Proceedings of ITC/ISHW2007

Simulation study of ICRF wave propagation and absorptionin 3-D magnetic conﬁgurations
T. Yamamoto1) , S. Murakami1) and A. Fukuyama1)
1)Department of Nuclear Engineering, Kyoto University, Kyoto 606-8501, Japan(Received 15 October 2007)

ICRF wave propagation and absorption are investigated by TASK/WM, in which Maxwell’s equation for RF
wave electric ﬁeld with complex frequency is solved as a boundary value problem in 3D magnetic conﬁgurations.
Magnetic ﬂux coordinates based on the MHD equilibrium by VMEC code are considered. The wave propagation
is solved in the tokamak conﬁguration (JT-60U) and also in the helical conﬁguration (LHD) in the minority ion
heating regime. Characteristics of the ICRF wave propagation and absorbed in the 3D magnetic conﬁgurations
are shown.
Keywords: Simulation, ICRF heating, wave propagation, helical conﬁguration

1Introduction

sional dumping, the dielectric tensor isThe ICRF heating has been successfully studied and theeﬃciency of this heating method has been shown in LHD.
The numerical analysis of ICRF heating has been carryedout by GNET[1] and also have shown the eﬃciency anddetail information about the energetic tail ion distribution
generated by ICRF heating. However, a detail analysis of
the wave propagation and absorption process has not been
carryed out enough in 3D magnetic conﬁgurations.
In this paper we study the ICRF wave propagation andabsorption by TASK/WM[2, 3], in which Maxwell’s equation for RF wave electric ﬁeld with complex frequency issolved as a boundary value problem in 3D magnetic conﬁgurations. We consider VMEC coordinate as the magnetic coordinate obtained from MHD equilibrium. We ﬁrst
solve the wave propagation in the tokamak conﬁguration(JT-60U) and , then, the propagation in the helical conﬁguration (LHD), where the minority ion heating regime isassumed.

2Simulation model

TASK/WM solves Maxwell’s equation for the electric ﬁeld
E with complex frequency ω as a boundary value problem
in 3D magnetic conﬁgurations.
∇×∇×E=

ω2 ↔
ϵ · E + iωµ0 jextc2(1)Here, the external current jext represents the antenna current in ICRF heating. Assuming cold plasma and colli-−iD 0 S0  ,0P
21 ∑ ωpsω + iν s S
ϵ =  iD
0↔

S =1−
D=ϵ0

ϵ0

P=1−,

Ωs,
ω (ω + iν s )2 − Ω2ss

∑ ω2ps1,ω ω + iν ss(2)where particle species s, the dielectric constant in vacuum
ϵ0 , the collisionality ν, the plasma frequency ωp and thecyclotron frequency Ω.
We rewrite the Maxwell’s equation (1) in a magnetic
coordinates (ψ, θ, ϕ). LHS of (1) is written as(∇ × ∇ × E) p()()
grq ∂E p ∂Er
1 [ ∂ { grp ∂Er ∂Eq=− r +− pJ ∂xq J ∂xq∂xJ ∂xr∂x()}()
grr ∂Eq ∂E p
∂ { gqp ∂Er ∂Eq+− q − r− rJ ∂x p
∂x
∂x J ∂xq∂x()()
gqq ∂E p ∂Er
gqr ∂Eq ∂E p }]+− p +− q ,J ∂xr∂x
J ∂x p∂x(3)where the metric coeﬃcient gi j = ei · e j , ei = ∂r/∂xi ，Jacobian J, (x1 , x2 , x3 ) = (ψ, θ, ϕ). The indexes (p, q, r) are(1, 2, 3), (2, 3, 1), (3, 1, 2).
The dielectric tensor is written as↔−1

author’s e-mail: yamamoto@p-grp.nucleng.kyoto-u.ac.jpω (ω + iν s )2 − Ω2ss2
∑
ω
1ps↔

↔

↔−1

ϵ̃i j = g i j · µ i j · ϵ i j · µ i j ,(4)Proceedings of ITC/ISHW2007↔

where the rotation transform tensor µ is1
µ11 = √ ,g11
1 { Bθ
(g23 g12 − g22 g31 )µ12 = √J g11 B+µ13 =(5)The electric ﬁeld E and metric coeﬃcient gi j are evaluated at grid points ψl in radial direction and expanded toFourier series in poloidal and toroidal direction.
∑E(ψk , θ, ϕ) =Emn (ψk )ei(mθ+nϕ) ,mn∑′′gi j (ψk , θ, ϕ) =(gi j )m′ n′ (ψk )ei(m θ+n Nh ϕ) ,(6)m′ n′where Nh is rotation number of helical coil in ϕ. The antenna current jext is given as
j1 = ( j1 )mn ei(mθ+nϕ) Θ(ψd − ψ),
j2,3 = ( j2,3 )mn ei(mθ+nϕ) δ(ψ − ψd ),(7)where step function Θ(x), delta function δ(x), and the radial antenna position ψd . These equations satisfy ∇ · jext =0.
We assume that the plasma is surrounded by a perfectconductor and that there is a vacuum layer between theplasma surface and the perfect conductor wall. Then, thetangential electric ﬁeld at the wall is set zero as the boundary condition;Eθmn = 0,

Eϕmn = 0.

