J. Plasma Fusion Res. SERIES, Vol. 9 (2010)

Plasma rotation effects on neoclassical tearing modes
D. Chandra, A. Sen and P. K. Kaw
Institute for Plasma Research, Bhat, Gandhinagar - 382428, India(Received: 20 November 2009 / Accepted: 6 May 2010)

The effect of an equilibrium sheared ow on the nonlinear evolution of a neoclassical tearing mode is investigated by estimating the inuence it has on the inner and outer layer dynamics of the mode. Two complementary
approaches are adopted. A generalized Rutherford model calculation is carried out to estimate the ow contributions to the polarization current term in the inner layer. For the outer layer, ow induced changes in the stability
parameter Δ� are estimated with the help of a 3D initial value reduced MHD code (NEAR). For realistic parameters it is found that the outer layer modication is the dominant one and the scaling of Δ� with the ow shear
parameter appears to agree with recent experimental observations.
Keywords: sheared ow, neoclassical tearing mode, Rutherford model, stability parameter

1. Introductionof the ow shear matters and hence a tailoring of theow prole can be an important factor in the control or
suppression of NTMs. Theoretical understanding of theseexperimental observations is still very limited. While itis known from past model studies that ows and owshear can bring about profound changes in the mode
dynamics near the resonant layer (inner layer) as well as
in the stability index Δ� (outer layer effect), a quantitative
estimate of these effects is not available to date. On thebasis of heuristic arguments, experimetal observations of
the DIII-D machine have been interpreted in terms of achange in the stability index Δ� and an empirical linear
scaling of the stability index with the ow shear has been
projected [16]. Comparison of multi-machine data toarrive at a better understanding and identication of theunderlying physical effects is currently being carried out
by various groups and is also a high priority study topic
for ITER.

The inuence of plasma rotation and rotation shear
on the excitation and stability of various MHD modes
has been an important topic of investigation in tokamak
research for a number of years now [1–12]. In particular
their effect on tearing modes has attracted a great dealof past theoretical work and has led to the identicationof many basic results. More recently the attention has
shifted to neoclassical tearing modes (NTMs) which posea potential threat to achievement of high β in present day
long pulsed tokamak experiments and in future operational
scenarios of ITER [13–15]. With the availability of better
diagnostics and means for controlling the direction andamount of plasma ow, several recent tokamak experiments have addressed the issue of establishing the natureof correlation between the amount of ow (and owshear) and the threshold β for the onset of an NTM aswell as the size of the saturated magnetic island. The
experimental data show a clear evidence of ow (and owshear) induced modication of both the onset thresholdβN and the saturation size of the island. On DIII-D
experiments [16], through controlled variation of co andcounter NBI beams, it has been found that, in general,
decreasing toroidal plasma ow impairs NTM (tearingmode) stability. With decreased ow, existing 3/2 islands
can become larger and stable 2/1 modes can get excited.
Thus with lower rotation the NTM onset has a stronger
drive (lower βN ). Although in most cases it is difcult toexperimentally separate the ow effects from ow shear
effects, on many shots where it has been possible to obtaina good measure of the ow prole it is observed thatow shear has a strong inuence on the mode stability.
By increasing the amount of ow shear it is possible toreduce the size of an existing 3/2 magnetic island. Suchobservations have been made for a variety of operational
scenarios (H mode plasmas, sawteething plasmas as well
as reversed shear plasmas) and on multiple machines(JET, AUG, NSTX etc.) [14, 16]. There is also strong
evidence (though not entirely unambiguous) that the sign

Our present work is motivated by a desire to provide
some theoretical understanding of the observed oweffects through a combination of analytical and numericalmodeling. To gain some insight into the ow induced
modications of the inner layer dynamics we have carriedout a model two uid calculation and derived a generalized
Rutherford equation which incorporates ow contributions
to the polarization current term. To study the inuenceon Δ� we have used a 3D initial value reduced MHD
code (NEAR) and investigated the evolution of a single
helicity NTM for various ow proles and estimated thechanges in the stability index from measurements of thesaturated island width. From a numerical comparison of
the inner and outer layer modications, for a set of realistic
parameters, we nd that the dominant contribution to the
NTM growth rate and saturation width arises from theΔ� modication rather than the inner layer effects. The
numerical simulations also display a near linear scalingof Δ� with the amount of ow shear - a result that is inqualitative agreement with the observed experimental
results on DIII-D.

