Screening of Resonant Magnetic Perturbations by flows in tokamaks.
M. Becoulet1, F. Orain1,2 , P. Maget1, N. Mellet1, X. Garbet1, E. Nardon1, G. T. A.
Huysmans3, T. Casper3, A. Loarte3, P. Cahyna4 , A.Smolyakov5, F. L. Waelbroeck6,
M.Schaffer7, T. Evans7, Y. Liang8, O. Schmitz8, M. Beurskens9, V. Rozhansky10, E.
Kereeva10.
1
CEA/ IRFM Cadarache, F-13108, St. Paul-lez-Durance, France.
PHELMA, Grenoble INP, Minatec, 3 Parvis Louis Néel, BP 257, 38016 Grenoble, France.
3
ITER Organization, Route de Vinon, CS 90046, 13067 Saint Paul lez Durance Cedex
4
Institute of Plasma Physics AS CR v.v.i., Association EURATOM/IPP.CR, Prague, Czech
Republic.
5
Physics and Engineering Physics Department University of Saskatchewan, 116 Science
Place, Saskatoon, SK S7N 5E2, Canada
6
Institute for Fusion Studies, University of Texas, Austin, TX 78712, USA
7
General Atomics, P.O. Box 85608, San Diego, CA 92186-5688, USA.
8
Association EURATOM - Forschungszentrum Jülich GmbH, D-52425 Jülich, Germany
9
Euratom-UKAEA Fusion Association, Culham Science Centre, Abingdon OX113 3EA,
United Kingdom.
10
St Petersburg State Polytechnical University, Polytechnicheskaya 29, 195251 St
Petersburg, Russia.
2
Abstract. The non-linear reduced four-field RMHD model in cylinder was extended to
include plasma rotation, neoclassical poloidal viscosity and two fluid diamagnetic effects.
Interaction of the static Resonant Magnetic Perturbations (RMPs) with the rotating plasmas in
tokamaks was studied. The self-consistent evolution of equilibrium electric field due to RMPs
penetration is taken into account in the model. It was demonstrated that in the pedestal region
with steep pressure gradients total poloidal rotation including E  B and electron diamagnetic
components: (V , EB  V*,e ) plays essential role in RMP screening by plasma. Generally,
screening effect increases for lower resistivity, stronger rotation and smaller RMP amplitude,
Strong screening of central islands was observed limiting RMPs penetration to the very
narrow region near the separatrix. However, at certain plasma parameters and due to the nonlinear evolution of the radial electric field produced by RMPs, the poloidal E  B rotation can
be compensated by electron diamagnetic rotation (V , EB  V*,e ) ~ 0 locally. In this case
RMPs can penetrate and form magnetic islands. Typical plasma parameters and RMPs spectra
on DIII-D, JET and ITER were used in modeling examples presented in the paper.
1.Introduction.
Type I ELMs represent a particular danger for Plasma Facing Components (PFC) and divertor
materials in ITER due to the fast (~0.25ms) transient release of energy (up to 20MJ) in
ELM crash [1,2,3]. The promising active method of ELMs control by Resonant Magnetic
Perturbations (RMPs) produced by specific coils permitted total Type I ELM suppression on
DIII-D [4,5] and AUG [6] or strong mitigation of the ELM size [7] on JET. At present RMPs
are intensively studied on many tokamaks which are developing new facilities (coils) for
RMPs generation (DIII-D, JET, MAST, TEXTOR, NSTX, AUG) [8]. However, at present the
underlying physics of ELM suppression is not totally understood up to the point to give the
reliable predictions for Type I ELM suppression by RMPs in ITER. As it was shown in DIIID experiments ELM are essentially suppressed if according to vacuum modeling for r/a>0.9
magnetic islands overlap (Chirikov parameter >1) creating ergodic region at the plasma edge
[11] . The present RMP system foreseen for ITER was essentially designed on this semiempirical and vacuum (without plasma response) modelling background [9, 10, 11].
However, based on the experimental data from different tokamaks recently it became clear
that “vacuum” criterion [11] is not sufficient [8] . In particular at similar to DIII-D a priori
“vacuum “ edge ergodisation [11], the application of RMPs demonstrated a variety of ELM
responses on different machines. In particular, ELM suppression on DIII-D [4,5,11] and
AUG [6] was observed, ELM mitigation was demonstrated on JET [7], Type III ELM
triggering is seen in ELM free discharges on NSTX [12], triggering small Type IV ELMs are
typical for MAST [13]. Moreover the ELM reaction on RMPs for a single machine depends
on equilibrium, plasma parameters and RMP spectrum.
This indicates the present lack of
understanding of plasma response for the reliable extrapolations of RMP method to ITER.
With this respect the non-linear MHD theory and modelling can provide further physical and
numerical improvements to refine knowledge of basic ELM dynamics and related ELM
control techniques [14]. The present studies of plasma response to static RMPs based on nonlinear MHD theory and modelling [15-22] are still in the initial stage and far from selfconsistent complete picture, but they demonstrated already the particularly important role of
plasma flows with respect to RMP penetration into the plasma. Depending on the plasma
parameters and RMP spectrum, the actual RMP field could be very different in rotating
plasmas where the generation of current perturbations on the rational surfaces could prevent
reconnections and islands formation, leading to the effective screening of RMPs [15-20]. As it
was shown in [22], a radial electric field produces E  B poloidal rotation which is
particularly important in the RMPs screening in the pedestal region with strong pressure
gradients and strongly sheared radial electric field. Note also that recently the importance of
the diamagnetic electron and ion rotation in RMPs screening in the zone of strong pedestal
pressure gradients was demonstrated [22, 23]. In the present work the resistive MHD rotating
plasma response to RMPs was studied using a non-linear code RMHD based on the reduced
four-field resistive MHD model in cylinder [24, 25]. The RMHD code [24] was adapted to
the RMPs studies in [17,22] and further developed in the present paper to include neoclassical
poloidal viscosity tensor and two fluid diamagnetic effects, neglected in the previous RMP
modelling [17]. In the present work we introduced this new physics in order to describe selfconsistently the evolution of the radial electric field and the poloidal rotation due to RMPs
penetration in the pedestal region, which is the essential new step forward compared to [17,
22], where equilibrium electric field was taken from the experiment [5] and kept constant in
the modeling [22]. Generally in experiments of Type I ELMs suppression by RMPs on DIII-D
the radial electric field in the pedestal tends to more positive values at the edge with the
minimum of the “well” shifted towards the core [5]. In the model we used in the present work
this trend in the changes in the equilibrium electric field due to RMPs can be reproduced.
The paper is organized as follows. In Sec. 2 the derivation of the numerical non-linear RMHD
model with diamagnetic and neoclassical effects is described. Sec. 3 is devoted to the
description of the generic features of the single RMP harmonic and RMP spectrum
penetration into the plasma with poloidal and toroidal flows. In this section DIII-D like
parameters were used. In Sec. 4 more realistic JET equilibrium, plasma parameters and Error
Fields Correction Coils (EFCC) spectra for toroidal number n=2 were used in modeling of
RMP screening in JET. In Sec.5 modelling results of rotating plasma interaction with RMPs
are presented for the standard H-mode scenario and actual design of in-vessel RMP coils in
ITER
2. Model.
2.1. Four field reduced MHD model with neoclassical poloidal viscosity. The well known
four-fields non-linear reduced resistive MHD model [24,25, 26] was used as a starting set of
the equations:

  J  || (   p)
t
W
 W ,    p   || J     p,    
t
p
  p,  
t
V
1
 V ,   
|| p
t
2
(1.1)
Compared to [24] we neglected electron inertia and finite -terms. Here normalized variables
are :  - poloidal magnetic flux, p - electron pressure ;  -electrostatic potential,
J    2  (
1  
1  2
(r
)  2 2 ) - parallel current, W    2 - is the vorticity;
r r r
r 
dimensionless parameter  
eB
1
c
, where  ci  0 is the ion gyrofrequency,

mi
2ci A 2 pi a
ratio of ion temperature to electron one  
VA 
a-is a minor radius, Alfven velocity
B0
is taken for central electron density, ne  Zni ; Z  1 . Cylindrical coordinates
0 mi n0
r; ; z were used in the model,
one, R0 
the
Ti
,
Te
where z  R0 .  -is the toroidal angle, - is the poloidal
RM
is the normalized major radius. The magnetic field normalized to the value on
a
axis
(B0)
B / B0  (ez    ez )  (
is
represented
1 

;
;1) .
r 
r
Beq  B0 (0, b ,0 (r ), bz ,0 ); bz ,0  1; b ,0 (r )  
in
cylindrical
Equilibrium
approximation
magnetic
as:
field
is:
 0
r
. Note that in this approximation both

r
qR0
equilibrium and perturbed magnetic fields are divergence free:  B  0 . For any scalar
function we introduce parallel gradient operator:
|| S 
1
S
S
1 S   S
B S 
   ez S 
  S ,  , where  S ,   (

).
B
z
z
r r  r 
Notifying physical parameters by the superscript (ph) , the normalization of the variables used
here is following:
B ph  B0, z B  B0 B;  ph  aB0 ; r ph  ar ; R0ph  RM  aR0 ; neph  n0 ne ; 0  Zmi n0 ;
V ph  VAV ; t ph   At ;  A 
p ph 
aB 2 0
a
;  ph 
;
VA
0  0
(1.2)
aB 
B02
a2
p;  ph  0 0  ;  ph (//, )   (//, )
2 0
A
0
In the derivation of (1.1) in [24] the fluid velocities for species s (s=e,i- is for electron and
ions) were approximated as :
Vs  VE  V||,s  Vs*
(1.3)
where V||, s 
1
  ez
EB
;
(Vs B) B  Vs b ; VE 

