Plasma and Fusion Research: Regular Articles

Volume 14, 3401134 (2019)

Simulation of Electromagnetic Wave Propagation in a
Magnetospheric Plasma∗)Takahiro MORI1) , Masaki NISHIURA1,2) , Zensho YOSHIDA1) , Naoki KENMOCHI1) ,Shotaro KATSURA1) , Kaori NAKAMURA1) , Yuuki YOKOTA1) ,Toru I. TSUJIMURA2) and Shin KUBO2)
1)

Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba 277-8561, Japan
2)
National Institute for Fusion Science, Toki 509-5292, Japan(Received 8 January 2019 / Accepted 8 May 2019)

The Ring Trap 1 (RT-1) device creates a laboratory magnetosphere, that is motivated by the Jovian magnetosphere, which contains self-organized, high-beta plasmas. In the RT-1 plasmas, the density limit for 8.2 GHz electron cyclotron heating (ECH) occurs at the electron densities 8.0×1017 m−3 and for 2.45 GHz ECH, 1.6×1017 m−3 .
We have used full-wave simulations to study the propagation and absorption of electromagnetic waves in the RT-1
plasmas in an eﬀort to understand the observed density limits as well as the over-dense state in which the actual
density exceeds the cutoﬀ density of the magnetospheric plasma. The simulation results cannot explain the experimentally observed over-dense states and density limits in the view of the power absorption. We discuss possible
reasons for the gap between the experiments and the simulation results.
c 2019 The Japan Society of Plasma Science and Nuclear Fusion ResearchKeywords: magnetospheric plasma, ECH, electron cyclotron wave, electric field, over dense plasma, electromagnetic wave propagation
DOI: 10.1585/pfr.14.3401134

1. Introduction
A dipole field is a basic structure in nature for plasma
confinement, and it is an appropriate subject of study for
extracting universal physical principles. The Ring Trap 1(RT-1) device is motivated by the Jovian magnetosphere,
and it enables us to study both magnetospheric plasma
physics and advanced fusion. The RT-1 device has previously demonstrated plasma confinement in a dipole fieldusing a levitated superconducting ring magnet [1], which
produces the dipole fields necessary for plasma confinement. The plasmas are produced by electron cyclotron(EC) heating using either or both 2.45 GHz and 8.2 GHz
electromagnetic (EM) waves. The plasma so produced hasa peaked density profile, which is a self-organized structurethat is also observed in naturally formed planetary magnetospheres. The measured density in the RT-1 device isshown below in Fig. 2. The aim of the present work is toextend the operation regime of the device.
For 8.2 GHz EM wave, we found that the line averaged electron density is limited to the range below the
cutoﬀ density. However, we found experimentally that a
2.45 GHz EM wave usually overcomes the 2.45 GHz cutoﬀ
density, even when the same interferometer chord is used
to measure the density. Understanding the propagation
and absorption of the EM waves is thus essential to disauthor’s e-mail: mori.takahiro18@ae.k.u-tokyo.ac.jp∗) This article is based on the presentation at the 27th International Toki
Conference (ITC27) & the 13th Asia Pacific Plasma Theory Conference(APPTC2018).

till the underlying physics from the apparent density limits
in the 8.2 and 2.45 GHz EC heating schemes. It is unclear
whether or not these phenomena are caused by the inherent magnetic structure and/or the inhomogeneous medium
produced by the dipolle confinement system.
The purpose of the present work is to understand thedensity limit in a dipole plasma confinement (due to overdense heating, thermal particle transport, etc.). As a first
approach, we studied the propagation and absorption ofelectromagnetic waves numerically for the inhomogeneous
region of a dipole plasma confinement system that contains
a self-organized plasma.