author’s e-mail: abhijit@ipr.res.in457©2010 by The Japan Society of Plasma
Science and Nuclear Fusion Research

D. Chandra et al., Plasma Rotation Effects on Neoclassical Tearing Modes

where the perpendicular component of the total current is
proportional to the plasma inertia through the ion polarization drift. With the substitution for J⊥ , (6) takes the form,pi �
c2 d 0 2 �
∇
=0
(7)
Φ
+
∇� J� −
en
4π v2A dt ⊥Our paper is organized as follows. In the next section
we present our analytical calculations for the ow effects
on the inner layer dynamics. Section 3 is devoted to an
estimation of changes in Δ� through numerical simulations
on the code NEAR. In the nal section we discuss andcompare the results of sections 2 and 3 and make someconcluding remarks.

where the operator d0 /dt can be written as,
d0dt

2. Flow effects on the Inner Layer dynamics=

We consider a single helicity magnetic perturbation
moving across an equilibrium magnetic eld. In the vicinity of a resonant surface where the helicity of the perturbation matches the pitch of the equilibrium eld (i.e.
q(rs ) = m/n, q is the safety factor), the magnetic eld canbe expressed in terms of an effective ux function ψ ,ψ =−B 0 x2+ ψ̃ cos ξ
Ls 2�

4Ls ψ̃
B0

� π

−π

−π

d ξ cos ξdxJ� =c �
Δ π ψ̃
4π c(3)d ξ sin ξ� ∞dxJ� =c �
Δ π ψ̃
4π s

(4)

−∞

−∞

Φ=

(8)

B0 ω x+ f (ψ )ckθ

(10)

where f (ψ ) the integration constant is an arbitrary function of ψ , and needs to be determined from the boundary
conditions. In carrying out the integration one also uses
the identity,� �ψ̃ sin ξ
∂x
=
(11)
ψx
∂ξ ψ

where the matching parameters Δ�c,s are determined fromthe outer (linear) region and are assumed to be given by
ideal MHD equations. The longitudinal current needs tobe obtained from the parallel Ohm’s law,��1 ∂μe c d p(5)ψ (t) −
J� = σneo −∇� Φ +c ∂t
νei Bθ dx

To account for the ow shear we write Φ and v�0 as,Φ = Φ�0 x + Φ��0x2
+ φ̃ (x, ξ )2v�0 = v�s + v��sx

where μe is the viscosity coefcient, νei is the electronion collision frequency, Bθ is the poloidal magnetic eld,
p is the plasma pressure and σneo is the neoclassical conductivity. The last term on the RHS is the perturbed bootstrap current which is responsible for driving the neoclassical tearing modes. For the low frequency tearing mode,quasineutrality condition holds so that we have,∇� J� + ∇⊥ · J⊥ = 0

�

where kθ = m/rs is the poloidal wave vector and derivativewith respect to ξ is to be evaluated at constant ψ .
Using the above expression for ∇� and integrating E� ≈ 0,in ξ we get,

(2)

� ∞

kθ ψx v�0 ∂B0 ∂ ξkθ c
Φ,B0

To solve (7) we need to get an expression for Φ, theelectrostatic potential. Note that to the lowest order in
Ohm’s law, E� = ∇� Φ + 1c ∂∂t ψ (t) ≈ 0, which implies anear cancellation of the electrostatic eld by the inductive electric eld. By transforming the coordinates from(x, y) → (ψ , ξ ) we can write ∇� as,� �kθ ∂ ψ ∂∇� =(9)B0 ∂ x ∂ ξ ψ

The nonlinear evolution equation of the magnetic island is derived from the matching conditions obtained byintegrating Ampere’s equation across the nonlinear region.
� π−ψs (ω − ω∗pi) x −Here Poisson bracket {α , β } = ∂∂ ψα ∂∂ βξ − ∂∂ αξ ∂∂ ψβ , ω∗pi is
the ion diamagnetic frequency, ω∗pi = kθ cTi p�0i /(eBp0i )[17, 18], vE is the E × B drift component and v�0 is theparallel component of the equilibrium sheared ow.

