at the midplane and throat, respectively.

5. Summary
The density dependence of a slow wave excitation in
the GAMMA 10/PDX central cell configuration is investigated based on a full wave calculation using TASK/WF3D
code to assess the applicability of a beach heating for ion
heating of a future high-density linear plasma device. The
shielding eﬀect, which a slow wave severely receives due
to its low frequency, is clearly demonstrated in a threedimensional configuration by solving Maxwell’s equation
as a boundary-value problem, and is well understood in
the limit of cold plasma approximation. The shield-

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Fig. 4 Density dependence of the spatial distribution of left-handed electric field strength. All the panels have the same color scale.

Fig. 5 (a) A left-handed electric field profile for 63.6 MHz RF
current and 1 × 1020 m−3 peak density. (b) The ion absorption power dependence on the wave frequency.

ing eﬀect becomes visible from the middle in 1018 m−3
for the 6.36 MHz wave. When the core density reaches
1 × 1019 m−3 , the left-handed electric field inside the half
plasma radius is significantly reduced, and thus, ion heating at only the periphery is expected. The squared electricfield intensity that is averaged over the resonance region
for the density of 1 × 1020 m−3 is less than one thousandthof that for 1 × 1018 m−3 .
This study based on a full wave calculation indicatesa diﬃculty in the application of a slow wave to ion heatingof a high-density linear plasma. Although increases in awave frequency, antenna current and the number of antennas help to increase a total left-handed electric field power
inside a plasma, it will be still mainly localized in a lowdensity peripheral region. However, eﬀect of ion heating
at plasma periphery on a core plasma, namely a radial energy transport in a high-density regime, is not clarified. In
addition, some favorable instabilities may appear and average the energy distribution in the radial direction, or there
may be some methods to intentionally increase a radial energy transport. Furthermore, kinetic eﬀects may change
the details of the results shown in this paper. Thus, to exactly determine what extent ion heating can be expected,
experimental assessments are still required. Experimentsof a slow wave excitation and its measurement on a higher

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and will be reported in another paper.

Acknowledgments

This work was partly supported by the bidirectional
collaborative research program of the National Institute for Fusion Science, Japan (NIFS14KUGM086 andNIFS17KUGM132).
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