2. Propagation and Absorption of
EM Waves in an Inhomogeneous
MediumFigure 1 shows the EC heating system and main diagnostics of the RT-1 device. The inside of the vacuum
vessel is evacuated to 10−6 Pa. The working gas (hydrogen or helium) is introduced into the vacuum vessel up toa range of more than 10−4 Pa, and we then use electron cyclotron resonance heating (ECRH) to produce and sustain
the plasmas.
For the ECRH, we introduce an EM wave from an
8.2 GHz klystron (Toshiba, nominal power 100 kW) into
the vacuum vessel through the transmission lines L#1 and#2 (up to 25 kW/line). The EM wave is launched fromthe upper ports obliquely toward the fundamental ECR

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

Plasma and Fusion Research: Regular Articles

Volume 14, 3401134 (2019)from Maxwell equations:
1 ∂D
,
c ∂t
1 ∂B
∇×E=−
,
c ∂t

∇×B=

Fig. 1 (a) The equatorial plane, and (b)a poloidal cross-sectionof the RT-1 device. The transmission lines L#1 and L#2
are used for 8.2 GHz EC heating and L#3 is used at
2.45 GHz.

(1)(2)

where B is the magnetic flux density, D =  · E is the electric flux density and c is the speed of light. After Fouriertransforming Eqs. (1) and (2) in time and space, we combine them to obtain the homogeneous-plasma wave equation with n = kc
ω.
n × (n × E) +  · E = 0.

(3)Here, ω is the angular frequency and k is the wavenumber
vector. The cold-plasma dielectric tensor is given by [3].
⎞⎛⎞⎛⎜⎜⎜ S −iD 0 ⎟⎟⎟ ⎜⎜⎜ E x ⎟⎟⎟⎟⎜⎟⎜S
0 ⎟⎟⎟⎟⎟ ⎜⎜⎜⎜⎜ Ey ⎟⎟⎟⎟⎟ .
 · E = ⎜⎜⎜⎜⎜ iD
(4)
⎠
⎝
⎠⎝
Ez
0
0
P
where
1
1
(R + L), D = (R − L),
2
2

ω2ps
R≡1−
,
ω (ω + Ω s )
s
S =

L≡1−
P≡1−

Fig. 2 Density limit observed in the RT-1 for 8.2 GHz ECH,
where the working gas is hydrogen or helium.

ω2ps

s

ω (ω − Ω s ) ω2pss

ω2ps ≡
layer, which lies across the levitated magnet. The EM
wave propagates via the edge region of the plasma to access the absorption layer. In the RT-1 experiment, wemeasured the line integrated density using interferometer
1 (IF 1). We determined the 1.6 m chord length of IF 1
from the two points at which the last closed flux surface
crosses the chord. The line integrated density divided by
the chord length gives the line averaged density. To determine the density limit for 8.2 GHz ECH, we varied theinjected power from 0 to 40 kW. The pressure of the hydrogen fill gas ranged from 1 mPa to 80 mPa and for helium
from 3 mPa to 28 mPa. For 8.2 GHz ECH, the electron
density limit is 8.0 × 1017 m−3 [2], as shown in Fig. 2. According to the results of the interferometric density reconstruction, the core density reached a so-called over-dense
state. We also introduced an EM wave from a 2.45 GHz
magnetron (nominal power 20 kW) into the vacuum vessel from the equatorial port through transmission line L#3.
The EM wave propagates perpendicular to the magnetic
fields. In this case, the fundamental ECR layer lies acrossthe plasma in the confinement region, and the EM wave has
to cross the peak-density region to reach the ECR layer.
To simulate the propagation of the EM waves, we start



ω2

,

,

4πn s q2s
qs B.
, Ω s = ωcs ≡ms
ms c

m s is the and n s is the number density of particles of species
s with charge q s , and Ω s = ωcs is the cyclotron or gyrofrequency for particles of type s.
We have simulated the propagation of 2.45 GHz and
8.2 GHz EM waves in plasmas using a finite element
method, full-wave calculation with COMSOL software.
The specific solver we used is the direct solver MUMPS.
The EC waves for 8.2 GHz are injected from the diagonal port and the 2.45 GHz waves from the horizontal port.
The mesh size is less than or equal to one twentieth of the
wavelength (0.6 ∼ 6 mm at 2.45 GHz, 0.18 ∼ 1.8 mm at
8.2 GHz). The center stack is defined to be an absorption
boundary, and the other boundaries are treated as perfectelectrical conductors.
Figure 3 (a) shows the magnetic field lines producedby the superconducting ring magnet (115 A, 2160 turns)
and the levitation magnet (430 A, 68 turns). We assumed
the electron density profile given by Eq. (5) and shown in
Fig. 3 (b):3401134-2

ne (r, z) = n0
⎞
⎛
2
−b ⎟
⎜⎜⎜
B(r,
z)
,
0)
ψ(r,
z)
−
ψ(r
⎟⎟⎟ (5)max
× exp ⎜⎜⎝−a⎟.
ψ(1, 0)B0 (r, z) ⎠Plasma and Fusion Research: Regular Articles

