J. Plasma
J.
Plasma Fusion
Fusion Res.
Res. SERIES,SERIES,Vol.
Vol.6 6(2004)(2004)119–124
000–000

Double Internal Transport Barrier Triggering Mechanism
in Tokamak Plasmas
DONG Jiaqi, MOU Zongze, LONG Yongxing and MAHAJAN Swadesh M.1
Southwestern Institute of Physics, P.O. Box 432, Chengdu, China
1
Institute for Fusion Studies, University of Texas at Austin, USA(Received: 9 December 2003 / Accepted: 3 April 2004)

Abstract
Sheared flow layers created by energy released in magnetic reconnection processes are studied with the magnetohydrodynamics (MHD), aimed at internal transport barrier (ITB) dynamics. The double tearing mode induced byelectron viscosity is investigated and proposed as a triggering mechanism for double internal transport barrier (DITB)
observed in tokamak plasmas with non-monotonic safety factor profiles. The quasi-linear development of the mode issimulated and the emphasis is placed on the structure of sheared poloidal flow layers formed in the vicinity of themagnetic islands. For viscosity double tearing modes, it is shown that the sheared flows induced by the mode may
reach the level required by the condition for ITB formation. Especially, the flow layers are found to form just outside
the magnetic islands. The scaling of the generated velocity with plasma parameters is given. Possible explanation for
the experimental observations that the preferential formation of transport barriers in the proximity of low order rationalsurfaces is discussed.
Keywords:
internal transport barrier, double internal transport barrier, viscosity tearing mode, double tearing mode,
sheared flow, rational surface

1. Introduction

and confinement resulting from the formation of ITB in AT
plasmas. The required power may also be lowered throughadjusting profiles of plasma parameters such as pressure,temperature, velocity and current. Therefore, in recent years,
the conditions, especially the radial position, for triggering
the formation of an ITB have been intensively investigated intheory and experiment.
There is evidence that ITBs are preferentially formednear low order rational flux surfaces (especially, q = 2, 5/2
and 3). It was first reported from JT-60U that the radial
position of the ITBs for both ion temperature Ti and plasma
toroidal velocity vf most likely coincides with the q = 3 flux
surfaces [1]. In DIII-D too the ITB growth events are wellcorrelated with integral q, although there are several counterexamples [2]. The “type II” confinement transition in TFTR
discharges requires lower NBI power than the threshold forenhanced reverse shear (ERS) transition and often occurs
when qmin, the value of q at the surface of shear reversal,
crosses rational values, q = 3 and 5/2 in particular [3]. In
electron cyclotron heated L mode discharges on the RTP
tokamak, the electron temperature (Te) profile can be wellsimulated with a thermal diffusivity, ce model which hasalternating layers of low and high ce located near and away

Internal transport barriers (ITBs) of ion or electronenergy are the regions where corresponding temperature
gradient is reduced with respect to the values in low (L) andhigh (H) mode tokamak discharges. The latter are governedby micro-instabilities and exhibit temperature profile
stiffness.
In experiments ITBs are formed when neutral beam
injection (NBI) or radio frequency (RF) heating power ishigher than a minimum threshold in discharges. The minimum
power required depends on plasma conditions when the NBI
or RF waves are launched. Besides other advantages such as
stable for a variety of instabilities, advanced tokamak (AT)operations with non-monotonic safety factor (q) profiles havelower thresholds than plasmas with monotonic q profiles. As
a result, the AT mode of operation is highly desirable for a
tokamak reactor.
Whereas reduced turbulent transport in the stationary
phase after the formation of an ITB can be explained invoking
the sheared perpendicular rotation resu1ting from theimproved confinement within the lTB itself, the dynamics ofthe formation of ITB is not yet well understood. The
delineation of the physics of ITB triggering mechanism willassist the ultimate goal of active control of the onset, duration©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: jiaqi@swip.ac.cn
1
119Dong J. et al., Double Internal Transport Barrier Triggering Mechanism in Tokamak Plasmas

