J. Plasma Fusion Res. SERIES, Vol.5 (2002) 36-41

Theoretical Studies on Neoclassical Tearing Modes
and Their Stabilization
YU Qingquan, CUNfeR Sibylle and LACKNER Karl
Max-Planck-Institut filr Plasmaphysik, EURATOM Association, Garching, Germany(Received: ll December 2001 / Accepted: 10 April2002)

Abstract
Numerical simulation results for the nonlinear evolution of neoclassical tearing modes (NTMs) andthe stabilization of NTMs by localized RF current drive are presented. It is shown that the saturated
island width does not increase linearly with the local bootstrap current density fraction. The NTMs arefound to be stabilized by a localized RF current drive, if a sufficiently large RF current is deposited at therational surface. However, the modulated current drive to deposit the RF current at the o-point of theisland has approximately the same stabilizing effect as a continuous RF current drive.

Keywords:
neoclassical tearing mode, magnetic island, nonlinear growth, RF current, modulated current drive,
stabilization

1. lntroduction
It is well known that classical tearing mode
instability is driven by a positive A', the tearing mode

perturbation of the equilibrium current density profile.

index determined by the plasma current density profileand the mode numbers [1]. For a high temperaturedensity at the island's o-point and decays away from thetokamak plasma, the perturbed bootstrap current usually

positive magnetic shear, similar to the usual island withan additional noninductive cuffent drive near its o-pointin the direction opposite to the equilibrium plasma

The perturbed current density takes the negative valueof the original local equilibrium bootstrap current
island. This perturbed current is destabilizing forplays an important role in the nonlinear growth of thetearing modes [2,3]. The inclusion of the perturbed
bootstrap current leads to the so called neoclassicaltearing mode (NTM). If the fraction of the bootstrap

curTent.

In recent years both experimental and theoreticalefforts have been devoted to NTMs. In the present paper
numerical modelling results on two aspects of NTMs,

current density is sufficiently large, the saturated
magnetic island width is determined by perturbed
bootstrap current and A'. NTMs have been found to be
the most severe limitation of the achievable plasma

their nonlinear growth and saturation and theirstabilization by localized RF current drive, arepressure in tokamaks [4-8].

presented. Two dimensional transport is included in the

The driving mechanism of a NTM can be easily
understood for a sufficiently large magnetic island. Due

calculations which is essential for modelling NTMs. It
will be shown that the saturated island width is notto the large ratio between the parallel and the

proportional to the local bootstrap current density
fraction for a sufficiently large bootstrap current
fraction. NTMs are found to be stabilized bv localized

perpendicular transport coefficients the plasma pressure

profile is flattened inside the island, and the bootstrap
current disappears there. This corresponds to a

@20O2 by The Japan Society of Plasma
C o rre s pondin g autho r' s e -mail : q iy @ ipp. mp g. de

Science and Nuclear Fusion Research

36

YuQ. et al., Theoretical Studies on Neoclassical Tearing Modes and Their Stabilization
RF current drive with an accurate RF wave deposition.
However, a modulated current drive to deposit the RF
current at the island's o-point has approximately thesame stabilizing effect as a continuous RF current drive.

0.10

0.08

2. Nonlinear evolution of NTMs

0.06

The equations describing the NTMs are Ohm's law,

the equation of motion, and the pressure evolutioneequation,=(l)!*tvO=E-n(i- i,),o(*. , v)v'O= B'vi + pttvoy

0.04

0.02
,

(2)

0.00
0.000

0.010

t

;($. , v)n=v (xnv,r)+V.(7,V.r) + e,

0.005

Fig. 1 The normalized 4/3 island width versus normalized time for a local bootstrap current density fraction of 0.22 (solid curve) and 0.038 (dotted curve).