(1)

�1/2

�
∂ �
+ vE + v�0 · ∇∂t �+Here B0 is the average equillibrium toroidal magnetic
eld, x = r − rs is the distance from the rational surface, Ls = qR/s is the shear length, s = rs q� /q and ξ =�mθ̂ − ω (t � )dt � where θ̂ = θ − ζ /qs is the helical coordinate with θ denoting the poloidal angle and ζ the toroidal
angle. Note that for m ≥ 2, when the constant ψ̃ approximation holds, the magnetic island halfwidth is given by,
W=

≡

(12)
(13)

Equating both the expressions for Φ (i.e. using expressions(10) and (12)) we can write,

φ̃ =

B0
B0 ωE� 2
x + f (ψ )(ω − ωE )x −
ckθ
ckθ 2

(14)

where ωE = kθ cΦ�0 /B0 is the drift frequency due to the radial equilibrium electric eld created by the magnetic perturbation moving across B0 and φ̃ is the perturbed Φ. An

(6)

458

D. Chandra et al., Plasma Rotation Effects on Neoclassical Tearing Modes

appropriate choice of f (ψ ) such that φ̃ vanishes for large
x (i.e. far away from the island) is,

0.00012NEAR
1.06e-4(1-W/W s)

0.0001B0
B0 ωE� 2(ω − ωE )λ (ψ )−λ (ψ ) (15)f (ψ ) = −
ckθ
ckθ 2

8e-05

dW/dt

Substituting for f (ψ ) we get,
B0
B0 ωE� 2(x − λ 2 ) (16)φ̃ =(ω − ωE )(x − λ ) −
ckθ
ckθ 2

6e-05

4e-05

Here the function λ (ψ ) is chosen to be zero inside themagnetic separatrix and λ (ψ ) → x in the region x >> W
[17, 18]. We have taken the following form of λ (ψ ) in theouter region,

2e-05

0

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

W

�
��
�
ψ 1/2
W
λ (ψ ) = √
−1
−
ψ̃
2

Fig. 1 Rate of change of island width vs island width(17)

where W is the magnetic island half width as dened in (2).

⎡√L�εβθ Lqp∂W
W2
neo ⎣ Δc
= DR
+ G2
G1
2
∂t
4
W
W + Wχ2

Following standard procedure we now integrate Eq.
(7) using Eqs. (12), (13) and (16) to obtain the parallel
component of current as,

J�

=
+
−

DI
L2 (ω − ωE )(ω − ωE − ω∗ )− G3 �
+ G4 2 s 2
W3kθ vA
W 2 + 0.65Wχ2
L2 ω � 2
Ls v�s ωE�
+G5 2 s 2 E − G6kθ vA vA W
kθ vA W

A(ψ )(cos ξ − < cos ξ >)σ� ∂ ψ̃< cos ξ >B(ψ )(x3 − < x3 >) +c ∂t��μe c ∂ ψ ∂ p(18)
νei Bθ ∂ x ∂ ψ�Ls v�s (ω − ωE − ω∗ )
c2 1)
−G7
− G8 2skθ vA vAW
vA W

A(ψ ) =

B(ψ ) =−as,kθ v��s � ω � ∂ 2 λ 2
cB40
E
2 ∂ ψ24π v2Akθ2 L2s B0�∂ 2λ(ω − ωE − ω∗pi)
∂ ψ2

(19)

(20)

RqL

�· · · � =(··· )

dξ

1

dξ

∂ψ
∂x
∂ψ
∂x

1/4

χ⊥
q
, below which
critical island width Wχ =mχ�radial transport becomes signicant and the pressure is nolonger attened across the island. In the above expression
for the island width evolution equation (22), the term proportional to the ωE� 2 arises purely from the perpendicularow shear contributios and the term proportional to v�s ωE�
is due to the combination of parallel ow and perpendicular ow shear.
In eqn. (22), the term proportional to G2 is the usual

and the ux surface average operator �· · · � is dened

�

(22)