Simulation results

3.1

Tokamak conﬁguration (JT-60U)

We ﬁrst study the ICRF minority heating in the tokamak
conﬁguration, taking JT-60U plasma as example. The assumed conﬁguration parameters are as follows; plasma
major radius R0 = 3.5m，plasma minor radius a = 0.98m，plasma shape elongation κ = 1.3，plasma shape triangularity δ = 0.31，magnetic ﬁeld at magnetic axis B0 = 3.3T.

}Bϕ(g33 g12 − g23 g31 ) ,BBθBϕ
g12 +g13 , µ21 = 0,BBϕ √BBϕBθµ22 =J g11 , µ23 =g22 +g23 ,BBB√Bθ
µ31 = 0, µ32 = − J g11 ,B
Bϕ
Bθ
g32 +
g33 .
µ33 =BB3(8)The poloidal electric ﬁeld at the magnetic axis ψ = 0 iszero. The toroidal electric ﬁelds of the modes m , 0 arezero. That of mode m = 0 is ﬁnite and the radial derivationis zero. These are boundary condition on the magnetic axis
ψ = 0;
∂Eϕ0n
 m=0∂ψ = 0(9) m , 0 E mn = 0.
ϕ
The Maxwell’s equations (3)-(7) are solved under theboundary conditions (8) and (9) on the 3D conﬁguration
given by numerical equilibrium data of the magnetic ﬂux
coordinates.

The obtained results for the minority ion heating
regime is showen in Fig.1. The assumed plasma parameters are as follows; temperature at magnetic axis T 0 =3.0keV，temperature on plasma boundary T s = 0.3keV，density at magnetic axis n0 = 1.0 × 1020 /m3 ，density
on plasma boundary n s = 0.1 × 1020 /m3 , minor ion ratio 5%, ration of collision frequency to wave frequency
ν s = 0.003 . The temperature and density proﬁles are given
by T (r/a) = (T 0 −T s )(1−(r/a)2 )+T s ，n(r/a) = (n0 −n s )(1−(r/a)2 )1/2 + n s respectively. Also, the used antenna parameters are antenna current density jext = 1.0A/m，wave frequency fRF = 45.0MHz, The upper side ﬁgures in Fig.1
assumes the wave frequency fRF = 45.0MHz and the lower
side ﬁgures is fRF = 48.0MHz.
Three lines drawn in Fig.1 (b), (f) represent ion cyclotron layer (a green line), two-ion-hybrid cut-oﬀ (a blueline) and resonance (a red line) layers from the outside. It
is observed that the absorption region of minority ion locates near the minority ion cyclotron layer in Fig.1 (b), (f).
The coherent waves are observed (c), (d), (g), (h). The
E+ component (right-circularly polarized component) of
the electric ﬁeld is absorbed and the amplitude of coherent waves is damped near the minority ion cyclotron layer(c), (g). While, the damping of amplitude of the E− component (left-circularly polarized component) not observed
near the minority ion cyclotron layer. The reason is that
the E− component of the electric ﬁeld do not provide theminority ion at the minority ion cyclotron layer under theassumption of cold plasma. The absorption increases at
the two-ion-hybrid resonance and cut-oﬀ layers near theminority ion cyclotron layer near z = 0 (a), (b). Since thewidth between resonance and cut-oﬀ is narrow, the wave is
reﬂected or transmitted at the cut-oﬀ layers. The reﬂected
waves is superposed incoming waves and the amplitude ofE+ component is enhanced (c). The amplitude of the transmitted wave is enhanced at the resonance layer (c). These
are the reasons for increasement of the absorption.
The lower in Fig.1 are results for the same parameters as those of upper except fRF = 48MHz. Since thewave frequency increases from 45.0MHz to 48.0MHz, theion cyclotron layer and the two-ion-hybrid cut-oﬀ and resonance layer shift toward the higher magnetic ﬁeld side.
Therefore, the absorption region moves from 4.0m (b) to
3.75m (f).