(3)

where B=Bot+YYLe, and v=-YQxe, are the magnetic
field and velocity, and Y and @ the magnetic flux function and the stream function, respectively. j is the cur-rent density along the e, (toroidal) direction, jo=-gd;/\)dpldr the bootstrap current density. g is afunction of the minor radius r depending on thecollisionality, *r/R the inverse aspect ratio, Bo thepoloidal field, p the mass density, p the viscosity, p thelimit v">>v1.ft6, Eq. (4) reduces to )(il=X1".In the oppo-

pressure, Q the heating power, E the equilibrium electric

magnetic island width (normalized to the minor radiusa) is shown in Fig. 1, where the solid (dotted) curve isobtained with a local bootstrap current density fraction
0.22 (0.038\ at the rational surface. The time t issite limit vs<<vrekb, f,17v7"/k6 [9]. The 711 given by eq.
(4) takes its maximal value at the rational surface anddecreases away from it.

The nonlinear time evolution of an mln=4/3

field, and Z the transport coefficient. The subscripts ll
and i- denote the parallel and perpendicular components,

respectively.

normalized to the resistive time tR=a2Uol n.
a2 | r*, 7 t=2.4a2 I t*, and, S=r,.rl t a=2x106 ar e

Equations (1)-(3) agree with the reduced MHD
equations if the bootstrap current and the parallel andperpendicular transport terms are neglected. The
equilibrium toroidal magnetic field is approximated to
be a constanl, and the toroidal mode coupling is

X tt"=8.5x10e
taken, where ?A is the Alfven time. It is seen that the 413

mode is stable for a low bootstrap current density
fraction but grows and saturates for a higher one.

In Fig. 2 the saturated mln=4/3 island width isshown as a function of the local bootstrap current
density fraction at the rational surface. The saturated

neglected. The bootstrap current is the only toroidal
effect included. Since the difference between the particle
and the heat transport is not distinguished, eqs. (l)-(3)are valid only 1or large islands.
According to previous theories [9,10], thefollowing form for 711 is utilized,

Xr=fr"flisland width increases strongly for lower local bootstrap
current density fraction but slower for a larger one. Sucha behavior suggests that, in addition to the m/n=413

helical bootstrap current perturbation, additional
nonlinear effects are also involved in the mode

(4)

saturation mechanism.
In Fig. 3 the local current density profile is shown.
The solid curve shows the original equilibrium current

where yx"-3.16v7""/v" is the classical parallel electron
thermal transport coefficient and kb=Bl.klB is the parallel wave vector, v7"=(T"lm")tt2 is the electron thermalvelocity and v" is the electron collision frequency. In thedensity profile, and the dotted curve is the m/n=O/O

current density profile at island's saturation,corresponding to the solid curve of Fig. I at r=0.01?n.
37

Yu Q. et al., Theoretical Studies on Neoclassical Tearing Modes and Their Stabilization

surface shifts inwards towards the magnetic axis, andthe local magnetic shear is increased. Such a change inthe m/n=0/0 current density profile and the 4 profile isfound to have a more significant effect on the saturated
island width in a small magnetic shear region [11].
It is therefore clear that, the disappearance of thebootstrap current inside the island leads to both a helicalcomponent and a mln=010 component current density

0.10

0.08

0.06

o
=

0.04

perturbation. The helical component is destabilizing,
while the m/n=010 component, leading to a new
equilibrium current profile, is stabilizing. This explains
the slower than linear increase of the saturated island
width in the large bootstrap current density fraction
region, as shown in Fig. 2.

0.02

0.00

0.0 0.1 0.2 0.3

0.4
bootstrap current density f raction

Fig. 2 The saturated 4/3 island width versus thelocalbootstrap cu rrent density fraction.

ootl

3. Stabilizing NTMs by localized RF current
drive
Because NTMs limit the confinement of tokamak
plasmas, one should try to stabilize these modes. A
straightforward stabilization method is to use localized
RF current drive to fill the current hole caused by thedisappearance of the bootstrap current inside the island.
It has recently been shown on ASDEX Upgrade that
localized ECCD can reduce the mode amplitude to zero

-0.20

E

[r2,r31.

'6coOhm's law is modified to the form

EE

To model the effect of RF current on NTMs,
-0.40

* *,

o

.YV= E - n(i - i"- i^) ,

(s)

:J

C)

where ja is the RF drive current source term.

