Fusion Res.
Res. SERIES,SERIES, Vol.
Vol. 6 (2004) 000-000
192–195
J. Plasma Fusion

Effect of Ion Viscosity on Neoclassical Tearing Mode
YOSHIDA Shigeki, ITOH Sanae-I.1, YAGI Masatoshi1 and AZUMI Masafumi2
Kyushu University, Interdisciplinary Graduate School of Engineering Sciences, Kasuga, Japan
1
Kyushu University, Research Institute for Applied Mechanics, Kasuga, Japan
2
Japan Atomic Energy Research Institute, Ibaraki, 311-0193, Japan(Received: 9 December 2003 / Accepted: 30 April 2004)

Abstract
Linear stability analysis of neoclassical tearing mode (NTM) is performed on the basis of four-field reduced
magnetohydrodynamic (MHD) model which takes account of fluctuating ion parallel flow and ion neoclassicalviscosity. The dependence of the growth rate on the kinetic effects is investigated. It is shown that the linear NTM isstabilized by ion neoclassical viscosity and that the stabilizing effect of ion parallel compressibility is weak in thebanana-plateau regime. It is found that not only ion neoclassical viscosity but also both ion and electron diamagnetic
effects are important for the stabilization of NTM.
Keywords:
neoclassical tearing mode, free energy source, ion neoclassical viscosity, ion neoclassical flow, plasma b value,pressure gradient, four-field model

1. Introduction

tearing parameter Dl [2]. The dependence of the growth rate
on Dl , plasma b value, kinetic effects is systematicallyinvestigated. The numerical analysis is described in Sec. 2.
Summary and discussions are given in Sec. 3.

The magnetic islands are often observed in high b (theratio of plasma pressure to magnetic pressure) tokamak
plasmas [1]. To achieve high performance in fusion plasmas,
it is necessary to understand the physical mechanism ofisland excitation and its saturation level in high b plasma andthe associated collapse phenomena. Several works on
neoclassical tearing mode (NTM) have been done [2-4] and it
is found that saturated island width, which is determined by
the competition between the driving term due to thefluctuating bootstrap current and the change in thelogarithmic derivative of the outer solution across theresistive layer Dl is not inconsistent with experimental
observations [5]. However, the onset condition or these
dynamics are not fully understood from the view point of
conventional approach. The previous analysis of NTM isessentially based on three-field model [2-4], in which theelectron neoclassical viscosity is only taken into accountassuming that the collision frequency is larger than therotation frequency of the NTM, i.e. ~ < n i . This assumption
is inappropriate for the case of the high temperaturecollisionless plasma and in the time scale of the island
evolution, where ~ > n i is held. In addition, the parallel
compressibility is also neglected. In this paper, we analyze
the linear stability of NTM based on the four-field reduced
MHD model which includes the ion parallel flow and ion
neoclassical viscosity [6,7]. Here, we call linear NTM for thetearing mode with flucutating bootstrap current andneoclassical flow damping independent of the sign of the

2. Four-field reduced MHD model equationand numerical analysis
We consider a cylindrical plasma of major and minor
radii R 0 and a with a toroidal magnetic field B0 in thecoordinate (r, i, z) . To analyze the linear stability of NTM,
the four-field reduced neoclassical MHD model [6,7] isintroduced. This model consists of the vorticity equation:

2 d 2 F + [F, d 2 F] - a d $ [p,d F]i 9992t 9= - d d92 A +n icl d 94 F(1)q2 U pi q m e neo 2 U pe,- e n ineo
- e mi n e
2r
2r
Ohm’s law:

2 (A -a 2 m e d 2 A)
mi 92tclm=- d [F - (a i + a e ) p] + a 2 mei [z ,d92 A] + h d92 A (2)mm- 4 n ecl a 2 mei d94 A + a h BS mei n eneo Upe ,©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: shigeki@riam.kyushu-u.ac.jp1192