Proceedings of ITC/ISHW2007
(b)

(c)

(d)

(f)

(g)

(h)

(a)

(e)

Fig. 1 Radial H absorption distributions (a), (e), contour plots of H absorption (b), (f) and E+ component (c), (g) and E− component (d),(h) of the electric ﬁeld on the poloidal cross section; upper ( fRF = 45.0MHz), lower ( fRF = 48.0MHz); JT-60U

3.2

Helical conﬁguration (LHD)

We study the ICRF minority heating in the Helical conﬁguration, taking the LHD plasma as an example. The
conﬁguration parameters of LHD are as follows; plasma
major radius R0 = 3.6m，plasma minor radius a = 0.58m，magnetic ﬁeld at magnetic axis B0 = 2.75T.
The magnetic surface structure obtained by VMEC
code is shown in Fig.2, where a vertically elongated crosssection is plotted.
ψ (VMEC)

θ (VMEC)

jext = 1.0A/m，wave frequency fRF = 38.5MHz. Also,
the temperature and density proﬁle are given T (r/a) =(T 0 − T s )(1 − (r/a)2 ) + T s , n(r/a) = (n0 − n s )(1 − (r/a)8 ) + n s
respectively. These parameters is used in an experiment onLHD.
The lower in Fig.4 are results used the same parameters as those of upper except n0 = 0.7 × 1020 /m3 ,fRF = 36.0MHz, the center mode number of the toroidal
wave mode number nph0 = 0; n = nph0 + n′′ Nh , wheren′′ = integer (n in Eq.6), is the toroidal mode number of thewaves exited by antenna current. The toroidal mode number dependency is showed in 3. It shows that nph0 = 0
is the leading mode number. Therefore, the lower in Fig.4
was analyzed with nph0 = 0.

900

pabs

800

Fig. 2 Conter plots of a magnetic surface structure，ψ (left ﬁg.),θ (right ﬁg.)

700

600

The obtained results for the minority ion heating
regime in the LHD plasma is showed in Fig.4. Upper sideﬁgures in Fig.4 are results with following plasma parameters; temperature at magnetic axis T 0 = 2.0keV，temperature on plasma boundary T s = 0.2keV，density atmagnetic axis n0 = 0.1 × 1020 /m3 ，density on plasmaboundary n s = 0.01 × 1020 /m3 , minority ion ratio 5%, ration of collision frequency to wave frequency ν s = 0.003. The used antenna parameters is antenna current density

pabs

500

400

300

200

100

0-4-3-2-10123

nph0Fig. 3 Toroidal mode dependency

45

Proceedings of ITC/ISHW2007
(b)

(a)

(c)

(d)
Region 1(f)

(e)

(g)

(h)
Region 2Fig. 4 Radial H absorption distributions (a), (e), contour plots of H absorption (b), (f), E+ component (c), (g) and E− component (d), (h)of the electric ﬁeld on the poloidal cross section; LHD

Three lines drawn in Fig.4 (b), (f) represent ion cyclotron layer (a green line), two-ion-hybrid cut-oﬀ (a blueline) and resonance (a red line) layers from the outside. Itis observed that the absorption region of minority ion locates near the minority ion cyclotron layer in Fig.4 (b), (f).
These are the same tendency as that of tokamak plasma.
But, it is not clear whether coherent waves are observedin the contour plots of the electric ﬁeld on upper Fig.4 (c),(d). While, coherent waves is observed in lower Fig.4 (g),(h). In upper ﬁgures, the wave length is comparable to theplasma width. In lower ﬁgures,the wave length is shorter
than the plasma width, because plasma dencity is higher.
This is the reason for coherent waves observed. We observe that E+ component of electric ﬁeld is damped at region 1 and 2 in Fig.4 (c), (g) respectively. Region 1 and 2
locates at ion cyclotron layers. Region 1 is larger than ion
cyclotron layer, because wave length is conparable to theplasma width.

4Conclusions

We have studied the ICRF wave propagation and absorption in a 3D magnetic conﬁguration using TASK/WM, inwhich Maxwell’s equation for RF wave electric ﬁeld issolved as a boundary value problem. The magnetic ﬂux
coordinates based on MHD equilibrium by VMEC code isconsidered. We have solved the wave propagation in thetokamak conﬁguration (JT-60U) and in the helical conﬁguration (LHD) in the minority ion heating regime.
The obtained results shows that ICRF wave is absorbed the region near the ion cyclotron layer and the twoion-hybrid cut-oﬀ and resonance layer.

These analyses were carryed out under assumption ofcold plasma. Not only the basic harmonics but also thesecond and higher harmonics are important. Therefore, weneed to analyze including ﬁnite larmor eﬀects. Furthermore, we need to analyze time evolution of velocity distribution function using GNET code.
[1] S. Murakami et al Nucl. Fusion 46 (2006) S425-S432.
[2] A. Fukuyama et al Proc. 18th Int. Conf. Fusion Energy 2000
THP2-26.
[3] A. Fukuyama and T. Akutsu Proc. 19th Int. Conf. on Fusion
Energy 2002 THP3-14.