Volume 14, 3401134 (2019)

Fig. 3 (a) Magnetic flux surfaces and (b) electron density profile
to calculate the propagation of an EM wave in plasmas.

Here, B(r, z) is the magnetic field at position (r, z), B0 (r, z)
is the magnetic field at (z = 0) which is on the flux fanction ψ at (r, z). The quantities (r0 , a, b, rmax ) are fitting
parameters. The fitting parameters are determined as thepeak density n0 = 3.5 × 1017 m−3 , other parameters are
a = 9.33, b = 1.084, rmax = 0.547 m by the results from interferometer 1, 2, and 3. The white line in the figure shows
the separatrix.

Fig. 4 Density limit observed in the RT-1 at 2.45 GHz ECH. The
working gas is helium.

3. Simulation Results
3.1

Density dependence

We investigated the experimental density limit for2.45 GHz ECH, with the result shown in Fig. 4. For
2.45 GHz ECH, we varied the injection power from 0
to 18 kW. The gas pressure of the helium fill ranged
from 0.28 mPa to 3.2 mPa. The density limit appeared at
ne = 1.6 × 1017 m−3 , which exceeds the cutoﬀ density for2.45 GHz. Compared with the case for 8.2 GHz ECH in
Fig. 2, there is a substantial diﬀerence between the cutoﬀ
density and the density limit for 2.45 GHz ECH. For theline averaged density of ne = 1.6 × 1017 m−3 , the peak density was n0 = 3.5 × 1017 m−3 . In the following simulation,
we changed n0 while keeping the other profile parameters
the same.
The EM wave propagates and reflects at R = 0 for the
X-mode or at P = 0 for the O-mode, as shown in Figs. 5 (a)and 5 (b). When an EC wave is launched from the equatorial port, the propagation and reflection for both the O- andX-modes is similar because the P = 0 layer is close to thatfor R = 0. The vacuum vessel also influences the reflected
EM waves.
We simulated the dependence on the electron density for three typical cases :below, equal to, and above
the cutoﬀ density. The results are shown in Fig. 6. The
2.45 GHz EM wave can reach the ECR layer directly whenn0 = 3.5 × 1016 m−3 , as shown in Fig. 6 (a). When n0 exceeds 7.0 × 1016 m−3 , the EM wave is divided into twopaths by the cutoﬀ layer, which makes it diﬃcult for the
EM wave to reach ECR layer. However, multiple reflections from the vacuum vessel and the three dimensional

Fig. 5 2.45 GHz EM wave propagation when the plasma density
is close to the limit shown in Fig. 4. The red line is R = 0,
the pink line is P = 0, the green line is L = 0, the yellow
line is S = 0 and the gray line is ECR layer.

Fig. 6 Propagation of 2.45 GHz O mode EM wave. (a) n0 =3.5 × 1016 m−3 , (b) n0 = 7.0 × 1016 m−3 , (c) n0 = 2.5 ×
1017 m−3 .

geometory actually make this possible.

