Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 (2004)
6 (2004)626–629
000–000
J. Plasma Fusion

Distribution Function of Higher Harmonic ICRF Heated Plasma
Calculated with Bounce-averaged Fokker-Planck Equation on LHD
SAITO Kenji, KUMAZAWA Ryuhei, MUTOH Takashi, SEKI Tetsuo, WATARI Tetsuo, TORII Yuki1,
TAKEUCHI Norio1, SHIMPO Fujio, NOMURA Goro, YOKOTA Mitsuhiro, KATO Akemi,
WATANABE Tsuguhiro, ZHAO Yangping2 and LHD Experimental Group
National Institute for Fusion Science, Toki, 509-5292, Japan
1Nagoya University, Faculty of Engineering, Nagoya, 464-8603, Japan
2Institute of Plasma Physics, Chinese Academy of Science, Hefei 230031, People’s Republic of China(Received: 9 December 2003 / Accepted: 24 March 2004)

Abstract
Higher harmonic heating is one of the effective ion cyclotron range of frequency (ICRF) heating scenarios. Highenergy particles are more effectively produced by this heating method than by minority heating. To study the heating
efficiency and the high-energy ion tail formation in the Large Helical Device (LHD), the distribution function was calculated using a bounce-averaged Fokker-Planck equation, and the absorbed and the lost power were estimated for thehigher harmonic ICRF heating. It was found that the higher harmonic ICRF heating enhanced the D-D reaction rate.
Keywords:
higher harmonic ICRF heating, LHD, high-energy particle, bounce-averaged Fokker-Planck equationof the LHD plasma. A wave number parallel to the magnetic
line of force is assumed to be zero for simplicity. The
strength of the radio frequency (rf) electric field |E 0| isassumed to be constant over the whole plasma. A dispersion
relation of a cold plasma approximation is used to calculate
the polarity of E0 and a wave number perpendicular to themagnetic line of force. The electron density and temperature
profiles are assumed as ne = ne0(1 − ρ8) and Te = Te0(1 − ρ 2),
respectively, where ρ is the normalized minor radius and ne0
and Te0 are electron density and temperature on the magnetic
axis, respectively. A bounce-averaged Fokker-Planck equation is obtained by integrating the local rf kick term Q andthe collision term C along the magnetic line of force in theapproximation of no deviation of particle-orbit from the magnetic flux surface [2].

1. Introduction
It was thought that in the helical device high-energy particles with a perpendicular pitch angle had a poor confinement performance because the trapped particle orbits are
deviated from the magnetic flux surface. Ion cyclotron rangeof frequency (ICRF) heating was thought to work well onlyin the electron heating regime. However, in the Large HelicalDevice (LHD), the high-energy tail was observed up to 500
keV in minority ICRF heating with an inward shifted magnetic configuration [1]. In this configuration, the deviation ofthe orbit from the magnetic flux surface is small. For producing higher-energy particles in the LHD, a higher harmonic
ICRF heating is thought to be more effective than the minority heating because under the former the particles are accelerated due to the finite Larmor radius effect. The distributionfunction of higher harmonic heated ions is calculated using abounce-averaged Fokker-Planck equation to determine towhich energy the ion energy is extended and to estimate theabsorbed power in this heating regime. In Sec. 2, the methodof calculation is introduced. In Sec. 3, calculated results aredescribed, and Sec. 4 is a summary.

∂ f0−−= C ⎛⎝ f0⎞⎠ + Q ⎛⎝ f0⎞⎠ + S∂t1−C ⎛⎝ f0⎞⎠ =τbdl1⎛ ∂µ∂ ⎞

∫ v C f = v mv ⎜⎝ ∂ E + E ∂µ ⎟⎠||⎛ ⎞⎝ 0⎠2⎛ ∂⎡µ ∂ ⎞ ⎛ 2 ⎞⎤1
mv ⎜+× ⎢− α v2f0 +⎟ ⎝β v f0⎠⎥
⎝ ∂E2
E ∂µ ⎠⎣⎦

2. Calculation of Fokker-Planck equation
For simplicity of the calculation, a helical symmetricalmagnetic configuration is employed as shown in Fig. 1,
where the bold solid lines represent the ion cyclotron resonance layers. The plasma size is selected to be similar to that

©2004 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchCorresponding author’s e-mail: saito@nifs.ac.jp1626

Saito K. et al., Distribution Function of Higher Harmonic ICRF Heated Plasma Calculated with Bounce-averaged Fokker-Planck Equation on LHD|x| < 0.308. ψn is expressed as follows:mγ∂ ∼ ∂ f0cµ22v τ b ∂µ∂µ+1−Q ⎛⎝ f0⎞⎠ =τb∫dl ⎛ ⎞ 2πmq2q Lres f0,
Lres ~Q ⎝ f0⎠ =v||τbψn=Eleft Jn−1(b) + Eright Jn+1(b)(1)

2,k⊥v⊥, and Eleft and Eright are left- and right-handΩpolarized components of the rf electric field, respectively. ψnwhere b =wherev||at the cyclotron resonance layer is approximately expressedin the following relation in the case of the second harmonic
heating:µ B | 2 | 2/3 2∼
Ai (x)|ψ n|2,π|Σq = res2n Ω¨ |·(n Ω )2 | 2 | 4/3
x=−| n Ω̈ | ,4|ψ 2|2 ∝ J12(b) + 6J1(b)J3(b) + 9J32(b).

