J. Plasma Fusion Res. SERIES, Vol. 5 (2002) 4'78482

Analysis of ICRF Heating by Three Dimensional Calculationin LHD
SEKI Tetsuo, KUMAZAWA Ryuhei, MUTOH Takashi, FUKUYAMA Atsushil. SAITO Kenji,TORII Yuki2, TAKEUCHI Norio2, WATARI Tetsuo and LHD Experimental Group
National Institute for Fusion Science, Toki 509-5292, JapanlKyoto (Jniversity, Uji 611-0011, Japan
2Nagoya Ilniversity, Nagoya 464-8603, Japan(Received:ll December 2001 / Accepted: 3l July 2002)

Abstract
Ion cyclotron range of frequency (ICRF) heating has been used in LHD and proved to be effective
heating method for helical plasmas. For analysis of ICRF heating, full wave code has been used fortoroidal plasmas. However, two-dimensional calculation, which has been carried out so far, is not
sufficient for helical configuration. Then, development of three-dimensional calculation code has beenstarted. In initial calculation, power deposition profile is similar to the experimental result. Dependenceof power distribution on position.of cyclotron resonance layer also agrees with the experiment.

Keywords:
ICRF heating, LHD, full wave code, TASK/WM, three-dimension

1. Introduction

codes cannot include the complexity of magnetic field

configuration and magnetic flux surface adapted inhelical devices. These points are needed to consider in

Ion cyclotron range of frequency (ICRF) heating is

one of the effective tools for plasma heating inusing the calculation result of these codes. Then, themagnetic-confined plasma experimental devices. InLHD, it is demonstrated that ICRF heating is also
effective in helical device [1,2]. By ICRF heating, theprofiles of the plasma parameters can be changed
through change of the power deposition profile. Controlof the profiles of plasma parameters is important for
improvement of plasma confinement. Behavior of highenergy particles generated by ICRF heating is also
important for understanding of plasma confinement andthe related instabilities. Therefore, evaluation of ICRF
heating is important for promoting the plasma
confinement research of ICRF heated-plasmas.
To estimate the ICRF heating, global wave code
has been used as one of the calculation codes. One-calculation code based on more actual theoretical modelis necessary. Due to progress of the computer
technology in recent years, three-dimensional full-wave
codes have been developed. TASK/WM code is one ofthree-dimensional full-wave code.

2. Coding of TASK/WM code
TASK/WM code is modeled by Vdovin et al. inref. [5] and developed by Fukuyama. Plasma
configuration is given by VMEC (Variational Moments
Equilibrium Code) [6] and non-symmetric magnetic flux
coordinate is used. Finite Larmor radius expansion isadopted and fast wave is mainly described. Toroidal andpoloidal components of wave electric field are solved

dimensional and two-dimensional codes have been usedfor tokamak and helical devices [3,4]. However, these@2002 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchC o r r e sp o ndin g autho r' s e - mai I : s e ki @ lhd. nifs. ac.j p478

Seb.tT. et al., Analysis of ICRF Heating by Three Dimensional Calculation in LHD

simultaneously. More details are described infollowings.

2.3 Response of Plasmas
The dielectric tensor E is usually described inCartesian coordinates with static magnetic field along2.1 Magnetic Flux Goordinates

the z axis. The local normalized orthosonal coordinates

Flux coordinates is non-orthogonal, i.e., the minor

is expressed as

radius direction (poloidal magnetic flux) is defined as t4,^YvBn0,=t-J' t, 0o=0nx0., 0n= ua.

" v,rlthe poloidal dircction is 0, and the toroidal direction is g
as shown in Fig.l. Co-variant expression ofE is

E = Eret + Ere2 + Ere3,lBoI

Variable transformation, ! is defined as(1)/r',\ /r"\t,l.
\",/ \n^)l c,l= p .l

where contra-variant basis iser =YV, e2 =Y0, e3 =Ye.

