J. Plasma Fusion Res. SERIES, Vol.5 (2002) 270-273

Self-Sustained Turbulence of Current-Diffusive Ballooning
Mode and Drift Wave Instabilities
UCHIDA Morihisa and FUKUYAMA Atsushi
Department of Nuclear Engineering, Kyoto University, Kyoto 606-8501, Japan(Received: 11 December 2001 / Accepted: 10 May 2002)

Abstract
Starting from two-fluid equations, we derive a set ofequations which describes the. current diffusive
ballooning mode (CDBM) and the ion temperature gradient (ITG) mode. We numerically solve theequations toobtain real frequency, growth-rate and spatial structure ofthe eigen mode. The magnitude oftransport coefficients in a saturated state are evaluated from the marginal stability condition. We reportmodel dependence of ITG and a contribution of compressibility on CDBM.

Keywords:
turbulent transport, ITG, ballooning mode, drift wave

1. Introduction

fluctuation amplitude. Additionally, we concentrate on
the influence of the diamagnetic drift on the CDBM andignore the kinetic effect due to the finite Larmor radius

It is important to understand the mechanism of theturbulent transport driven by micro-instabilities toimprove confinement of tokamak plasma. Among(ftrPi << l).

various turbulent transport modes, we consider a model
driven by two kinds of micro-instabilities to explain theparticle and the energy transport across the magneticfield. One is the self-sustained turbulence model [] ofcurrent diffusive ballooning mode (CDBM) [2], which
has successfully described the L mode transport andimproved confinement associated with the formation of
the internal transport barriers [3]. The other one is theion and electron temperature gradient modes (ITG andETG) on which fluid [4] and kinetic [5] analyses are in

2. Basic Equations
2.1 Full two-fluid equations
We start from collisionless two-fluid equations, i.e.

the equations of continuity, motion and state, andPoisson's and Ampere's eqs. (4), (7)-(10).

0n,,
Dr'/

\+ V_, .lnoivrir+ ntivoit)+ V,,(n,-uo-,, + no,vr,,,)=o(1)progress.

We analyzed the kinetic effect on CDBM discussed
in the previous conference [6], but the role offinite drift
frequency was not clearly presented. In this paper, westart from full two-fluid equations to analyze both theelectromagnetic drift mode, CDBM, and the electrostatic
drift mode, ITG, including the turbulence coefficients
based on the self-sustained turbulent theory [1] andevaluate the transport coefficients from a saturated

Yp,xb
vrit=E,xb-sfiEhn,,lYpo,xb) 1 dv1,xb,-Corresponding author' s e -mail: uchida@ p- grp.nucleng.kyoto-u.ac.jp
270

4rn'orBo-g=t(2)@2002 by The Japan Society of Plasma
Science and Nuclear Fusion ResearchUchicla M. er al., Self-Sustained Turbulence of Current-Diffusive Balloonins Mode and Drift Wave Instabilities

mjnoj+ +Vnpr1- einsiElr=o* ro,. Yp,i+ v,i.Ypoi(3)+ y,(po,Y,,vr.,, + p,rV,,vor,,) = 0(4)*^,r0,* = -Y

* n,no,(r,,,.+) (8)11pri*=rn,no,arr(t + n,)O- TipoiYnvri, (9)2

Y A, = - trto rn o (r r,,,-Y'O=q5?

(s)vzA = -- ltoir

(6)

(10)

- r r,,.)

where n6 = noi = t?'e is assumed and ve,1 and Es are ignored, for simplicity. Now we consider the drift waveapproximation, where all fluctuations are expressed aswhere D - BlBn, the unit vector of magnetic field, E =-YQ- dAldt, the electric field, @, the electrostatic po-an eikonal type,/r(r) =l'(x)exp(i(ftry + knz)), and unperturbed quantities are constant in the direction to y or 0and z. Using the first order of the equation of motion,
vojt= -Ypoj x bl(qtnstBs), we can reduce the advection
term in eq.(4), rs;1 .Ypri + vrjL'Ypoi, to iq,nqal(l +4)Qby ignoring the time derivative.

tential, A, the vector potential, j1 = Z/4fl{)ivu +Qinuvs), the fluctuation current, Y t = -b x (D x V).