3.2 Power absorption of EM waves for2.45 GHz
We calculated the wave propagation based on thecold plasma approximation introduced in the previous section. We used these calculated wave propagation is re-3401134-3

Plasma and Fusion Research: Regular Articles

Volume 14, 3401134 (2019)

Fig. 7 The spatial profile of power absorption from a 2.45 GHz
EM wave, in units of W/m3 , when n0 = 3.5 × 1016 m−3 .

sults to evaluate the absorption of the EM waves at theresonance point. According to the hot-plasma model, thepower absorption per unit volume in a plasma is given by
Eq. (6) [4, 5].
⎛⎞ω2pe⎜⎜⎜ (ω − ωce )2 ⎟⎟⎟ 0
π2⎜⎟⎠
exp ⎝−
Pabs =
E x − iEy .
22
2kz vT e 22
2kz vT e(6)
Here, ω pe is the electron plasma frequency, and vT e =√2kB T e /me is the electron thermal velocity, where kB is
Boltzmann’s constant. The z-direction is parallel to themagnetic field, and k is the wavenumber vector, which isparallel to the angle of incidence of the EM waves, and kz
is the z-component of that vector. The quantities E x and
Ey are the x- and y-components of the wave electric field.
We calculated the local values Pabs on the mesh points, andwe obtained the total power absorbed Ptotal
abs by integratingover the entire calculation space. The eﬃciency of power
absorption is defined as the ratio of Ptotal
abs to Pin , where Pinis the incident EM wave energy. We calculated the profileof the power absorption using n0 = 3.5 × 1016 m−3 , assuming a spatially uniform electron temperature T e =10 eV. As
shown in Fig. 7, the absorption concides with the fundamental ECR layer. (In this case, we ignored the absorption of the second harmonics.) The eﬃciency of power
absorption reaches a maximum value of ∼ 70% at n0 =2.5 × 1016 m−3 , and it vanishes at n0 = 4.2 × 1016 m−3 ,as shown in Fig. 8, even though the EM wave reaches the
ECR layer.

3.3

Fig. 8 The absorption eﬃciency of 2.45 GHz O-mode EM wave
as a function of the line averaged density for T e = 10 eV.

Fig. 9 The propagation of 8.2 GHz, X-mode EM waves for (a)n0 = 3.5 × 1017 m−3 , (b) n0 = 1.0 × 1018 m−3 , (c) n0 =1.0 × 1019 m−3 .

Power absorption from 8.2 GHz EM
waves

We also simulated the propagation of 8.2 GHz EC
waves in the RT-1 device. Figure 9 shows three casesof propagating 8.2 GHz EM waves. When n0 = 3.5 ×1017 m−3 , the 8.2 GHz EM wave reaches the ECR layer.
For n0 > 1.0 × 1018 m−3 , however, the EM wave is interrupted by the cutoﬀ layer before reaching the ECR layer, as
shown in Fig. 9 (b). When n0 = 1.0 × 1019 m−3 , the cutoﬀ
layer moves toward the outer wall of thevessel, and the EM
wave does not reach the ECR layer, as shown in Fig. 9 (c).
The ECR layer attains a power absorption per unit volumeof ∼ 1MW/m3 for both 2.45 and 8.2 GHz EM waves.

Fig. 10 The absorption eﬃciency of an 8.2 GHz X-mode EM
wave as a function of the line averaged density with
T e = 10 keV.

The eﬃciency of power absorption drops to zero atn0 = 9.0 × 1017 m−3 , which is close to the cutoﬀ density for 8.2 GHz which is also the density limit, as shownin Fig. 10. To produce over-dense plasma at 8.2 GHz, we

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Plasma and Fusion Research: Regular Articles

Volume 14, 3401134 (2019)must therefore propose another method of introducing the
EM wave.