The RF driven current is caused by the asymmetricheating of electrons in the velocity space. The resulting
fast electrons have a much longer slowing down timethan the thermal ones. The corresponding time scale isusually in the range 0.1-lms for ECCD, and evenlonger for LHCD for high temperature tokamak plasmas
t10,141. The fast electrons will diffuse in space while
they slow down, and therefore the driven current profile
depends on both the wave deposition profile and the fastelectron transport. In the presence of a magnetic island
both the parallel and the perpendicular transport affectthe profile of the driven current. To take into accountthese effects, the fast electron density is described by thetwo dimensional transport equation [10]

-0.60
0.3

o.4

0.5

0.6

ila
Fig. 3 The original equilibrium current density profile(solid curve) and the mln=OlO current density pro-file at island's saturation (dotted curve), corresponding to Fig. 1 at t=0.01.

The equilibrium plasma current is in the negative e,

direction in the opposite direction to the toroidal
magnetic field, and therefore it is negative. The
equilibrium rational surface is at r"=Q.46a. The local ml
n=0/0 current density and the current density gradient isreduced around the rational surface due to zerobootstrap current inside the island. The change of thelocal current density profile leads to a corresponding
change of the local safety factor q proflle. The rational! =v '(xu,Y un,) +Y '(tt',Y 'n,)+ vr(nr"- rr) ,(6)

where n1 is the fast electron density, )(n1 and )( rr are the38

YtQ. et al., Theoretical Studies on Neoclassical Tearins Modes and Their Stabilization
parallel and perpendicular transport coefficients of thefast electrons, and yal is the slowing down time of thefast electrons. n1, is the fast electron source due to the

0.10

RF waves given by [10,14]

I l. .. \21nf,= nra"^p'| l- zl ' :.' o' | | n (nt wmlI ,I wa. I I

^,

(s

Q)

=

0.05

where rzL6 is the magnitude of the source, w6. is the radial half-width of the fast electron density profile, and
16. is the deposition radius where ni, has its maximal

value.

The square box function II(ho,Lh) in eq. (7) takes
into account the effect of island rotation [0,14]. We
assume that the island does not rotate, and its o-point

0.00
0.00

0.01 0.02 0.03t

and x-point are at the helical angle h=m9+nfuU and *n

respectively, while the instantaneous wave deposition
width along the helical angle, L,h=mL9+nLQ, rotateswith respect to the island at an angular frequency rrr.
II(ho,Lh) is defined as II1ho.Lh1=l for lh-hol<Ah andhon<ho1ho11, and II(hs,L,h)=0 elsewhere, wherc, ho=at

Fig. 4 The normalized 3/2 island width versus the normalized time before (solid curve) and after (dottedcurve) RF current is turned on.

and fton (/r"s) is the helical angle at which the RF wave isturned on (off). When hon=-7t12 and ho1pnl2, the wave

71"=2.lxl0ea2lTp are taken. The minor radius of the RF
current peak, 16", is chosen to be at the q=312 rationalsurface r.. It is seen that a sufficient large RF current
can stabilize the NTM. The MCD can also stabilize the
3/2 mode with the same parameters as given above.
In Fig. 5 the time evolution of the island width isshown after the RF current is turned on at t=0.01?n,with 16./10=0.01 for both the MCD (solid curve) and the
NMCD (dotted curve), and all the other parameters
being the same as in Fig. 4. It is seen that the island
width is reduced to approximately the same level for the
MCD or the NMCD case. The reason for such results isthat, although the RF source cunent is modulated for the
MCD to deposit the RF current at the o-point of theisland to have a larger mln=3/2 component, for the
NMCD the mln=3/2 component ofi6 is also formed dueto the geometry and the transport, since the fastelectrons generated near the x-point of the island diffuse
along the field lines to a larger area around the island,
and the driven current density there is therefore small.
While the fast electrons generated around the o-point of
the island can only diffuse away by perpendiculardeposition and the resulting fast electron source arelocated around the o-point of the island, and we will callthis case the modulated current drive (MCD). When
hon=-lt and ho1pv, the fast electron source rotates along
all the helical angle corresponding to a continuous RF
current drive in time, and we will call this case the nonmodulated current drive (NMCD).