2
where, Dneo
R = c /4πσneo is the magnetic diffusion coefcient calculated using the neoclassical resistivity, βθ =8π pe /B2θ , L p = −(d ln p/dr)−1 , Lq = (d ln q/dr)−1 , Ls =qR/s, DI = −2(q2 − 1)rp� /(s2 B20 ) is the resistive interchange parameter, s = rq� /q is the magnetic shear andv¯�0 is the average parallel ow velocity. The coefcients
G1 to G6 are dened in the Appendix and their numerical values are: G1 = 0.41, G2 = 0.58, G3 = 6.35, G4 =5.7, G5 = 0.24, G6 = 0.77, G7 = 2.05 and G8 = 1.9. For
evaluating the neoclassical contribution we have adopted√
the standard procedure outlined in [18] where μe � ενei
for the long mean-free-path regime has been used andω∗ = ω∗pi + kω∗T [18], with ω∗T = kθ cTi� /eB0 . The factor
W 2 /(W 2 + Wχ2 ) in the neoclassical term is the usual effectassociated with nite radial�thermaland sets a
� diffusion�

where,
� �
cB40W 2
ωE ∂ λ 2∂λ− (ω − ωE )222
∂ψ
8π vAkθ Ls 2 ∂ ψ���kθ v�s ωE� ∂ 2 λ 2∂ 2λ+− (ω − ωE − ω∗pi)
B0
2 ∂ ψ2
∂ ψ2

�

(21)

We now use Eq. (18) in the matching condition (3), toarrive at the following island evolution equation,459

D. Chandra et al., Plasma Rotation Effects on Neoclassical Tearing Modes

NTM driving term arising from the perturbed bootstrap
current and the term with the coefcient G3 is the
Glasser-Green-Johnson (GGJ) contribution representing astabilization effect due to the toroidal curvature. The nextterm involving G4 is a ow term which is destabilizing
in nature except for the window 0 < ω − ωE < ω∗pi andhas been discussed in several past works [18–21]. The
term involving G5 arises purely due to perpendicular ow
shear on tearing modes and from the sign in front of it,
is always destabilizing in nature. The next two terms
proportional to G6 and G7 depend upon parallel shear
ow. These terms are proportional to both ow shear aswell as the ow velocity at the resonant surface and canchange sign depending on the signs of the ow shear andow. These last two terms also differ in other ways fromthe G4 term. The G4 term goes as W13 . So for NTMs which
begin with a nite threshold island size, the effect of thisterm diminishes as the island evolves as its magnitudedecreases more rapidly than the driving NTM term.
The numerator of the term also remains small because
island rotation frequency ω is usually close to the ow
frequency ωE i.e. (ω − ωE ) is small. But the shear terms
go as W1 which are similar to the way the NTM driving
term varies. So their effects remain similar throughout
the evolution of NTM and can inuence the full NTM
dynamics from threshold to saturation. Secondly, the G4term depends mainly on the ow frequency whereas the
new terms depend also on the ow gradient. We should
also mention here that the terms parallel and perpendicular ows in the context of a toroidal ow velocity
in a tokamak, refer to the parallel and perpendicular(to the magnetic eld) projections of the ow velocity
and are therefore nite quantities. In other words for apurely toroidal ow (with zero poloidal velocity) there is
still a perpendicular component of the ow because themagnetic eld lines are not purely toroidal. We defer thenumerical estimates of these ow induced contributions
to the inner layer to the nal section when we carry out
a comparison with the changes in the outer layer dynamics.

10

9

8

Δ’

7

6

5

4

3

2
-0.05

-0.04

-0.03

-0.02

-0.01

0
0.01
dΩ/dr(r=rs)

0.02

0.03

0.04

0.05

0.06

Fig. 2 Stability index Δ� vs ow shear.

eralized reduced MHD equations that are valid for arbitrary aspect ratios and can correctly handle sub-Alfvenic
toroidal equilibrium ows. Details of the equations and thecode have been extensively discussed in the past and can befound in [13,23]. The code has been well benchmarked forboth linear and nonlinear classical tearing modes as well as
NTMs and has also been also employed previously to study
the nonlinear evolution of NTMs in the presence of toroidal
sheared ows [13]. These studies have shown that sheared
toroidal ows can signicantly alter the stability and saturation properties of single helicity NTMs. Our objectivein the present work is to extract the information about thecorresponding changes in the stability index Δ� as a function of the ow characteristics. To accomplish this we haveexamined the temporal evolution of the mode upto its saturation. In Fig. 1 we have plotted the rate of change of theisland width as a function of the island width. Earlier analytical studies [24] have shown that close to saturation theisland evolution equation for a tearing mode can be represented by a simple equation of the form,��
1.66Δ�
W
dW
=
1−
(23)
dt
S
Ws

