J. Plasma Fusion Res. SERIES, Vol.2 (1999) 105-108

lon Temperature Gradient Modes in Helical Systems
KURODA Tohru, SUGAMA Hideo', KANNO Ryutaro' and OKAMOTO Masao'
Graduate University for Advanced Studies, Toki 509-5292, Japan
lNational Institutefor Fusion Science, Toki 509-5292, Japan(Received: 18 December 1998 / Accepted: 19 February 1999)

Abstract
Ion temperature gradient (ITG) modes in helical systems are studied. The gyrokinetic equation forions, the adiabatic assumption for electrons, and the charge neutrality condition are used with theballooning representation to derive a kinetic integral equation, which is solved numerically to obtain thelinear growth rate, the real frequency, and the eigenfunction of the ITG modes. Using a simple helicalfield model, cases with L = 2 and 2 < M < 10 are investigated where L and M are the poloidal andtoroidal polarity numbers characterizing the helical field ripple, respectively. The effects of the toroidal
polarity number M on the dispersion relation and the mode structure of the ITG modes are clarified.

Keywords:
ITG modes, helical systems, ballooning representation

1. lntroduction
The ion temperature gradient (ITG) modes areelectrostatic drift waves and have been studied as acandidate to explain anomalous heat transport in highright-hand side represents the helical ripple with thepoloidal and toroidal polarity numbers denoted by Z, andM, respectively. For example, we have L = 2 and M =10 for LHD and L = 2 and M = 8 for CHS. Since thetemperature plasmas []. For toroidal systems such as

helical magnetic ripple affects the particles' drift
motion, the properties of the ITG modes stability intokamaks and stellarators, the properties of the ITG
modes are significantly affected by the magnetic field
geometry through the VB and curvature drift motion ofparticles.
For tokamaks, the magnetic field strength is given

helical systems can be different from those in tokamaks
t5l.

In the present work, we investigate the propertiesof the ITG modes for the helical systems with I = 2 and2 < M < l0 which are compared to the tokamak case.
Especially, the effects of the toroidal polarity number M
on the dispersion relation and the mode structure of theITG modes are studied.

by the large-aspect-ratio approximation as B/Bo = I t,cos0 where 0 is the poloidal angle and e, = r/R is theinverse aspect ratio representing the toroidal ripple
where r and R denote the minor and major radii,
respectively. Many studies have been done on the ITG
modes in tokamaks 12,3,41. In this case,the ITG modes
are confined mostly in the outside of the torus, -nl2 < 0< nl2, which corresponds to the bad curvature region.
For helical systems, the magnetic field strength is

2. Kinetic ITG Mode Equation for HelicalSystems
Here we consider a high-temperature collisionlessplasma and assume that, in the presence of theelectrostatic perturbation @, the perturbed electron

approximately given by

Bl Bo= I - €tcos 0- en cos (Le - M€),(1)@1999 by The Japan Society of Plasma
Science and Nuclear Fusion Researchwhere ( is the toroidal angle and the term with e6 in the105

Kuroda T. et al., Ion Temperature Gradient Modes in Helical Systems

density is described by the adiabatic (or Boltzmann)withresponse 6n.= (eQ/T")ns. The perturbed ion distribution

velocity for the ions with the mass m; and thetemperature Ti, P = b x vlQi @ = B/B) is the ion
gyroradius vector, and Oi = eBl(mic) is the ion

Xgyrofrequency. The non-adiabatic part of the distribution

Tfunction ft is determined by the linear gyrokineticequationfI o qr, pyno-iffir*.

? ft,k't2t4la),t,L-!<')2t -trn,+n{k2 '4aA.