Yoshida S. et al., Excitation of convective cell mode by universal drift wave turbulence

where rs is the rational surface. We set a = 3.0 , b = 1.0 ,q (0) = 1.2 and rs = 0.6 . In this case we have Dl= 10.6 .
Figure 1 shows collisionality dependence of the growth ratec of (2,1) mode in cases with h BS = 1.0, 2.0, 3.0, 4.0 . It is
confirmed that the fluctuating bootstrap current destabilize
the tearing mode in the banana-plateau regime (NTM), while
the destabilization effect is weak for the classical tearing
mode in the high collisionality regime.
The dependence of the growth rate on Dl is analyzed
and the result is compared with that of the three-field model.
Dl is calculated by changing q (0) and fixing a, b, rs in themodel q profile. Figure 2 shows the dependence of thegrowth rate on Dl in cases with collision frequency o e = 2.4(plateau regime) and h BS = 1.0 . The relation between n ineo,eand oe is shown in ref. [7]. The dashed line and solid line
indicate the results of the three-field model and the four-fieldmodel respectively. In the four-field case, the threshold valueof the instability Dlc (. 10.2 > 0) is a finite positive number
while c > 0 holds for Dl > 0 in the three-field case. Although
the absolute value of collision frequency where the growthrate is equla to zero, depends on the value of Dl , ion
neoclssical viscosity is an essential factor to the stabilizationof the NTM, which does not include in the three-field model.

ion parallel momentum equation:

2 v + [z , v ] = -d p + 4 n cl d 2 v
i92tm-n ineo Upi - me n eneo Upe ,(3)iand the electron continuity equation:

2 p + [z , p] = bt - d (v + a d 2 A) + h cl d 2 p
=(4)
9
9
92t
q2Umpe- a m e e n eneo,i2r G
where F = z + a i p is the generalized potential.
The closed set of equations (1)-(4) is called as ‘four fieldmodel’ in this paper. On the other hand, the approximations
d v = 0 and Upi = 0 give the three-field closed set ofequations, whereq
Upi = v + e 2F ,
2r

(5)

q
Upe = v + a d92 A + e 2 (F - a p).
2r(6)

Upi and Upe are ion and electron fluctuating neoclassical flowwhich include the trapped particle effect in the tokamak
plasma. The poloidal Alfven time a/( ev A ) and the minor
radius a are used for normalization [7]. The variables offour-field {z, A, v , p} are the fluctuating electrostatic
potential, vector potential parallel to the magnetic field,parallel velocity and electron density, respectively. In thismodel, the ion and electron temperature T i and T e areassumed to be constant and uniform. The coefficients{n icl , n cle , h cl, h 9cl }are classical ion viscosity, electron viscosity(hyper-resistivity), parallel and perpendicular resistivity. n neoeand n ineo are neoclassical electron and ion viscosity expressedby the interpolation formula [6,7]. a is the normalized ion
skin depth ( c/a~ pi ), a e = a/( 1 + x) and a i = ax/( 1 + x)
with x = T i /T e . The last term of right hand side of equation(2) is the fluctuating bootstrap current term, where h BS is aparameter to control the absolute value of flucutatingbootstrap current arbitrarily. We set h BS = 1.0 as a default
value. b is the local plasma beta value and bt = b /( 1+ b ) .
q and e indicate safety factor and inverse aspect ratio
respectively. On the basis of the liniearized four-field model,
the dependence of the growth rate of NTM on various
physical effects is investigated solving eigen value problem.
To check the validity of the code, we also developed theinitial value code and compared the growth rate obtained by
two codes.
At first, tearing mode is analyzed changing the absolutevalue of fluctuating bootstrap current.
The model safty factor profile and the pressure profile
are given by

0.0014BS

h =2.00.001

h =3.00.0008

h =4.0BS

γBS
BS0.00060.00040.00020-0.0002
0.1

1

10100

νeFig. 1 The dependence of c on collisionality in cases withh BS = 1.0, 2.0, 3.0, 4.0 .

0.000150.0001

3field4field5 10 -5γ0-5 10 -5-0.0001

q (r) =

h =1.00.0012

q (rs) - q (0)q (rs) - q (0)[1+(r/rs )a ] + q (0) ,2b - 12b- 1(7)bp (r) = e (1- r 2 )2,

0

51015

∆'
Fig. 2 The dependence of c on Dl in cases with three-field
and four-field models. Parameters: b = 0.01, a = 0.01,
a i = a e = a/2 are used.

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Yoshida S. et al., Excitation of convective cell mode by universal drift wave turbulence

electron neoclassical viscosity.
At first, we confirm that the NTM is destabilized by thefluctuating bootstrap current in the banana-plateau regime.
It is found that the stable regime of the NTM exists even
if Dl> 0 in four field model which implies the careful
examinations are necessary for experimental interpretations.
The threshold value, Dl c , depends on the collisionality
and we obtain Dlc . 10.2 for the case with o e = 2.4 (plateauregime). To clarify whether ion neoclassical viscosity or ion
parallel flow play the more important role on the stabilizationof the NTM, the dependence of the compressibility on thegrowth rate without ion neoclassical viscosity is investigated.
It is found that the ion neoclassical viscosity stabilizes theNTM rather than the ion parallel compressibility does in thebanana-plateau regime. The stability effect is weak fortearing mode in the collisional regime. It is concluded thatthe synergetic effect of ion neoclassical viscosity and both
ion and electron diamagnetic effects stabilize the NTM.
Finally, the finite- b effect is shown to stabilize the tearing
mode and NTM in plateau regime.
More detailed analysis is necessary to discuss the0.00154field without ion neoclassical viscosity
4field with neoclassical viscosity
3field0.001