3. Flow effects on the Outer Layer dynamics

As is clear from Fig. 1 the island evolution rate indeed satises such a model form close to saturation and the slope of�and hence Δ� .
the linear curve then directly gives us 1.66ΔS�We have evaluated Δ with and without equilibrium ow
and thereby estimated the changes brought about by ow.
In Fig. 2 we have plotted the values of Δ� as a function of
the amount of ow shear and Fig. 3 provides a variationof the saturated island width with ow shear. Both gures clearly show that increase of positive ow shear has
a stabilizing effect on the stability index as well as on thesaturated island width. Interestingly the scaling of Δ� with
ow shear shows a linear dependence - a trend that is inqualitative agreement with the empirical scaling seen in theexperimental data of DIII-D. However a detailed quantita-

As discussed in past model calculations, equilibriumows can also affect the stability index of a tearing modeby modifying the shape of the eigenfunction of the outerlayer MHD equations. One way of estimating this inuence is to set up an appropriate modied Newcomb equation which incorporates the inertial contribution of ow
and solve this equation for various ow proles. Sucha calculation was done for a cylindrical model in [7, 22]and earlier in the slab approximation [5]. However for a
toroidal conguration such a formulation is quite difcultand we have instead chosen to address this issue throughdirect numerical simulations using a fully toroidal reduced
MHD code called NEAR. This code solves a set of gen-460

D. Chandra et al., Plasma Rotation Effects on Neoclassical Tearing Modes

choice of parameter values. This makes it possible for theow to inuence the outer layer dynamics.

2/1 NTMs
NEAR
0.684/(1+3.6 dΩ/dr)

0.14

To conclude, our analytic calculations based on
the generalized Rutherford model and the numericalsimulations on NEAR provide separate estimates of theeffect of shear ow on the inner and outer layer dynamics
of a neoclassical tearing mode. Approximate numericalestimates of the two contributions show that the outerlayer modications are more signicant. The resultsfrom NEAR further show that positive ow shear has
a stabilizing inuence and the stability index follows alinear scaling with the amount of ow shear. These results
are in qualitative agreement with recent experimental
ndings [14, 16]. These preliminary and qualitative
results which provide useful and encouraging insights
into present experimental observations will we hopelead to more detailed investigations on improved models
coupled to state-of-the-art codes like NIMROD for a better
quantitative extrapolation of these effects to ITER.

W sat

0.12

0.1

0.08

0.06

0.04
-0.15

-0.1

-0.05

0
dΩ/dr (r=rs)

0.05

0.1

0.15

Fig. 3 Saturated island width vs ow shear

tive comparison with the experimental results is beyond thescope and ability of the model calculations presented here.
In the next section we will however carry out a comparison of the ow induced effects in the inner and outer layer
dynamics to get some quantitative idea of their relative importance.

Appendix: Calculation of the numerical coefcients in the island evolution equation

The G coefcients in the equation (22) have been
evaluated numerically. Here G1 is given by the integral,

4. Discussion and Conclusion

As demonstrated in Secs. 2 and 3, sheared ow caninuence the dynamics of a tearing mode or an NTM in
two different ways. It can affect the dynamics near themode resonant surface (the inner layer) by its contributions
to the polarization current term and it can alter the stability
index Δ� by inuencing the outer layer MHD mode
structure. We now carry out an approximate quantitative
estimate of these two effects to assess their relative
importance. As a common physical parameter space wechoose the typical values we have used for our numericalsimulations on NEAR, namely, rs /a=0.565, β0 =0.002,
R/R0 =1.001, qs =2.0, s=0.743, Ωs τA = 3.53 × 10−3,d(Ωs τA )/d(r/a) = 6.34 × 10−2 and Ws /a=0.103. For
such a case we have Δ� a=2.108 while in the absence ofow we get Δ� a=6.385 - a change of about 4.2. For thesesame parameters, the contributions from the ow shear
terms which are related to G5 , G6 and G7 coefcients−4−4−4, 1.3×10
and 3.5×10
reare approximately, 5.6×10(W /a)(W /a)(W /a)
spectively. So even for W /a = 0.01 at the beginning of
the Rutherford phase the contributions from these terms
remain of the order of 10−2 and are quite small comparedto the corresponding change in Δ� . Thus from our model
calculations it appears that the outer layer modications
are more pronounced compared to the inner layer ones in
the parametric space that we have examined. It should be
mentioned here that although the toroidal ow velocity
in our numerical code is sub-Alfvenic its perpendicular(to the magnetic eld) component is comparable to thepoloidal Alfven velocity - the latter being low for our