τb =∫ v , ∼c =∫ B dl,
dl||Lres =This equation shows that ions are not accelerated in the low
energy range, |ψ 2|2 takes the maximum value at b ≈ 3.4, andthe rf kick term vanishes again at b ≈ 6.1; ions are selectively
accelerated in some energy region. Therefore, the higherenergy ions are produced in the higher harmonic heating
rather than in the minority heating at the same absorption
power.
Ions exist in a triangular region in the µ-E space becauseof µ < E/Bmin, where Bmin is the minimum magnetic fieldstrength along the magnetic line of force. The coordinate istransformed from the µ-E space to the v-ξ space, where v is avelocity of ions and ξ is a cosine of the pitch angle at B =Bmin. The steady state distribution function is obtained by setting the left-hand side of eq. (1) to be zero. In the calculation,
only the balance-injection of neutral beam injection (NBI) isconsidered, then the beam source is symmetry. The sink ofNBI particles is located at v = 0.

1 ⎛ ∂nq ⎞+⎟ , and⎜m ⎝ ∂ E mω ⎠α = 〈∆v||〉 +1〈(∆v⊥ )2〉, β = 〈(∆v||)2〉, γ = 〈(∆v⊥ )2〉,2v

where S is a source or sink term of ions. 〈∆v||〉, 〈(∆v||)2〉,
and 〈(∆v⊥)2〉 are the Coulomb diffusion coefficients, and m
and q are the mass and charge of resonant ions, respectively.
n is a harmonic number of the ICRF heating. ω and Ω are theapplied frequency and the ion cyclotron frequency, respectively. E and µ are the energy and the adiabatic magnetic
moment of ions, respectively. The summation of the kickterm Σ is carried out at each resonant point on the magnetic
resline of force. Airy function is approximated as follows [3]:3. Result of calculation
In Fig. 2 is shown the calculated contour plot of the distribution function of ions (H+) heated by the second harmonic
heating. In this case, the normalized minor radius ρ is 0.56,

1⎧
for |x| > 0.308⎪Ai (x) = ⎨ 2π |x|⎪⎩ 0.287 for |x| < 0.308.
2The resonance layer exists near the turning point whenFig. 2 Contour plot of distribution function at the position ofminimum magnetic field on ρ = 0.56. The plasma isheated by the second harmonic heating.

Fig. 1 The helical symmetric plasma model. Bold solid lines
indicate the ion cyclotron resonance layer.

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Saito K. et al., Distribution Function of Higher Harmonic ICRF Heated Plasma Calculated with Bounce-averaged Fokker-Planck Equation on LHD

the magnetic field strength on axis is 1.325 T, and the frequency is 38.47 MHz, which conditions are similar to thoseof the experiment. The values employed for electron density
and temperature on the axis are ne0 = 1 × 1019 m-3 and Te0 = 2keV, respectively. The rf electric field strength |E 0 | isassumed as 3 kV/m. The ions with a pitch-angle of 82.3° arereflected just at the ion cyclotron resonance layer. The perpendicular velocity of the ion passing through the resonancelayer is statistically increased until the reflection point reaches the resonance layer. Therefore, a ‘butterfly’ structure isformed. The solid line in Fig. 3 is the distribution function
along the pitch-angle of 82.3°. The ion-energy is found toextend over 3 MeV in this calculated case. The power density
is calculated to be 77 kW/m3.
There exists a limitation of the maximum energy ofions in the real LHD. Figure 4 shows the confinement limitation obtained by the calculation of the particle-orbit in theLHD [4]. The start point is ρ = 0.5, which is on an innerport side of the LHD, with the magnetic field on the axis of