4. Discussion
In order to clarify the limitations of our present modeling eﬀorts, we discuss here the simplifications and/or assumptions we used in the present model that should be improved in future studies. This discussions is intended to fill
the gaps between the experiments and the simulations.
We calculated two-dimensional EM wave propagationin order to understand the density limits observed in the
RT-1 magnetospheric plasma experiments. We reduced thespatial dimensions from three to two. Thus, we are unable to calculate the spread of the propagating waves in thetoroidal direction. Waves hitting a wall are scattered partially into the same plane but mostly away in the toroidal
direction. Therefore, our model treats the boundary at thecenter wall as an absorbing boundary, while the other, surrounding, walls are assumed to be perfect electrical conductors. Multiple reflections from the stainless-steel wall
surfaces severely influence the distribution of the wave,
which aﬀects the quantitative analysis. Thus, it is possible that eﬀects not included in this calculation may cause
at least some disagreement between the simulations ans theexperiments.
An Electron Bernstein Wave (EBW) is a longitudinally polarized electrostatic mode that propagates nearly
perpendicular to the magnetic field. To excite an EBW,
mode conversion from electromagnetic to electrostatic
must occur in the plasma [6, 7]. For example, the X-mode
wave can access to an upper-hybrid resonance layer where
it can be converted into an EBW. Three scenarios for exciting EBWs are well known [8].
1. Perpendicular injection of X-waves from the highfield side (slow X-mode - EBW (SX-B) conversion)
2. Perpendicular injection of X-waves from the low-fieldside (fast X-mode - EBW (FX-B) conversion)
3. Oblique injection of O waves from the low-field side
(O-mode - slow X-mode -EBW (O-X-B) conversion)
Our experiments achieved a density exceededing the cutoﬀ
density, but we have not yet investigated EBW heating orother eﬀects mentioned above for the case in which weused the 2.45 GHz X-mode injection from the equatorial
port. Further studies are thus required to understand theresult shown in Fig. 8.
The power absorption in our simulation is not a selfconsistent solution, because we solved the propagation of
the EM waves and the power absorption independently.
In addition, Eq. (6) for the power absorption is calculated
from the second approximation for the hot-plasma dielectric tensor.
The locations of the ECR layers are diﬀerent for2.45 GHz and 8.2 GHz. The 8.2 GHz ECR layer lies

across the levitated superconducting magnet. From theviewpoint of electron heating, this situation is disadvantageous compared with 2.45 GHz-ECH. However, the density limit occurs at 8.0 × 1017 m−3 for the 8.2 GHz-ECH.
The actual density limitachieved is still higher than that at
1.6 × 1017 m−3 for the 2.45 GHz-ECH, and it is dominatedby the cutoﬀ density rather than by the location of the ECR
layer. The over-dense state we observed for 2.45 GHzECH cannot be explained by the heating location, because
the absorption eﬃciency at both frequencies is as high as
60 - 70%.

5. Conclusion
We have studied the propagation of EM waves in plasmas generated in the RT-1 device both through experiments
and through full-wave simulations of the production andheating of a magnetospheric plasma. We solved the propagation ofthe EM waves using the cold plasma approximation, and we evaluated the power absorption. Density limits occured in the magnetospheric plasma experiments forboth 8.2 GHz and 2.45 GHz ECH. The actual density limits
observed in the experiments exceeded the cutoﬀ densities
at both frequencies. The simulated EM waves are absorbed
below the cutoﬀ densities at both frequencies, because thedensity is inhomogeneous and its gradient deflects the EM
waves. The present simulation cannot explain the experimentally observed over-dense states and density limits inthe view of the power absorption. The discrepancy may be
due to multiple reflections of the EM waves, which makes
the ECR layer accessible from the high-field side. Anotherpossibilitiy may be mode conversion to an EBW [8]. In a
dipole plasma confinement, the peak of the density profile
exceeds the cutoﬀ density at both frequencies. This feature
may also indicate the generation of self-organized plasmas
with up-hill diﬀusion [1]. Further study is thus needed tounderstand the density limits in the magnetospheric plasmas produced in the RT-1 device.

Acknowledgments
Authors would like to thank Dr. H. Saitoh of the
University of Tokyo for discussion. This work was
supported by the NIFS Collaboration research program(NIFS15KOAH034, NIFS19KBAR026), and JSPS KAKENHI Grant No. 17H01177.
[1] Z. Yoshida et al., Phys. Plasmas 17, 112507 (2010).
[2] M. Nishiura et al., Nucl. Fusion 55, 053019 (2015).
[3] T.H. Stix, Waves in Plasmas (AIP press, 1992).
[4] K. Miyamoto, Plasma Physics for Controlled Fusion(Iwanami Book, 2012).
[5] I. Fidone et al., Phys. Fluids 21, 645 (1978).
[6] H.P. Laqua et al., Phys. Rev. Lett. 78, 3467 (1997).
[7] R. Ikeda et al., Contrib. Plasma Phys. 50, 567 (2010).
[8] K. Uchijima et al., Plasma Fusion Res. 6, 2401122 (2011).

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