G1 =

1
√
2 2π

� ∞

−1

d α �cos ξ �2� πdξ−π (cos ξ + α )1/2
(24)

where, α = −(ψ /ψ̃ ). Now by putting α = 2k2 − 1 we havegot,
4π
where,

G1 =

� 1
0

kdk �cos ξ �2in K(k2 ) +

4
π

� ∞
1

1
dk �cos ξ �2out K( 2 )k

E(k2 )− 1, k < 1
K(k2 )
� 1
�
E( k2 )
2
− 1 ,k > 1
�cos ξ �out = 1 + 2k
K( k12 )�cos ξ �in = 2

and E,K are the elliptic integrals of rst and second kindrespectively. Here, the inner region of the island is correspond to k < 1 where the magnetic eld lines are closed
and the outer region is correspond to k > 1 with an open
magnetic eld lines.
Similarly other coefcients are given by,√ ���∂ g ∂ 2g
8 2 ∞d α ( cos2 ξ − �cos ξ �2 )G4 = −π −1∂ α ∂ α2� πdξ×−π (cos ξ + α )1/2
�
8 ∞ dk � 2 �=−( cos ξ out − �cos ξ �2out )π 0 k2
�
�
∂g ∂
1 ∂g
1
K( 2 ) (25)×∂ k ∂ k 4k ∂ k
k461

D. Chandra et al., Plasma Rotation Effects on Neoclassical Tearing Modes

√ �
�
�
2 ∞
∂ g2 ∂ 2 g2
d α ( cos2 ξ − �cos ξ �2 )π −1
∂ α ∂ α2� πdξ×−π (cos ξ + α )1/2
� ∞dk � 2 �
1=( cos ξ out − �cos ξ �2out )π 1 k2��
1∂ g2 ∂
1 ∂ g2K( 2 )
×∂ k ∂ k 4k ∂ k
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G5 =

√ �
�
�
2 ∞
∂ 2 g2d α ( cos2 ξ − �cos ξ �2 )
π −1
∂ α2
� π
dξ
×
−π (cos ξ + α )1/2
� ∞4
dk � 2 �( cos ξ out − �cos ξ �2out )
=
π 1 k
�
�
1
∂
1 ∂ g2K( 2 )
×∂ k 4k ∂ k
k

(26)

G6 =

(27)

√ �
�
�
2 ∞
∂ 2g
d α ( cos2 ξ − �cos ξ �2 ) 2
π −1
∂α
� π
dξ
×
−π (cos ξ + α )1/2
� ∞4
dk � 2 �( cos ξ out − �cos ξ �2out )
=
π 1 k
�
�
∂
1 ∂g
1
K( 2 )
×∂ k 4k ∂ k
k

G7 =

(28)
�
�
√
W
g(α ) and g(α ) =α − 1 forWhere, λ (ψ ) = √2outer region. All the integrals are contributed only in theouter regions because g(α ) is zero in the inner region.
Here,
�

1
�
4 E( k2 ) 2 28k (2k − 1) + k2 (k2 − 1) + 1
cos2 ξ out = −3 K( 12 )3k

For neoclassical term, we have calculated G2 as,√ �∞
G2 = −1.46 2π
d α �cos ξ � � π= −1.46π

� ∞
1

−1

Θ(α − 1)dξ(cos ξ + α )1/2
−πdk
�cos ξ �outE( k12 )(29)

Here the step function Θ is taking care of the fact that thepressure gets attened inside the island.
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