from low q rational flux surfaces, respectively [4]. In ASDEX
Upgrade reversed shear experiments, the onset of ITB is
usually accompanied by MHD activity. In the 80 discharges
studied, there is not a single one in which a clear ITB forms
without the presence of MHD activity [5]. It was observed in
JET experiments that the radial position of ITBs always
coincided with the q = 2 or q = 3 surface in low core magnetic
shear discharges. In addition, an appropriate q profile, and inparticular the location of the q = 2 surface, is essential for
triggering of ITBs. With negative central magnetic shear, theanalysis of ITB triggering reveals a correlation between theformation of the ITB and qmin reaching an integral value (q =2 or 3). The time of ITB emergence was also found to coincide with the time when qmin reaches 2 [6]. A comprehensive
recent review on ITB in tokamaks, including experiment andtheory, is given by Wolf [7].
The minimum power threshold represents an external
condition that must cause required changes in local (at theposition of ITB formation) as well as global (profile) plasma
and magnetic configuration parameters in order to form an
ITB at the position. An increase in the E ¥ B shear (in theregion of ITB formation) has been identified as one of thesechanges. In addition, it has been shown that a self-organizedlayer with the features of an ITB may exist in collisionlessplasmas. One of the necessary conditions for the formationof such a layer is a poloidal velocity corresponding to apoloidal Mach number of order unity [8]. However, we still
have to establish the relevant relationship between the position where such dramatic increase of flow velocity takes place
and plasma properties. In other words, the reason that a flow
layer emerges and an ITB forms at a specific position andnot elsewhere in a discharge has not been addressed in detail.
Especially, the observations that the ITBs form preferentiallyin the proximity of low order rational surfaces has not beensatisfactorily explained.
It is well known that low order rational flux surfaces areprone to excitation of ideal and dissipative MHD instabilities
and that magnetic energy released in the development of themodes may create significant plasma flows. Therefore, besides
fishbone oscillations, other MHD instabilities, forming
magnetic islands are proposed as plausible triggering
mechanisms for the formation of ITBs in the proximity of
low order rational surfaces in ASDEX Upgrade reversed shear
discharges [5]. External kink modes coupled to inner rationalsurfaces have been shown to be able to trigger ITBs in
positive shear discharges in JET [6]. However, the double ITB(DITB) structure observed in JET reversed magnetic shear
discharge has not been addressed in detail [9].
The linear double tearing mode (DTM) mediated byanomalous electron viscosity was studied in Ref. [10] where
the possibility for such modes to drive DITB was alsosuggested. By simulating the quasi-linear development of thismode, we demonstrate in this paper the creation of sizable
sheared poloidal flow layers in the vicinity of the magnetic
islands. The nature and magnitude of the generated flows
makes the double tearing mode a strong candidate for thetriggering of ITBs in tokamak plasmas with non-monotonic
q profiles.

2. Physics model and MHD equations
We consider a plasma slab of length a in the x-direction,with a current in the z-direction, and zero equilibrium flow
velocity V0 = 0 embedded in the standard sheared magnetic
field

B0 ( x ) = B0 y ( x ) yˆ + B0 z ( x ) zˆ,(1)where B0y(x) equals zero at x = ±xs. The stability of this initialconfiguration will be examined with respect to two-dimentional, incompressible perturbations. The vector fields are
expressible in terms of two scalar potentials: the flux functiony (x, y, t),B^ = —y ¥ zˆ,(2)and the stream function f (x, y, t),V^ = —f ¥ zˆ.

(3)With electron viscosity, the Ohm’s law becomesmm
1
E = h j - V ¥ B - e 2e — 2 j .
c
ne e

(4)

It is straightforward to write the z-component of Eq. (4) as

m m c2yc2= - V  —y +h— 2y - e e 2 — 4y ,t
4p4pne e(5)

after using Eq. (2) and Faraday’s law. Here, h is the plasma
resistivity, me is the parallel electron viscosity diffusion
coefficient, me the electron mass, ne the electron density, ande and c are, respectively, the electron charge and the speed oflight. The z-component of the vorticity equation may be
written as