K,+K,2(l+ a)c,.#hl)]., (k''k'')'l, - ,r+ iv,,b'Y) h= (a - a..r)f:f0

o*,dt ^l ze
K(k.k')=-il
J-Ja(l +function is written as dl - -@QlT")nsF\a + h exp (-i k1. p) where Fy: v-3t2 v7] exp
1-v2lvl-r1is the Maxwellian
distribution function, vri = (2Tilm) r/2 is the thermal(5)where l, - (a-,r)2 (3 e,lq)2lr"a, 0 = kli ka, 0' = k'13 ku
lo = 1o &&i/Kl + a)r"l) exp [-(t? + k'21)121" (l + a)),k2L = kZ + k2, k'2L = kZ + k'2, andQ)

Here ar is the frequency of the perturbation, arp = kt. vo
is the ion VB-curvature drift frequency, Js is the Bessel
function of order zero, and @* = a*r [l + r1i{(vlvfi2 -a = | + i2(L, I r) t;ta-"t | @ - e')x (st [(s + 1) (sin 0 - sin9')

3/2ll where 4r = dlnl ldlnns is the ratio of the ion
temperature gradient to the density gradient, (D*i= -T ,lro*" is the ion diamagnetic drift frequency, t": T"lTiis
the ratio between the electron and ion temperatures, rrl*e-i10cos0- 0'cos9')+ (Len | (L - Mq)) t{s/ (Z - Mq) += cksT.l(eBZ,) is the electron diamagnetic drift

x {sin ((L - Mde - Md)

frequency, Ln = -(dlnns/dr)-t is the density gradientscale length, and /<e is the poloidal wavenumber.
We use the ballooning representation and write theperpendicular wavenumber vector as k1 = ko (Y a +?eYq) where 4 is the safety factor, a = ( - 40 is the- O'cos ((L - Mq)9'- Mo"))ll).

ll- sin ((L - Mde'- Ma))l- f{d cos ((L - Mq)O - ua)(6)

Here the wavenumber variables ks, k, and k' arenormalized by p,' (p, = VzrJ*.Ja,) and e5 * rL isused. The integral equation (4) with the boundary
conditions Q@ -+ l-) = 0 determines the complex-

label for magnetic field line, and ko= -n represents thetoroidal mode number, which is related to the poloidal
wavenumber as ft6 - nqlr. ln the present work, weassume that 0p- 0. We consider a large aspect-ratio andlow B toroidal system and use Eq. (l) to give the ion
VB-curvature drift frequency asvalued eigenfrequency and eigenfunction of the ITG
mode for the helical system. If we put €l = 0, the integral ITG mode equation (4) with Eqs. (5) and (6)reduces to the one given by Dong, et al. for the tokamak@o = 2(L,l r) o - i (v? + v2, | 21 v2r,

case [3].

I e, {cos 0 + s0 sin 0} + len {cos (Le - M()+ fd sin0 (L0 - M()ll,

3. Numerical Results(3)

Here we numerically solve the integral equation (4)with the boundary conditions @(0 -+ +-; = 0 to obtain

where .i = (rlq)dqldr is the shear parameter.

Neglecting the trapped ions, integrating thethe growth rate, the real frequency, and the

gyrokinetic equation (2) along the field line with theeigenfunction of the ITG mode for the L = 2 helicalboundary conditions h(0 --> +*1 = 0, and substituting it
into the charge neutrality condition eQl7" = 6n"lns =6ni/ns=-eQlTi+ I&vJo&e)h, we obtain the integral
equation which is written as

concerned with the effects of the helical ripple, we
neglect the toroidal ripple by putting 6t = O (straighthelical system) for simplicity. Typical parameters usedI\systems with various M numbers. Since we aren+-here are q = 2, ke pri = 0.75, Ti/7"= l, L,lRo= O.2, L,J-Jzn

e6/r = O.2, and .i = -l (negative shear). We also treat asingle field line labelled by a= 0.

