J. Plasma Fusion Res. SERIES, Vol.2 (1999) ll7-120

Analysis of Current Diffusive Ballooning Mode in Tokamaks
UCHIDA Morihisa, FUKUYAMA Atsushir, ITOH Kimitaka2, ITOH Sanae-I.3 and YAGI Masatoshi3
Faculty of Engineering, Okayama University,tDepartment of Nuclear Engineering, Kyoto lJniversity,
2National Institute
3Research Institutefor Fusion Science,

for Applied Mechanics, Kyushu lJniversity(Received: 9 December 1998 / Accepted:21 July 1999)

Abstract
The effect of finite gyroradius on the current diffusive ballooning mode is examined. Starting fromthe reduced MHD equations including turbulent transports, coupling with drift motion and finite
gyroradius effect of ions, we derive a ballooning mode equation with complex transport coefficients. The
eigenfrequency, saturation level and thermal diffusivity are evaluated numerically from the marginal
stability condition. Preliminary results of their parameter dependence is presented.

Keywords:
ballooning mode, current diffusivity, turbulent transport, finite gyroradius effect\_1. lntroduction

t+ +io.i - Ir,vilY".Q =The self-sustained turbulence model [] of the\ot

current diffusive ballooning mode (CDBM) is one of themost promising candidates to explain the particle andenergy transport in tokamaks. This transport model has
successfully described the L mode transport andimproved confinement associated with the formation of
the internal transport barriers [2]. Since the previousIV,, 7,r * (D xr; 'Y (i, + i),(1)*i -v,,0=- r[^v,,6,- Lviir, e)l r"- r"v'ri"+r '(v' 6rY'r*)=0, (3)analysis [3] based on the reduced MHD equations [4]neglected the finite ion gyroradius effect, we could notseparate the transport coefficients of ions and electrons.
In this paper, we describe the formulation and the resultsof numerical analysis of the ballooning mode equation
which includes the coupling with the drift motion andthe finite gyro radius effect.

fi n, - r,v'ri, +t (o, r v, p,o) = o, (4)pV2t; =i1 ,(5)where ?a - qRlv',lri the ion viscosity, Tiand 7" the ion

and electron thermal diffusivities, l. the current

2. Ballooning Mode Equation

diffusivity due to the electron viscosity p" and other notations are standard. The streaming function for ions p

We start from a set of reduced MHD equations, i.e.

the equation of perpendicular motion of ions, theextended Ohm's law, the energy equations of electrons
and ions [5],is reduced from the electrostatic potential fr by a factor

Corresponding author's e-mail: uchida@ nucleng.kyoto-u.ac.jp
117@1999 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchUchida M. et a/., Analysis of Current Diffusive Ballooning Mode in Tokamaks

where n is the toroidal mode number and f = ofrrzlc2.
When (fl'z) is smaller than critical values, we take thatthe square roots have negative imaginary values so thatthe turbulent transport coefficients vanish for (Q') = O.
A pure imaginary value for small (fr'z) implies nonlinear
frequency shift rather than diffusion. We should noteof Ao(b) = 1o(b)e-b where 1s is the modified Bessel function of the first kind and b = k2Lp2;. Length and time arenormalized by the minor radius r and the poloidal
Alfv6n time ?A, respectively. Using the ballooning
transformation [6], we obtaingf'd4 -ia+),naqaf2 d4\.(- i(,)*"-iat+7"n2q2a^dtLo- ia + 7"n2 q2f- ito + y,n2 q2fthat the electron thermal diffusivity 7" is less than 76iby
a factor of A6 for large ( Q'z;. Wlttr the increase of thef )aperpendicular wave number kt - ne, the finite

gyroradius effect reduces 71.
With the marginal stability condition (Imal = 0), we
numerically solve the ballooning mode equation (6) tofind the eigen modes and evaluate the real part of theH(D O+(io-ia.,- F,n2q2 7lflt'op=9,(6)where o = -q2R(dB/dr) is the normalized pressure
gradient, s = (rlq)(dqldr) the magnetic shear and 4 theballooning coordinates. Taking account of the Shafranov
shift of the magnetic surface, we take / =l + (sn asinq)2 and H(rD - K + cos4 + (r4 - asin4)sin4
where rc= (r/R)(1 - llq2) is the average magnetic curvature. The first term in eq.(6) has a stabilizing effect due
to the bending of the magnetic field and is reduced by
the current diffusivity ,1,. The second term coming fromthe pressure gradient destabilizes the mode in the bado. 120.08

o.04

curvature region and is reduced by the thermaldiffusivity. The last term represents the stabilizing effectof r0*1 and the ion viscosity.
The transport coefficients are determined by therelation for the nonlinear transfer r:ite T, e.g. X"= (Q')ly
and y- -iot + k2t7., where the bracket ( ) denotes theo.02

o.00

onFig. 1 Spatial structure of the eigen mode for theparameters s = 0.4, a = 0.4, Q = 20, plr = 0.01 andna =134'

average over the wave number and the 4 dependence of(f2) is neglected in the present analysis [1]. Since mode

frequency has a real part, the transport coefficients are

written asu,=t[=,'*\> " Y /I G')\' tl,lrd*n'o't\''l/\lv-\' n'q'

- 4lX",-u*\nooof'*ya(€+n'!'),
2s,5I

no qo-ft

s.4A)
x -'-

xo

M:a-a, . .'l +lF;=\ I --:-'l 4\n'q'f- "r. (8)^Z n'r'fn'q'V)-flQ^u*s.0 |n'q'f't20

100(e)140nqFig.2 nq dependence of the thermal diffusivity 7"

derived from the marginal stability condition forthe parameters s = 0.4, a = 0.4, q = 20 and plr =-f-'tro>q-r
n-0.01.