γ 0.0005
0-0.0005
0.1

1

10100

νeFig. 3 The dependence of c on collisionality in cases withthree and four-field models. The solid line, dashedline and long dashed line indicate the results of fourfield model, three-field model and four-field model
without ion neoclassical viscosity, respectively.

The effect of ion neoclassical viscosity on the growthrate is analyzed. Figure 3 shows the dependence of thegrowth rate on collisionality. The solid line, dashed line andlong dashed line indicate the results of four-field model,
three-field model and four-field model without ion
neoclassical viscosity, respectively. It is found that the NTM
is strongly stabilized by ion neoclassical viscosity and effectof parallel compressibility is rather weak.
Next, ion and electron diamagnetic effects on the NTM
are analyzed. Essentially, two terms contribute the stability.
One is the ion diamagnetic effect proportional to a i whichappears in the vorticity equation and the other is electron
diamagnetic effect proportional to a e in the Ohm’s law.
Figure 4 shows collisionality dependence of the growth ratein cases with various values of a i and a e fixing a = 0.01. In
this case, the relation a = a i + a e is not held except a i = a e .
The solid line, long dashed line and dashed line indicate theresults of a i = a e = 0.005, a i = 0, a e = 0.005 and a i = 0.005,
a e = 0 respectively. It is found that synergetic effect of ion
neoclassical viscosity and both ion and electron diamagnetic
effects give rise to the stabilization of the NTM.
Finally, the finite- b effect on the NTM is examined.
Figure 5 indicates the dependence of the growth rate on thecollisionality in cases with b = 0.005, 0.01, 0.02 . The solidline, long dashed line and dashed line indicate the results ofb = 0.005, 0.01, 0.02 , respectively. It is found that thegrowth rate becomes small as the b increases in collisional
and plateau regime. The NTM in the plateau regime isstabilized by finite- b effect. In the banana regime, this effectis week.

0.0015αi =αe=0.005

0.001

αi =0,αe=0.005
αi =0.005, αe=0

γ 0.0005
0-0.0005
0.1

110100

νeFig. 4 The dependence of c on collisionality in cases
with/without ion and electron diamgnetic effects. The
solid line, long dashed line and dashed line indicate
the results of a i = a e = 0.005, a i = 0, a e = 0.005 anda i = 0.005, a e = 0 , respectively.

0.00150.001

γ0.0005

β=0.005
β=0.01β=0.020-0.0005-0.001
0.1

3. Summary and discussion

110100νe

The linear stability of the NTM is investigated based on
four-field reduced MHD equations in which ion neoclassicalviscosity and fluctuating ion parallel flow are taken into
account. The results are compared with those fromconventional three-field model which only includes theFig. 5 The dependence of c on collisionality in cases withvarious values of b. The solid line, long dashed line anddashed line indicate the results of b = 0.005, 0.01, 0.02,
respectively.

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Yoshida S. et al., Excitation of convective cell mode by universal drift wave turbulence

threshold value quantitatively for various current andpressure profiles. In addition, to clarify the onset of magnetic
island or subcritical bifurcation in high b experiments, thenonlinear simulation of the NTM is necessary. It is left as afuture work.

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[2] T.S. Hahm, Phys. Fluids 31, 3709 (1988).
[3] A.I. Smolyakov, Plasma Phys. Control. Fusion 35, 657(1993).
[4] C.C. Hegna and J.D. Callen, Phys. Plasmas 4, 2940
(1997).
[5] Z. Chang et al., Phys. Rev. Lett. 74, 4663 (1995).
[6] J.D. Callen and K.C. Shaing, Phys. Fluids 28, 1845
(1985).
[7] A. Furuya, M. Yagi and S.I. Itoh, J. Phys. Soc. Jpn., 72,313 (2003).

Acknowledgement
This work is partly supported by the Grant-in-Aid for
Scientific Research of MEXT Japan,by the collaboration
programmes of National Institute for Fusion Science and of
the Research Institute for Applied Mechanics of Kyushu
University.

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