 2
1(— f ) = -(V  —)— 2f +[—(— 2y ) ¥ —y ]  zˆ ,
t
4pr

(6)

where r is the mass density of the plasma. Normalizing alllengths to a, time toth = a/vA, the poloidal Alfvén time of aplasma column of scale width a, and the magnetic field tosome standard measure B0, Eqs. (3), (5) and (6) may betransformed to:y1
1= {f ,y } + — 2y - — 4y + E ¢,
t
S
R

(7)

 2
(— f ) = {f ,— 2f} - {y ,— 2y },t(8)where S = tr/th is the magnetic Reynolds number with tr =4pa2/c2h, R = tv/th is the fluid dynamic Reynolds number,while tv = 4pa4nee2/c2meme = w 2pea4/c2me, and2120

Dong J. et al., Double Internal Transport Barrier Triggering Mechanism in Tokamak Plasmas

105, S = 9.4 ¥ 105, Bc = 0.233509, z = 2.68298 corresponding
to two rational surfaces at x = xs = ±0.25. The results arechecked to be independent of xw, the grid size and the timestep.

f y y f{f ,y } =x y x yis the Poisson bracket. Assuming the perturbation potentials•f=Â f ( x, t )sin(mky),m(9)

3. Numerical results

m =1

The effective growth rate g =  lnBx(0)/t versus time isshown in Fig. 1. The run was stopped at g = 0. For initialtimes, t <~ 30, g ~ 0.04 approximates the linear growth rate[10]. After that, the quasilinear effects begin to suppress theeffective growth rate-g decreases with time and reaches zeroat t @ 78. Here, g is the growth rate of x-component of themagnetic field at the origin (x = y = 0) point. Therefore, themode keeps growing after g reaches zero as shown in Figure
2 by the kinetic and magnetic energy.
The magnetic energy

and•y = dy ( x, t ) +Ây ( x, t )cos(nky),
n

(10)

n =1

we obtain the following coupled quasi-linear equations fromthe first harmonic perturbation,dyy 1 ˆ 1 Ê d 2y 0  2dy ˆk Ê f= - Á 1 y 1 + f1+˜+˜
2 Ë xtx ¯ S ÁË dx 2
x 2 ¯-1 Ê d 4y 0  4dy ˆ+
˜ + E¢
R ÁË dx 4
x 4 ¯=18p

Ú ( B + B ) dxdy
2
x

Ú

2
y

2
È
2 Ê y ˆ ˜  dxdy,
Í(ky 1 sin ky) + ÁË x ¯ 
ÍÎ


(16)

and the kinetic energy

Ek =

(12)

Equations (11-13) are solved as an initial value problem – E¢
is chosen such that the equilibrium does not dissipate due toresistivity and viscosity.
For the magnetic field, we employ the configuration used
in Ref. [10,11],(14)z x s = sech -1[1 / (1 + Bc )].

(15)

Ú

2
È
2 Ê fˆ Í kf1 cos ky + Á 1 sin ky˜  dxdy,
Ë x
¯ 
ÍÎ


Ú(

)

(17)

as functions of time are given in Fig. 2. Here, the magnetic
field is normalized to B0y (±•), while in the final expressionof Ek the velocities are measured in units of the poloidal
Alfvén velocity. It is clearly shown that the magnetic energy
released in the reconnection process, following the development of the DTM, converts to kinetic energy and can drive
large flows.
The flux contours at the end of the run are given in Fig.
3. The flux surfaces here resemble those for resistivity DTM.

(13)

B0 y ( x ) = 1 - (1 + Bc )sech(z x ),1r (u x2 + u y2 ) dxdy
2

1
=
8p

ˆ
Ê dy 0 dy ˆ Ê  2y 1 Ê  2f1
2 ˆ
2
+
˜Á
Á
2 - k f1 ˜ = k Á
2 - k y 1˜t Ë xx ¯ Ë x
Ë dx¯¯
Ê d 3y 0  3dy ˆ-kÁ˜ y 1.
3 +x 3 ¯
Ë dx

1
8p

(11)