118

Uchida M. et al., Analysis of Current Diffusive Balloonins Mode in Tokamaks

mode frequency at and the average amplitude of thefluctuation (Q').fig. I shows an example of the CDBM
eigenmode for s = 0.4, a = 0.4, q - 20, p/r= 0.01 andne = 734 . It has a small negative real frequenc! (@, =-0.05) and propagates in the direction of ion drift
motion. The saturation level of the fluctuation is (fr2) =6 x l0-5 and this mode is unstable for smaller (0r). The
eigen mode structure indicates that the dominant realpart localizes in the outer region (4 = 0), while the outof-phase imaginary part in the inner region (n = tn).
The saturation level of the average fluctuation
amplitude depends on the mode number nq. Figure 2
illustrates the n4 dependence of the electron thermal
diffusivity 7" derived from the saturation fluctuation
amplitude ( @'?). The maximum value of N,e near nq =\0.8

1.6cx,Fig. 3 Contour of the electron thermal diffusivity 7. onthe (s- o) Plane lot Plr= 10'3'

134 gives the thermal diffusivity necessary forstabilizing the ballooning modes for all nq.

3. Parameter Dependence
In order to study the parameter dependence of themode frequency and the thermal diffusivity, we look forthe maximum value of 7" by numerically solving theballooning mode equation (6) for wide range of nq andfor various values of s and a. We have been much
benefitted by parallel processing on a PC cluster of 8CPUs with a numerical code using the Message Passing
Interface (MPI) library.
The (s, a) dependence of the electron thermal
diffusivity is shown in Figs. 3 and 4 for { = lOa and q =103. Fig. 3 depicts the case of small gyroradius, p;/r =l0-3, while Fig. 4 the case of larger gyroradius, p1lr =l0-2. The (s, a) dependence in Fig. 3 is almost same as-0.4

ilil**'t.20.4

r.62..O0,Fig. 4 Contour of the electron thermal diffusivity 7" onthe (s - a) plane for plr = 0.01.

that of pilr = 0 [2]. For larger pi,7" is enhanced,
typically, by a factor of 3 and the shear dependence isslightly reduced.
For small d(d < 0.4), 7. is a strongly increasing
function of a(11'"e q3t27 and decreases for small andnegative values of s. For larger value of a(a> 0.4), 7"
is enhanced in the stronger shear region (s > 0.5) where
the ideal ballooning mode is unstable. The reduction ofy" in the weak and negative shear region is more
prominent for larger a (larger pressure gradient). Thesebehavior is similar to the result of the previous analyses
without coupling to the drift motion, 0)*i= o)*e= 0. Oneof the reasons is the large q value in the present

0.8

0.4

0.0

-0.4

calculation (4 = 103). For realistic values ofq, the effectof finite al*; ood ar-. affects the mode structure, butt.2

0.8

1.6

(I

conclusive results have not been obtained yet.
The mode frequency is usually negative except inthe region, )("> lO'4, where the ideal mode is unstable.

Fig.5 Contour of the rctio )(i/)(" on the (s- a) plane forolr = 0.01.

ll9Uchida M. et al., Analysis of Cunent Diffusive Ballooning Mode in Tokamaks

Science, Sports and Culture Japan, by the collaboration
programme of the Advanced Fusion Research Center ofThe mode number ,?4 decreases with the increase of 7".
Typical value of nq is 300 for 7,= l0-5 and 100 for 7" =Kyushu University, and by the Collaboration

l0-4. As the mode number increases, the finite
gyroradius effect enhances 1" and reduces 7i. Fig. 5
indicates the contour of the ratio fily" on the (s, a)plane. In the region where a > C.4 and , 0.5. X,,
approaches to 7", otherwise yilye is less than about 0.4.

programme of National Institute for Fusion Science.

References

tll K. Itoh et a/., Plasma Phys. Control. Fusion' 36,
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l2l A. Fukuyama et aI.,Plasma Phys. Control. Fusion.

4. Gonclusion

37,6rr (1996).

Taking account ofthe gyroradius effects, we foundthat )(e of the CDBM transport model is enhanced by afactor of 3 for p/r = 10-2 and that the ratio 7i/7" isconsiderably reduced for low a(L-mode) region. In thepresent calculation, the safety factor q is limited to arather large value. The calculation with realistic valuesof 4 as well as the effect of the E x B rotation shear will
be reported in future.
This work is partly supported by the Grant-in-Aid
for Scientific Research of the Ministry of Education,

t3l M. Yagi et a/., Phys. Fluids. B 5,3'102 (1993).
t4l H. Strauss, Phys. Fluids 20,1354 (1977).
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t6l J.W. Connor and J.B. Taylor, Nucl. Fusion 17,1047 (1977\.

l7l K. Itoh et al., Plasma Phys. Control Fusion 36,279(r99q.

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