(6)

IQ)e)

withJacobian isIr-I'-"t.dx"3-vw.ve*tp'I(3),=(Metric tensor is 8i1 = e,' e1 and co-variant basis is e; =0rl0x1.

2.2 Wave Equation
In full wave calculation, the wave equation issolved with poloidal and toroidal harmonic expansion.
Maxwell's equation for stationary wave electric field Edc",lj{

c28n+ cag:r,0crJ{8"c2922+ cag23

0-c,ilinc2932+ cag33(8)andcz= Bo lB , ct= Be /Bd= cz(gztgn- SzzBy) * ct({yBn- SyBz)(angular frequency ar, light velocity c) isVxVx n=4E.E+i0F0i",,,c"(e)glt =(gzzTn- 8rr8t)/J' '(4)Dielectric tensor in non-orthogonal coordinates is E = I

where E is dielectric tensor including effects of finite
temperature (cyclotron damping, Landau damping) andj,,, is antenna current. Wave equation in non-orthogonal
coordinates (radial components) isE,tn 'fr -t.

2.4 Fourier Mode Expansion
Fourier mode expansion in poloidal and toroidaldirections is utilized to solve the wave equation. Spatial
variation of wave electric field. medium and. the lefthand side of Maxwell's equation areI(v x v x E),= 1l=a- 1e+ /lt: - ?+ l"'-Jldx,\/\Dx,r,tn,,/d*')*' s-)2 ( Lt:L- q4) . s,, ( dE,- q4'l \J \ar3 dx' l',/ \dx' ar' Jl

E (W,0,q) =G (W|,q) =DE, \d I s,, (oE,-A;']-eF',l-rliE#]il'|E^,(V)ri(moorite* *(V)+ ns)+ kN pQ)"r(V x V x.E) = G (rtt,Q,rp) E (W),Q)= > t/(V xY xE)l^,^."i(n'0+n'$, (10)\lsu I A]- q4l * s,. (dE,- q4'l \**7traa-#J.7|'e7H(s)mnwhere No is rotation number of helical coil in g. Toroidal and poloidal modes are coupled separately and havewhere the coordinates is (xt, x2, x31 = 1ry,0,cp1. Similar
expressions hold for poloidal and toroidal components

following relations : n' =n+kN p, m' --m+1.

479

SekiT. et al., Analysis of ICRF Heating by Three Dimensional Calculation in LHD

2.5 Parallel Wave Number
Plasma dielectric tensor E (r1,@ e,kr("'") depends onk(""" tltrolugh the following plasma dispersion function

ZI@ - Na",1lkty"""rr,l, where N is the cyclotron
harmonic number arrd m" and n" are defined later ineq.(11). To account the effect of the poloidal magnetic
field influence on k11 wave number formation it is usuallyconsidered that the k11 is the component of the spatial
gradient in the direction of magnetic field. Then, wedefine kn as follows:
Krl- - i€,' r = - ie^' (v e$.o r&)- i€n(-#.r&)=*"ff.*""ff,(11)Fig. 1 Non-orthogonal and cylindrical coordinate sys-tems.

Fourier components of ,E in the local normalized orthogonal coordinates is expressed as

QE'Dj-Jg'. tt. E,uo. fr-''Ei.

Co-variant basis and metric tensor gii= et.€i a;ra ox-G2)

pressed as follows, respectively;

2.6 Boundary conditions
The plasma is assumed to be surrounded by,,= (3u, ,r,#)perfectly conducting wall and vacuum region exists
between plasma surface and the wall. In this boundary
conditions on the conducting wall, tangential
components of electric field vanishes. Co-variant",=(#,',#)

expression is expressed asr,'= (Pr)' .(ut)'

E =EtYtlt+ ErY0 + ErYg

Ez=0, Ez=0.

(13)''"=(3u,)(#).

For the boundary conditions on the magnetic axis (ye0),
finiteness of the wave magnetic field and the induced
charge density is utilized and they are expressed as follows:,-.laa az az aR\'-"10,/ U- av U)(16)

Finite difference method in yr is adopted and linear
mesh with respect to p is used. Where p is p =aE9:

m=O -a=0dvt

m*0 ET =O(ffi)(#){,//,/r(rurf""4 They are numerically solved by the band(r4)matrix equation.

