J. Plasma Fusion Res. SERIES, Vol. 5 (2002) 306-309

A Calculation of the Distribution Function ofICRF Heated Plasma in LHD
using a Bounce-Averaged Fokker-Planck Equation
SAITO Kenji', KUMAZAWA Ryuhei, MUTOH Takashi, SEKI Tetsuo, WATARI Tetsuo,YAMAMOTO Taror, TORII Yukir, TAKEUCHI Noriol, SHIMPO Fujio, NOMURA Goro,
YOKOTA Mitsuhiro, KATO Akemi, OZAKI Tetsuo, SASAO Mamiko,ISOBE Mitsutaka, OSAKABE Masaki and LHD Experimental GroupNational Institute for Fusion Science, Toki 509-5292, JapantNagoya University, Faculty of Engineering, Nagoya 464-8601, Japan(Received: 11 December 2O0l I Accepted:9 April2002)

Abstract
High-energy ions produced by the ion cyclotron range of frequency (ICRF) heating were observedon the Large Helical Device (LHD). The dependence of the count number of high-energy particles onpitch-angle was studied using the time-of-flight neutral particle analyzer (TOF-NPA) in the LHD. The
angle of the line-of-sight was scanned at five successive discharges sustained by ICRF heating only. A
unique "rabbit-ear" structure with a large population of high-energy ions at a certain pitch-angle, wasobserved. A bounce-averaged Fokker-Planck equation was applied to the LHD plasma in the helicalmagnetic configuration in order to explain the structure of the measured distribution function. The
calculation successfully reproduced the "rabbit-ear" structure.

Keywords:
ICRF heating, LHD, high-energy ions, bounce-averaged Fokker-Planck equation

1. Introduction
Ions are accelerated perpendicularly to theIn order to investigate the effect of ICRF heating inhelical systems, the dependence of the distributionfunction on pitch-angle was studied using the time-offlight neutral particle analyzer (TOF-NPA) [5] on thelarge helical device (LHD). A bounce-averaged FokkerPlanck equation was applied to the ICRF heated plasmaon the helical magnetic configuration in order to explain

magnetic line of force near the ion cyclotron resonancelayer by the ion cyclotron range of frequency (ICRF)
heating, resulting in an increase in the population oftrapped particles. The high-energy ions are accelerated
until they do not across the ion cyclotron resonancelayer. Therefore, the particles with a pitch-angle of 90'

the experimental result.

have the largest population near the ion cyclotron
resonance layer. The peak ofthe distribution function is

2. Experimental Result
The angle of the line-of-sight with respect to themagnetic axis, @, was changed as shown in Fig. I, and it
was scanned from 40.4" to 93.9" at five successive

shifted to a lower angle as these particles are observedat the location with lower magnetic field strength thanthat of the cyclotron resonance layer. This distribution
has a so-called "rabbit-ear" ("butterfly") structure [1-4].
C o rre sponding autho r' s e -mail : saito @ nifs. ac@2ffi2 by The Japan Society of Plasmajp

Science and Nuclear Fusion Research306

SaitoK. etaI., ACalculationof theDistributionFunctionof ICRFHeatedPlasmainLHD

plasma discharges sustained by ICRF heating in theminority heating regime in the He plasma having H
minority. The rnagnetic field strength on the axis ,86 was
2.75 T and the applied frequency / was 38.47 MHz.
Plasma parameters and an ICRF injection power werekept constant ne = O.9 191e --3, T"o = 1.5 keV and"Prcnr = 1.2 MW.
Figure 2 shows the contour of the neutral particle
flux shown in polar coordinate with the angle @ and themodulus ^lEnG, where E is particle energy. The labelsv, = ^'l?-Elm, cos @ and vL = ''[Elmtr sin @ on the abscissa

and the ordinate are parallel and perpendicular velocity

to the magnetic axis, respectively. In order tocompensate for the difference of viewing volume, which
depends on the line-of-sight, fluxes are normalized on

the assumption that the distribution function wasisotropic at the energy range of 6 keV (lowest channelof the TOF-NPA). It was found that the flux of highenergy neutral particles was largest at the angle ofQ = 70".It was thought that this was related to the socalled "rabbit-ear" structure of the distribution function.

3. Calculation of Fokker-Planck Equation
The bounce-averaged Fokker-Planck equation is asfollows [3]:dt^ -/?!=clt,)+Q(y,1dt\
The distribution function/e is independent of I, a length
along the magnetic line of force. The collision term,C(/o), and the quasilinear diffusion term for ion cyclotron heating, Q$d, xe averaged along the magnetic lineof force weighted by llvlland divided by the bounce period, z6:c.(r.) = +lo,,r,.

,Fig. 1 The TOF-NPA. The angle of the line-of-sight withrespect to the magnetic axis, {, was scanned from* c (r,) = #*,(#. #&){-40.4'to 93.9'.