The subscript I denotes a perpendicular component ofquantities (aL= b x (a x r)), the suffix 0 the unperturbed quantities, the suffix I the fluctuation and thesuffix j the species of fluids ("t = i, e).

2.3 Turbulent transport coefficients
Based on the self-sustained turbulence theory [],we can evaluate the turbulent transport coefficients due

2.2 Reduced set of equations
The main pulpose of starting from the full set offluid equation is to examine the effects of the various
approximations, especially quasi-neutrality andincompressibility. Since handiness and accuracy ofdescription are trade-off, we reduce the two-fluid
equations in the following. Subtracting the continuity

toExBdrift,

L (v,sxYx,,l --r,v1x,,.
t!

tt^

J'rr

LtJ

Non-liner transport rate yN(i.e. life time of fluctuation)
is defined by yr = -irro + Ttk2lwhere 41 is the mode frequency. Substituting the above equation to the transportcoefficients, )Ci = k1(f)l1NBil, in the case of finite realfrequency, the transport coefficients are evaluated as afunction of the fluctuation amptitude (t')

equation of electron from the continuity equation of ion

and substituting Poisson's equation for density
fluctuations and equation of perpendicular motion, we
obtain the vorticity equation,

ft

(11)

In the following, we neglect the kinetic effect due to thefinite Larmor radius for simplicity and assume I," = Li

= Frn,L= k.

(7)where Q = 4lBslmi is the cyclotron frequency and ris
the magnetic curvature. If we assume incompressibility

2.4 Ballooning expression
Using the ballooning transformation [14], we(dpryldt = iq1ns1(l + n)o.j|, where io)*; = (Tojlobtain from eqs. (7)-(9),eflsiBs)(Yrnst)Vo, the density drift frequency and 4; d(ln Zs;)/d(ln n6), the temperature scalelength) andiot-(16'Yl)k!-ir-u,^ , ---g:-'iPt' n, ^-t -Q*"- v?)vlszi \quasi-neutrality and ignore the electron inertia term (lQ1limB^.nnBo At r,r vr"r)t+'TH(€) p', (I2)=n"<< lO"l), the fanriliar vorticity eqs. (6), (11)-(13) can bea"6(obtained. Note that the right hand side of eq. (7) vanishes if the magnetic curyature is absent (i.e. a shearedslab plasma). Remaining equations are the equation ofm, n o, (- iro - p,,,Y

parallel motion, the equation of state and Ampere's2r)-r4

v r,,,=-&p_n,n,(e#_ ,o,,)equation:
271

(13)

Uchida M. er c/., Self-Sustained Turbulence of Current-Diffusive Ballooning Mode and Drift Wave Instabilities+ino a.

'lL+ n ,)geometry including the effect of diamagnetic drift using

Q,iqeqs. (10), (13)-(15). Figure 2 shows the s - adependence of the turbulent transport coefficient divided
by the dt2 for the several magnetic shear parameters in,.'4)

where m is the poloidal mode number, pr,= pti* prc, Be

the poloidal magnetic field, B, the toroidal magnetic
field, r the minor radius, Rs the major radius and { theballooning coordinate. Taking account of the Shafranov
shift of the magnetic surface, Yl --> -m272t1, we take f2=1 + (s6 - asin()2 and H(() = K * cos6 + (s6 -= #- =l- = f. =l= =f_ =. g..._...._...;;_:_:_ :9- _+.

'-"']"'.

T/O*easinf)sin(, where s = (rlq)dqldr the magnetic shear, a= -qzRdBldr the normalized pressure gradient and K=-(rlR) (l - q-21 the average magnetic curvature.