Since the driven current is caused by the fastelectrons, we assume the driven current densityTi to be

proportional to the fast electron density, ja=cflr. The
total driven curent 1o is obtained by integratingTi overthe plasma cross section. The constants c; and zs.6 in eq.
(7) can be determined from Ia. Similarly, the source
current, 10", can be obtained by assuming the source
current density ja"=crnr", and integrating j6. over theplasma cross section. The half width w6. and the total
RF source current 16, can be obtained from a ray-tracing
and Fokker-Planck code [4].

The growth and saturation of an mln=3/2 magnetic
island driven bv a perturbed bootstrap current is shownby the solid curve in Fig. 4. The equilibrium bootstrap

transport which is much slower than the paralleltransport. Therefore, for NMCD the driven current
density is also larger at the o-point of the island thancuffent densitl, fraction is 0.14 at the q=312 surface
Iocated at e0.58a. After the RF current is turned on at
t=0.01?n, the time evolution of the island width isshown by the dotted curve for the NMCD, where /6,/
Ip=O.O2l, wa"/ a=O.O4, Xr23a2/ rp, I,Lt =3.0a21 rs,
Lh=0.482, a>Sxl}alrp, v;=4x7}3/r*, and, 7x1=7u withelsewhere, and the mln=312 componentTi is close to thatof MCD. The slighrly larger stabilizing effect for the
NMCD is due to a larger change of the local mln=Ol}
component current density gradient, since 16=10" for the39

YuQ. et al., Theoretical Studies on Neoclassical Tearing Modes and Their Stabilization
0.11

0.11

0.09

I

0.10

g o.o7

(5

3

trll
la

!,

\l

la

t,

/LR ",,
0F10 3t-

0.09
0.05

0.08
0.010

0.015

0.03
0.010

0.020

o:0

0.015 0.020 0.025
tt

Fig. 6 The normalized island width versus the normalized time for a*1lalra (soild curve), 103/zr (dottedcurve), and 0 (dashed) after RF current is on.

Fig. 5 The normalized island with versus the normalized

time for MCD (solid curve) and NMCD (dottedcurve).

In Fig. 7 the normalized island width is shown as a

NMCD and 16=0.J7u. for the MCD.
Usually a magnetic island rotates with respect tothe laboratory frame with olvp>I. However, when c/v5<1, Ia will reach 16" in one cycle, and a larger
stabilizing effect is achieved as seen form Fig. 6, where
the time evolution of the magnetic island width is shown
for rrrl0a/rn (solid), l0r/r* ldotted) and 0 (dashed),
respectively, for.|6,/10=Q.QQ7 and NMCD. All other input
parameters are the same as those for Fig. 4. It is seen
that, for a larger a, or|}altp, the island width decreases
to a steady value similar to Fig. 5. When a=5x104/rn.

function of the normalized time for Lh=O.4 (solid curve)and 0.8 (dotted curve), respectively, where crO and 16,/
1p=0.014 are taken, and all the other input parameters
are the same as those for Fig. 4. It is seen that the island
width decreases as Ay'l decreases due to a larger helicalcomponent of j6 for a smaller Aft.

The major difficulty encountered in theexperiments of stabilizing NTMs using localized current

drive is to identify the exact location of the rationalsurface. If 16. is shifted from r,, the stabilizing effect isweaker as shown in Fig. 8, in which the time evolutionof the island width is shown for dr-(r6.-r.)/a=0.005

the result is approximately the same as that with 104/rp.

For a smaller a, oFl03lla., the island width oscillates intime because of the rotation of the RF culrent deposition
along the helical angle. The time averaged island width
is the same as that with olvp>|, but a minimal value ofthe island width exists which is helpful for thestabilization, as the perturbed bootstrap current is not(solid curve), 0.02 (dotted curve) and -0.015 (dashedcurve), respectively, where 16./10-0.03 is taken. It is seenthat the suppression of the NTM is possible if dr issufficiently small.

able to drive the mode unstable for an island width
smaller than the seed island width [8]. For arO the RF
current source is always on and localized around the opoint of the island with a deposition width Aft along thehelical angle, and 1a=1a, in a time scalev;1, which corresponds to a larger mln=312 component

4. Discussion and Summaryofj6 and therefore a stronger stabilizing effect.
For rrrlv1>>I, no significant effect of A/r on thestabilization has been found for both MCD and NMCD
with Ah varying from 0.1 to 1.0, while for rrlv61, thestabilizing effect is found to be larger for a smaller A/2.