^[' **lQro,=l +K&,k'\o&'). (4)\ 4i106

Kuroda T. et al., lon Temperature Gradient Modes in Helical Systems

Figure I shows the normalized real frequency orl(l)6oe* and growth rate @ilo)"* of the ITG mode as aIF

function of M for 11i = 3, 4,6, 8. Other parameters usedhere are the same as mentioned above. The real
frequencies obtained here are all negative, whichcorresponds to the ion diamagnetic rotation. The growthrate decreases with increasing M.
Corresponding to the cases for M =2,3,4,5,8, 10in Fig. 1, the profiles of the eigenfunction and thehelical drift frequency - cos [(L - Mq)O] + .i 0 sin [(t Mq)01 in the covering space (&space) are plotted in Fig.
2. The regions where the helical drift frequency isTh=3{,}--'----'Y'" rli:4 L-lh()
nC)

=d(,)

(a)

t\tr

n.:6\*

ft:8-3l0M

positive (negative) corresponds to bad (good) curvature.
As M increases, the eigenfunction p becomes more(.)

0.6.*

rippled in the covering space and has a larger amplitudein the good curvature region, which is related to thereduction of the growth rate for larger M.
In the cases with large M llke M = 8 (CHS) and M= 10 (LHD), a positive growth rate for the ITG modecannot be found for 4r < 4, with the other parameters asgiven above. Unstable ITG modes are found for M = 8and 10 with the very large temperature gradient (or verysmall density gradient) tli = 6, 8 and thgy have more

0.5

o0.4

cd0.3

I0.2

&Bo 0.1(,-negative frequencies and more rippled eigenfunctions asxff(b)

shown in Figs. I and 2.

l0M
Fig. 1 The normalized (a) real frequency a,/a"- and (blgroMh rate at'1a,. of the ITG mode as a functionof M tor various 4is. Here Q = 2, ke Pn = 0.75, TilT"= 1, L,lRo = 0.2, L,en/r = 0.2, e, = 0, and 5 = -1.

4. Conclusions
In the present paper, we have derived a kinetic
integral equation for the ITG modes in helical systems.
The ITG mode equation was numerically solved for astraight herical system with the poloidal polarity number
L = 2, and the effects of the toroidal polarity number M
on the dispersion relation and the mode structure of theITG mode were studied. Field ripple with larger M

Japanese Ministry of Education, Science and Culture.

References

tll W. Horton, M. Wakatani, and J.A. Wooton, A/P
Conf. Proc. 284, U.S.-Japan Workshop on Ion

reduces the growth rate of the ITG mode. Thisstabilizing effect is understood based on the structure ofthe eigenfunciton along the field line as follows. As M
increases, the connection length between the good andTemperature G radient-Driven Turbulent
Transport, New York: AlP,1994, ch. I, p. 3.
[2] F. Romanelli, Phys. Fluids B I, l0l8 (1989).

bad curvature regions becomes shorter and thet3l J.Q. Dong, W. Horton, and J.Y. Kim, Phys. Fluidseigenfunction invades the good curvature region, whichleads to the stabilization.
In this work, the toroidal ripple in the magnetic
field strength, the nonadiabatic electrons, and thetrapped ions are not considered. These effects on theproperties of the ITG modes are studied as a future task.

B 4,1867 (1992\.
t41 J.Q.Dong and W. Horton, Phys. Fluids B 5, 1581

(l993).

[5] J.L.V. Lewandowski, Plasma Phys. Control. Fusion
40,283, (1998).

Acknowledgments
The authors thank Dr. Rewoldt for his carefulreading of the manuscript and useful comments. Thiswork is supported in part by a Grant-in-Aid from thero-7

Kuroda T. et aL., lon Temperature Gradient Modes in Helical Systems

q=2, $--1, €n{.2,T.|Ii=I, hpr=0.75.

'* \rti-3,M-20tco{ (LMq)e )+30 sir{ (LMq)0 )Fig.2 The profiles of the eigenfunction @ and the helical drift frequency * cos[([ - Mql9l + 60 sin[([ -Mql1lin the covering space (&spacel tor M =2,3,4,5,8, 10. Other parameters are the same as in Fig. 1.