With the marginal stability condition, lm(a) = y0, we numerically solve the ballooning mode eqs. (10),(12)-(14) to find the lowest eigen mode and evaluate theaverage amplitude of fluctuation (t'). Then we seek themaximum;" which isneeded to stabilize the fluctuation;.tfor all m.

+

3 equations(0,v',pi)-..r... g equations (Q,vilr,pi,Al)

3. Numerical Results
3.1 ITG mode in a sheared slab plasma
Figure I shows ft1pi dependence of the eigen+ - 4 equations (Q.n1,v1r,vi"v;r,pj,A)
1

0.4

0.2
krPi

frequency for several models of the slab ITG mode. A
solid line illustrates the result of the one-fluid modelFig.1 krp, dependence of the mode frequency of thederived by Lee and Diamond t41 (3 equations). A
dashed line indicates the result of the electromagnetic
full two-fluid model described by eqs. (2)-(6) (14lowest order calculated by several models, where
Doi= flo" = 2 x 10rs m't, 7i' = 12.8 keV, 70" = 3.9 keV,
Bo=4.7 T, l,=6.36 m,\i= 5, tt.=O and Ls=2.75m.

equations). A doted line shows those of reduced modelobtained from eqs. (7)-(10) without curvature (6equations). These modes have a phase velocity directedto the ion drift motion and the real frequency decreases
with the increase of kpi. The growth rate in the onefluid model once increases and then decreases with theincrease of k1pi, while that of the two-fluid models aremonotonically decreasing function.

10

s=0.1, ti=5/3+
s=0.3, Yi=5/3'+
s=0.5, Ti=5/3#
s=0.1, t=0 ""O"'
s=0.3, 1i=0 "".}"'
s=0.5, 1,i=0 "---]"".

Nbx

From this figure, we find that the differences ofgrowth rate among several models are sufficiently small
for qualitative analysis and the electromagnetic models
can describe the ITG mode in a sheard slab plasma. Ino>athe 6 eqs. model, even if we neglect the electron inertia
termand assume the quasi-neutrality in eq. (7), the resultchanges little. The difference between the reduced 6equations model and the full 14 equations model is quite

10'small. Therefore, we may adopt the reduced two-fluid

21

model to consider the coupling between the drift
instabilities and the balloonine mode.

s- clFig. 2 s - o dependence of 6ld2 to various values of themagnetic shear parameter (s = -O.5 - -0.1), where
Bo= 1 T, Do= 1 x 10rs m-s, e =8, Ro= 3 m, I = 1
keV, I. = 1 keV' Ln = 3 m and r= 0.17 m.

3.2 CDBM with compressiblity
Next we consider the CDBM in the toroidal
2'72

Uchida M. er al., Self-sustained Turbulence of Current-Diffusive Baltooning Mode and Drift Wave Instabilities

the case without compressibility (yi = 0), doted lines,

turbulence effect. Additionally, the coupling between
toroidal ITG mode and the CDBM is an interesting
problem in progress.

and with compressibility (yi = 5/3), solidline.
Incompressible result corresponds to the previous
analysis [3,6] and are expressed by the relation, I = F(s,Odt2 Pl. With compressibility, the turbulent transportcoefficients are about one order higher than those
without compressibility. The net value of 7" reaches 5 10[m2ls] and is close to the experimental results. The
increase of 7" comes from that the role of the electron
thermal diffusivity changes from suppression of theAcknowledgments
This work is supported by Japan Society for thePromotion Science (Japan).

References

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4. Summary and Discussion
To describe both the electromagnetic (CDBM) andthe electrostatic (ITG) drift waves, we obtain a reduced
set of equation starting from the two-fluid equations. We
confirm that the full and reduced two-fluid models
correctly describe the ITG mode in asheared slabplasma. Taking account of the tokamak configurationand magnetic curvature, we derive the current diffusive
ballooning equation with compressibility. From

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the electric field felt by the ion, we need to solve thePoisson's equation. For general analysis, it is necessary
to start from the gyro-kinetic equations including thet041 (L971).

zI3