For a high bootstrap current density fraction, theIn the present paper it is shown that the increase of
the saturated island width with the local bootstrap
current density fraction is slower than linear because of
the change of the m/n=0,/0 current profile due to thedisappearance of the bootstrap cuffent inside the island.
saturated island width tends to saturate. The numericalresults agree with the experimental observation [7].

The NTMs are found to be stabilized by localized
RF current, if the RF wave deposition location is40

YuQ. et al., Theoretical Studies on Neoclassical Tearinq Modes and Their Stabilization
RF current deposition and the island. In this case if the

0.11

minimal value of the island width is smaller than theseed island width, the mode will be stabilized [8]. When

alv5l, a smaller instantaneous wave deposition width

0.09

Lh along the helical angle leads to a larger stabilizing
effect. These results suggest that a slow rotating NTM is
more easily to be stabilized by the RF current, especiallywhen Afr is small.
In conclusion, our numerical results indicate that:(1) The saturated island width tends to saturate for highbootstrap current density fraction.
(2) A continuous RF current has about the same stabilizing effect as a modulated one.

g o.o7
0.05

0.03
0.010

0.015

0.020

t
Fig. 7 The normalized island width versus the normalized time for Ah=0.4 (solid) and 0.8 (dotted).

Acknowledgment
Q. Yu was partially supported by the Institute of
Plasma Physics, Chinese Academy of Science (CAS)and the "one hundred elite" project of CAS during thework presented.

0.08

References

.:

tll

dt=0.02[2] National Technical Information Service Document,

0.06

No. D886008946 (University of Wisconsin Plasma
Report UWPR 85-5 by W.X. Xu and J.D. Callen).
t3l R. Carrera, R.D. Hazeltine, and M.Kotschenreuther,(U

3 o.o+:i""""""";;;;

Phys. Fluids A29,899 (1986).

t4l Z. Chang, J.D. Callen, E.D. Fredrickson et al.,Phys. Rev. Letts.74,4663 (1995).

o.o2

t5l H. Zohm, G. Gantenbein, A. Gude er al., Nucl. Fu-

dr=0.005

0.00
0.000

H.P. Furth, J. Killen and M. N. Rosenbluth, Phys.
Fluids ,{6, 459 (1963).

0.005

sion,41, 197 (2001\.
0.010

tFig. 8 The normalized island width versus the normalized time for de0.005 (solid), 0.02 (dotted) and-O.015 (dashed).

[6] R.J. La Haye, L.L. Lao, E.J. Strait er a/., Nucl. Fusion 37, 397(1997).
[7] S. Giinter, A. Gude, M. Maraschek et al., Plasma
Phys. Control. Fusion 41,767 (1999).

t8l S. Giinter, A. Gude, M. Marschek er a/., Nucl. Fusion 38, 1431 (1998).

l9l Z. Chang and J.D. Callen, Phys. Fluids B.4, 1167
(ree2).
sufficiently accurate. However, there is no advantage tomodulate the current drive to deposit the RF current at
the o-point of the island, since the transport along themagnetic field in the island geometry will lead to alarger RF current density at the o-point than that at thex-point of the island. This agrees with the experimental
results that MCD has approximately the same stabilizing
effect as a continuous RF current [2]. Our numericalstudies also show that, for alv5l the island width
oscillates in tinre due to the relative rotation betwee thetl0l Q. Yu, S. Giinter, G. Giruzzi et al.,Phys. Plasmas
7,3r2 (2OOO).
[1] Qingquan Yu and Sibylle Gtnter, Plasma Phys.
Control. Fusion 40, 1989 (1998).

[12] H. Zohm, G. Gantenbein, G. Giruzzi et al.,Nucl.
Fusion 39,577 (1999).
tl3l G. Gantenbein, H. Zohm, G. Girttzzi et al.,Phys.
Rev. Letts. E5, 1242 (2000).

[14] G. Giruzzi, M. Zabiego, T.A. Gianakon et al.,Nucl. Fusion 39,107 (1999).