Co-variant component E{n always vanishes on the axis.

3. Galculation Results
Figure 2 shows the profile of hydrogen absorbed
power at the poloidal cross section. Helium majority and2.7 Equilibrium calculation
VMEC (Variable Moments Equilibrium Code) isused as equilibrium solver. Cylindrical coordinates(R,Q,Z) shown in Fig.l is expressed in terms of (ry,0,E)

hydrogen minority plasma is assumed. The electron
density is 1x10le m-3 and the temperature is 1.5 keV. Ion
cyclotron resonance layer of hydrogen is also plotted.
The position of the layer is same as the experimentally
optimized place. In two-dimensional calculation,

as follows:* = * -,(rtt) cos (rn? - nrp)
Q= (2^

t =*t^,(r,r)sin(mO - nrp)

absorbed hydrogen power concentrated at the vicinity of(15)the cyclotron layer. However, it is spread in the flux
surface in the three-dimensional calculation.
480

Seki T. et a/., Analysis of ICRF Heating by Three Dimensional Calculation in LHD

zoq rs (a)shot No.l 341 3

10o-5=ooo.2 0.4 0.60.8p : normalized minor radius

4i*tn IPohe(b)zFig. 2 Deposition profile of minority ions (hydrogen) at
the poloidal cross section. The ion cyclotron resonance layer is also plotted.

o-2-4E530

d0n0G0.5,0Fig. 4 Radial deposition profile. (a): The experimental re-oo=20osult. Only electron absorption; (b): The calculation
result. Solid line, dashed line, and dashed-dotted-oLline correspond to electrons, majority ions (helium), and minority ions (hydrogen).

Ec)

a10qI(!

hydrogen absorption.

303234363840

Figure 4 shows the radial profile of the power
deposition obtained by the experiment [2] and the42frequency IMHz]

calculation. In the experiment, only electron absorption
is shown. The ion cyclotron resonance layer is locatednear the plasma edge in this case. The ion absorption isFig. 3 Dependence of power distribution on wave fre-quency. Solid line, dashed line. and dotted line
correspond to electrons, minority ions (helium),and minority ions (hydrogen), respectively.

very weak. Then, it is difficult to tell about the ion

Figure 3 shows the power distribution as a functionof wave frequency. The cyclotron resonance layer isabsorption in this case. Concerning to the only electron
absorption, the peak is located at around the p=0.7 andalmost same shape of profile is obtained in theexperiment and the calculation.

located at the magnetic axis around 42 MHz and theplasma edge al 31.5 MHz. The ratio of power absorption

4. Summary
To study the ICRF heating in helical device, threedimensional full wave code, TASK/WM, was developedby hydrogen ions is increased as the cyclotron resonancelayer approaches to the magnetic axis. This tendency
was observed in the experiment. The power absorption
by helium ancl electron is very small compared to theand applied to the LHD plasma. In the initial
calculation, dependence of ion absorption on the481

SekiT. et al., Analysis of ICRF Heating by Three Dimensional Calculation in LHD

position of the cyclotron layers is studied and theelectron absorption profile in the case that electron
heating is dominant is obtained. The study has justt21 K. Saito er a/., Nucl. Fusion 41,l02l (2001).
t3l A. Fukuyama et al., Nucl. Fusion 23, 1005 (1983).

begun and more parameter scan can be necessary.

t4l E.F. Jaeger er al., Nucl. Fusion 30, 505 (1990).

pean Physical Society (2000) CD-ROM.

t5l V.L. Vdovin et al., NIFS-469 (1996).
t6l S.P. Hirshman et aI., Comp. Phys. Comm. 39, 161

References

tll T. Seki et al., Controlled Fusion and Plasma Phys-(1986).

ics (Proc. 27th Eur. Conf. Budapest, 2000) Euro-482