L^,(# . #fi)(o,,r,)}.ffi&**

12atr,)=+l *,oV,) =?!# Ln"dL,."f10owhere,o.'u-Jrr,_ldl

I4=14" -* rl(,,,- t,,4-20246810

t,""=

12u r t)(a o r at)ll"*',(+)l',*(# . #"), a = (tv,,l * ;(1t,,,)'),

vr/vooF = ({u,,1'}. r= (1aFig. 2 Contour of count number of neutral particles de-tected with the TOF-NPA. The contour of the"')'),

wherc (Av1), ((Avt)z|, and ((1vJ2) are the Coulomb diffusion coefficients, and m and q are the mass and charge

count number is shown in a polar coordinate (0,^EEim"l.
307

Saito K. et al., A Calculation of the Distribution Function of ICRF Heated Plasma in LHDof the resonant particle, respectively. a and d) are the(a)applied frequency and an ion cyclotron frequency of theresonant particle, respectively. E and 1t are the energy
and the magnetic moment of the resonant particle, respectively, and 4"it is a left hand component of the radio
frequency (RF) electric field. The summation I is taken
at the resonant point on the magnetic line of force, i.e.,'.oo-Eo'8G'Nxo.+I-k"v"-Q=o)aI r=pa+ lmvl,

o2o.o0.ot2

Coefficients Tb, e, and Q are calailated along the magnetic line of force using plasma parameters.
In the helical magnetic configuration, there aremany helical ripples along the magnetic line of force.
The coefficients ?b, e, and { must be calculated at eachhelical ripple. The particle trapped in one helical ripple
is transferred to the other ripples via drift motion.
Therefore, the calculation of drift orbits was required inorder to deduce the number of bounces in i-th ripple, n;.
The coefficients ?b, e, and {were weighted by n,:(b) .,.o0.E€'Nd\-o-8o.+x6-1 11 ,to, /2n,o2

e -1",6 , tDn,o.oo.o-\-Q =zrn ,4,I

0.4 0.6
rcdQa

1.00.2

rcilQAlln,Fig. 3 Contour of magnetic field strength with (a) themagnetic line of force and (b) the drift orbit of theparticle which passes (p, 0, 9l=10.75, 0", 18") withan energy of 100 keV and a pitch-angle of 70'. The
abscissa is toroidal angle g and the ordinate ispoloidal angle 0. The solid circles are locations of
the ripple bottoms. The area is divided into twelve
regions by dashed lines.

The magnetic line of force at the normalized minor

radius p = 0.75 was employed in the following
calculation of the distribution function. Figure 3(a)shows the magnetic line of force and the contour of themagnetic field strength in the plane, where an abscissa
and an ordinate are a toroidal angle tp and a poloidal
angle 0. The bottoms of each helical ripple are indicatedby solid circles. The magnetic field is obtained bysolving the Biot-Savart equation. Figure 3(b) shows thedrift orbit of the particle which passes the position (p, 0,E) = (0.75,0", 18') with an energy of 100 keV and apitch-angle of 70". The poloidal angle is divided into
twelve regions separated by dashed lines as shown in

15\,Fig. 3(a). The trapped particles at the i-th region in Fig.
3(b) behaves as the particles on the magnetic line offorce in the i-th ripple in Fig. 3(a), and the bounce

10-10

number is given by

01020vilvrn

Fig. 4 Contour of the calculated distribution function bysolving the bounce-averaged Fokker-Planck equa-

tli=ti/Ti,tion. vt and v, are normalized to yth, the thermalvelocity of hydrogen calculated with IH= 0.75 keV
at P = 0'75'

where l; is the staying-time in the i-th region.
Figure 4 shows the contour lines of a calculated308

Saito K. et al., A Calculation of the Distribution Function of ICRF Heated Plasma in LHD

is close to the line-of-sight angle with respect to themagnetic axis, employing p - 0.75 is reasonable insteady state distribution function in the velocity space at@, e, 0 = (0.75, 0', l8') in which the magnetic fieldstrength is lonest. The solid lines indicate a pitch-angleof 70", and tho particles on the solid lines are reflected
at the ion cyclotron resonance layer. The dashed lines

comparison to the experimental result.

4. Summary
In order to observe the anisotropy of theare the boundary between passing and trapped particles,

above which particles are trapped. In the calculation of
Q the left han<l component of the RF electric field was
deduced from the elements of the dielectric tensor,

distribution function, the line-of-sight ofthe fast neutral
particle analyzer was scanned. The signal had a peak at

assuming a uniform RF electric field strength in thepropagating region. In the evanescent region, the electricfield was put at zero. The distribution function depended
on the power clensity. The power density was employedat 60 kWm3 with n" = I x 10re m-3, nu = 5 x l0r7 m-3,and Z. - Tp..= 0.75 keV in this calculation.
A large fi'action of the particles are in the trapped
region, and tht: particles whose banana tips are near theusing the bounce-averaged Fokker-Planck equation. Inthis calculation, the bounce-averaged Fokker-Planck
equation was modified in order to adapt it to the helicalsystem, which has toroidal and helical ripples. Theion cyclotrorl resonance layer have the largest

The authors wish to thank the technical staff of theLHD group of the National Institute for Fusion Sciencea certain angle. The result was compared with analysis

calculation successfully simulated the "rabbit-ear"structure.

Acknowledgements

population; the calculated "rabbit-ear" structure of thedistribution function is similar to the experimental result
shown in Fig. 2. However, the angle of the largest
population is a little lower than that of the solid line. A
left-hand coml)onent of the RF electric field is small in
the ion cyclotron resonance layer, but at slightly higher
magnetic fiel,l strength it increases in the minority
heating regime. Particles with slightly smaller pitchangle than that of particles on the solid line can reachthe strong abs<lrption region. Therefore, the angle of thelargest population is slightly smaller than that of thesolid line. The particles with a pitch-angles of 90ocannot cross ttre ion cyclotron resonance layer, but theirfor their helpful support during this work.

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