2
ˆ
y 1
Ê dy 0 dy ˆ 1 Ê  y 1= - kf1 Á+- k 2y 1 ˜
˜+ Á
2
Ë dx
t
x ¯ S Ë x¯2ˆ1 Ê  4y 1
42  y12- Ák
4
2 + k y 1˜R Ë x
x
¯

Em =

where

The constant Bc is chosen such that B¢0y(xs) = p/2. We do notneed to specify B0z(x) and plasma pressure P0 (x) since incompressible equations are used. The resistivity and the viscosity
are both assumed to be constant. The initial conditions for y– 1–
and f 1 are the linear eigen-functions multiplied with a small
number and dy (t = 0) = 0 [10]. The boundary conditions aredy (x) =  dy / x = 0, and the values provided by the initial––conditions such as y–1 (x) = 0, f1 (x) = f1/ x = 0 for x = ± xw,
the position of wall. The chosen parameters are k = 0.25, R =Fig. 1 Effective growth rate g = In Bx(0)/ t versus time.

3
121

Dong J. et al., Double Internal Transport Barrier Triggering Mechanism in Tokamak Plasmas

Fig. 2Magnetic energy Em (See Eq. (16)) and kinetic energy
Ek (See Eq. (17)) as functions of time.

Fig. 4 Profiles of (a) v– y and (b)v– y / x at the end of the run.

Fig. 5
Fig. 3 Flux contours at the end of the run.

The profiles of (a) v–y and (b)  v–y/ x contemporaneous–with Fig. 3 are presented in Fig. 4 while Bx for the same time
is given in Fig. 5. Two very important points emerge: 1) the
amplitude of the poloidal velocity v–y reaches the level requiredby the condition for ITB formation [8], and 2) the flow v–y> 0.5
and flow shear  v–y/ x remain at noticeable levels for x ~–where Bx is negligibly small. One cannot overemphasize the–Profile of the perturbed radial magnetic field Bx at theend of the run.

significance of the latter finding; because the velocity shear
layers are formed outside the magnetic islands on both sides,
the (magnetic) turbulence may be suppressed and DITB may
form in these layers. Here, shown are the magnitudes of theperturbations. The flow patterns have to be formed from Eqs.
(3) and (9) with the magnitudes given here.

4122

Dong J. et al., Double Internal Transport Barrier Triggering Mechanism in Tokamak Plasmas

4. Conclusions and discussion

E ¥ B stabilization and the force that drives turbulence
through instabilities like the ion and electron temperature
gradient (ITG and ETG) modes, and the trapped electron (TE)modes. The fact that the required threshold power in plasmas
with reversed magnetic shear is lower than that with positive
shear is attributed to the stabilization of the ballooning
instability, larger Shafranov shift and lower safety factor near
the magnetic axis [7]. In addition, the E ¥ B velocity shear
required to completely suppress the (ITG) modes in theformer is lower than that in the latter [12].
There is a common refrain in the literature that ITBs
occur under various conditions depending on the interplay
between the mechanisms that drive and suppress plasma
turbulence. It is quite natural to expect that the mechanism
proposed in this work is relevant only in plasmas with nonmonotonic q profiles. Other mechanisms such as fishbone
instability, mode coupling etc. could be relevant elsewhere [7].
It is even possible that the resonance mechanism does notplay a major role under certain conditions. At this stage there
is no unique or universal mechanism.
The electron viscosity induced DTM is shown to generate localized shear flows in plasma configurations with nonmonotonic safety factor. The proposed mechanism emerges
as a major contender for a DITB trigger because the generated
shear flows bear such striking qualitative and quantitative
similarity to the flows that accompany DITB formation(located in the proximity of low order rational surfaces, andjust outside the magnetic islands).