1.375 T. The confined ion is defined as not-escaped formore than 50 turns in the toroidal direction. The loss boundary is seen at the energy of 1 MeV. Therefore, the loss term
− ν0H(E − Ec)f0 is added to the right-hand side of eq. (1),where H is a Heaviside unit function with Ec of 1 MeV over
the whole plasma region. ν0 is employed 1 × 106 s-1 as theloss frequency, which is insensitive to the loss power Ploss.
Due to the function H(E − Ec), only the particles with energy over 1 MeV are lost. The dashed line in Fig. 3 represents
the distribution function calculated with this loss term. The
input power is calculated as 70 kW/m3, which includes theloss power of 6 kW/m3 of the resonant ions with an energy
over 1 MeV; the net power is reduced to 64 kW/m3. Whenions with the energy of 150 keV and a pitch angle of 0° are
injected from NBI with a power density of 121 kW/m3, thetail is enhanced as shown by the dotted line in Fig. 3. Ploss isincreased to 26 kW/m3, and the net power is increased only
by 10 % from 64 kW/m3 to 71 kW/m3.
The efficiency of the higher harmonic heating isimproved at a higher density, because the loss power can bedecreased. Figure 5 shows the distribution functions at thelower and the higher density, i.e., at ne0 = 1 × 1019 m-3 and ne0= 4 × 1019 m-3, respectively. At the higher density, the taildoes not reach 1 MeV as shown by the solid line in Fig. 5
because k⊥ increases with the density, and the energy of 925
keV corresponds to b = 6.1 (|ψ2|2 = 0). Therefore, the lost
power is decreased to 0. The power absorption is enhanced
and the power density reaches 1 MW/m3 when the rf fieldstrength is fixed at 3 kV/m. When the power density is fixedat 64 kW/m 3, then |E 0| decreases to 1 kV/m and the tailshrinks.
In 2001, the second harmonic heating experiment wasconducted in the NBI sustained plasma [5]. The absorbed
NBI power was 1.5 MW and the injected ICRF power was 2

Fig. 3 Three types of distribution functions. The turning
points of resonant particles are on the ion cyclotron
resonance layer.

Fig. 5 Distribution functions for different electron densities.
The tail of high-density plasma cannot extend over 1MeV.

Fig. 4 The boundary of confinement. Outside of this boundary is the region of particle loss.

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Saito K. et al., Distribution Function of Higher Harmonic ICRF Heated Plasma Calculated with Bounce-averaged Fokker-Planck Equation on LHD

planned perpendicular NBI on LHD. Figure 6 shows thereaction rate of the D-D reaction. ρ is the normalized minor
radius and V is the plasma volume inside ρ. The total reaction
rate is 5.4 × 1014 s-1 with only the NBI. The additional second
harmonic ICRF heating with a net rf power of 1.4 MW wouldenhance the reaction rate to 1.1 × 1015 s-1 if the NBI was atangential injection to the torus. The reaction rate is increased
to 2.5 × 1015 s-1 by changing the pitch angle of NBI particles
from 0 to 85°, and net rf power also increases to 1.9 MW.

4. Summary
The distribution function was calculated for the higher
harmonic heating, and the power absorption and the loss ofpower was estimated using it. In the calculation, a large tailcould be formed by second harmonic heating. However, withthe low electron density, the high-energy tail was limited
within 1 MeV due to particle loss. The population of highenergy ions produced by the ICRF second harmonic heating
was increased by the NBI. However, the loss of power wasalso increased. The loss could be decreased, and the heating
power efficiency therefore improved, at the higher density.
The D-D reaction rate was calculated from the distributionfunction. The ICRF heating increased the reaction rate as aresult of the high-energy ion tail formation, especially during
perpendicular injection of NBI.

Fig. 6 The profile of the reaction rate. ρ is the normalizedminor radius. V is the plasma volume inside ρ.

MW. The net absorbed ICRF power deduced by the decay ofthe stored energy was 0.6 MW. Using the same plasma
parameters as those in the experiment, ne0 = 0.8 × 1019 m-3
and Te0 = 1.2 keV, and, assuming the constant power densityof NBI, an effective rf field |E0| of 3.2 kV/m was deduced.
Then the calculated total net power was 0.6 MW and the lost
power due to the tail was 0.7 MW. The remaining rf power of0.7 MW was thought to be absorbed by the unconfined
peripheral plasma or the vacuum wall. When the electron
density was 4 × 10 19 m -3, the calculated loss of the taildecreased to 0.
A D-D reaction rate is also estimated from the calculated distribution function. The conditions are as follows: |E0| =3 kV/m, f = 38.47 MHz, B0 = 2.65 T, ne0 = nD0 = 4 × 1019 m-3,
Ec = 1 MeV, and the ion and electron temperature on the axis
is 3 keV. Injection energy and power values of the NBI are
50 keV and 4 MW, respectively. These are the values of theReferences[1] T. Mutoh et al., IAEA Conference 2002, Lyon, IAEAEX/P2-19.
[2] T.H. Stix, WAVES IN PLASMAS (AIP, 1992), p521.
[3] G.W. Hammett, Ph.D. Thesis, Princeton University(1986), p50.
[4] Y. Matsumoto, S. Oikawa and T. Watanabe, J. Phys.
Soc. Jpn. 71, 1684 (2002).
[5] K. Saito et al., Plasma Phys. Control. Fusion 44, 103(2002).

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