ITBs are generally accompanied by a strong E ¥ B shear
flow, low or negative magnetic shear, and turbulence suppression. It has been pointed out that E ¥ B shear flow may begenerated by a variety of mechanisms [3]. In positive magnetic shear discharges, for instance, it is the coupling of aninternal mode with an external MHD mode that leads to the
local braking of the toroidal and poloidal rotations and thenthe E ¥ B sheared flow results. This mechanism does notseem to work in discharges with central negative shear. Thevery stimulating observation that may provide supporting
evidence for the mechanism proposed in this work is that two
radially separated ITBs (DITB) simultaneously exist andfollow the two q = 2 surfaces in a section of JET discharge
pulse 51573 [9]. Moreover, it is confirmed that the DITB is
terminated by an m = 2 MHD mode which extends from theinner to the outer foot point location of the DITB. This isprecisely the defining theoretical characteristic of the proposed DTM.
ITB emergence is sensitive to local conditions and, inparticular, to the properties of the integral q surfaces withregard to MHD instabilities and other phenomena. Our initial
results, though by no means a final explanation for experimental observations, are highly encouraging as they stand up
to the test of a quantitative comparison with experiment. It isvery likely that additional experimental and theoretical investigations in this direction will help to expose the ITB triggering
mechanism.
For the viscosity DTMs, it is easy to estimate that thesaturated poloidal shearing velocity

Vy ~

2

4 /15

Acknowledgements
One of the authors (JQD) acknowledges academician Li,
Zheng Wu for continuous encouragements and Prof. W.
Horton for very helpful discussions. This work is supportedby the National Natural Science Foundation of China, Grant
No. 10135020 and by US Department of Energy, Grant No.
DE-FG03-96ER-54346.

B0¢ 4y / 5u A
~ 0.1u A ~ 0.1u ti ,1/ 5

( Rk )

for R ~ 5 ¥ 107 and k = 0.1 [10], here, v ti is ion thermalvelocity. The scaling of the generated velocity with plasma
parameters may be written as,

Vy µ

References:m e4 / 5 B0¢ 4y / 5[1] Y. Koide et al., Phys. Rev. Lett. 72, 3662 (1994).
[2] C.M. Greenfield et al., IAEA 1998 P.413.
[3] M.G. Bell et al., Plasma Phys. Control. Fusion 41, A719
(1999).
[4] G.M.D. Hogeweij et al., Plasma Phys. Control. Fusion
44, 1155 (2002).
[5] S. Gunter, G.D. Conway, A. Gude, J. Hobirk, M.
Marasccek, A.G. Peeters, R.C. Wolf and the ASDEX Upgrede Team, 28th EPS Conference on Controlled Fusion
and Plasma Physics, June 18-22, 2001, Paper P1.006.
[6] E. Joffrin, C.D. Charlis, T.C. Hender, D.F. Howell,
G.T.A. Huysmans, Nucl. Fusion 42, 235 (2002).
[7] R.C. Wolf, Plasma Phys. Control. Fusion 45, R1 (2003).
[8] S.M. Mahajan, Z. Yoshida, Phys. Plasmas 7, 635 (2000).
[9] E. Joffrin, G. Gorini, C.D. Charlis, N.C. Hawkes, T.C.
Hender, D.F. Howell, P. Maget, D. Mazon, S.E. Sharapov,

k 1/ 5 ne3 / 5 A 2 / 5

where A is mass number of plasma ions. The fully nonlinear
development of the mode is under investigation. Although thescaling with A and k seems in line with the experimental
observations [3], the relation between the mode behavior andthe minimum input NBI and RF power is a challenge for themodel. Possible directions to explore are: (1) macroscopic andmicroscopic electromagnetic perturbations that develop andcause anomalous electron viscosity do so only when theauxiliary heating power exceeds the threshold; (2) existenceof a “proper” magnitude of viscosity that does allow DTMs
to develop (and the velocity shear layer to form) but does notallow the DTMs to develop too fast creating violent MHD
activity. The threshold depends on the competition between

5123Dong J. et al., Double Internal Transport Barrier Triggering Mechanism in Tokamak Plasmas

G. Tresset and contributors to the EFDA-JET Workprogramme, Plasma Phys. Control Fusion 44, 1739
(2002).
[10] J.Q. Dong, S.M. Mahajan, W. Horton, Phys. Plasmas 10,3151 (2003).
[11] D. Schnack and J. Killeen, in Theoretical and Computational Plasma Physics (IAEA, Vienna, 1978), Page 337.
[12] J.Q. Dong, W. Horton, Phys. Fluids B5, 1581 (1993).

6124