2
2
B
B
B
Vs* 
B ps
p  e
  s z . In our
2
es ns B
ns ms cs
previous work [17] only toroidal rotation and the corresponding equilibrium radial electric
field Er ~ V B were taken into account, neglecting poloidal rotation and pressure gradient
terms similar to [15]. However, the poloidal component of the diamagnetic rotation Vs*, and
E  B rotation: VE , could play the most important role in the pedestal region where radial
electric field is large and the pressure gradient is very steep in H-mode plasmas [22-23]. In
the present work the non-zero poloidal rotation and diamagnetic effects are added into the
model to study their role in RMP screening in the pedestal region. Here we assume that
toroidal rotation is prescribed essentially by the external source, for example neutral beam
injection, similar to [17], and the poloidal plasma rotation without RMPs tends to the
neoclassical value proportional to the ion temperature gradient due to the large neoclassical
poloidal viscosity [27, 28] :
V  V neo 
ki Ti
mi ci r
(1.4)
Certainly, we are aware of the limits of this approach, since at present the predictive capacity
of neoclassical theory for poloidal rotation in tokamaks is rather limited. Usually poloidal
rotation is measured for carbon impurities (C4+,C6+) Doppler CXRS measurements [29,30].
The information about poloidal rotation for the main species is not directly available in
experiment, since the impurity flows differ from the bulk ions for different reasons,
demanding self-consistent modelling of flows in multi species plasma [29], taking into
account ionization and neutrals at the edge [31], plasma turbulence in the core etc. [32,33].
Note, that at present the self-consistent predictive first principles theory of poloidal rotation is
still missing. The measured poloidal rotation was shown to deviate from neoclassical
predictions especially in the plasma core [34] and in the presence of Internal Transport
Barriers [30]. Note also that drift-kinetic simulations [35, 36] pointed out the importance of
the direct ion-orbit losses in the pedestal region leading corrections to the neoclassical theory
also in the pedestal. However, in the pedestal region in spite of the extensive debates [31, 3536], the order of magnitude and the form of poloidal velocity profile is closer to neoclassical
predictions in many experimental cases [29]. This motivated us to include the poloidal
viscosity tensor in MHD modelling using its neoclassical form as a first step approximation to
capture main physics of interest here which is the change of the poloidal flow and the edge
radial electric field due to RMPs in H-mode plasmas.
In the following we present the derivation of new terms we added to the model (1.1).
Similar to [24] the equation of motion for species (s) was used in a form:
 (VE  V||,s  Vp )

s
ms ns 
 (VE  V||,s  Vs* ) (VE  V||, s  Vs* )     Ps  es ns ( E  Vs  B)  Fs
t


(1.5)
The pressure tensor can be represented as: Ps  Ips  s , where ps - is a scalar pressure for
species and viscous tensor can be represented in two parts: gyroviscosity and neoclassical
parallel viscosity: s  sg  sneo . The explicit form of sg is usually not needed in reduced
MHD because of gyroviscous cancelation [24]. For neoclassical part of pressure tensor here
we used the simplified heuristic closure from [27] :
  sneo  ns ms s
Here VT , s 
B2
(Vs e  ksVT ,s e )e
B2
(1.6)
B Ts
T  e
  s z  O( ) , the expressions of  s -neoclassical poloidal flow
2
es B
ms cs
damping rate and coefficient ki are given in [27,28]. The electron part of the neoclassical
pressure tensor enters in the Ohm’s law and describes the bootstrap current [28, 37]. The
bootstrap current evolution is not treated in the present work and is considered to be a part of
the total equilibrium parallel current profile given at the beginning of the modelling at t=0.
The equation of motion for ions is represented in a form:

B2
 

mi ne (  Vi )VE  (  VE )V||,i  (V||,i)V||,i )   P  mi ne i 2 (V ,i  kiV ,Ti )e  J  B
t
B
 t

current
is: J 
1
 B .
0
Note
also
(1.7)
Where
plasma
that
V ,i  kiV ,Ti 
 (Ti ne )
1 
r
1
1 Ti
. Neoclassical coefficient k i and
V

 ki
B r
qR0 mi ne ci r
mi ci r
i depend on equilibrium and plasma parameters and are calculated numerically using [28]
for equilibrium and plasma profiles used in modeling examples.
Ti (r ) and electron
Te (r )
In the present work ion
temperature profiles are fixed, for simplicity we take
Ti (r ) / Te (r )    const and pressure is evolving only due to density evolution. Total pressure
is P  Pi  Pe  pe (1   ) . Equation for the parallel flow is obtained from the scalar product of
(1.6) with unit vector b parallel to the total magnetic field:


 

 mi ne (  VE )Vb  bV ||V )   P 

 t

0
b

1 1 
r
1  (Ti ne )
1 Ti 
V

 ki
)e 
  mi ne i 2 (
b B r
qR0 mi ne ci r
mi ci r


(1.8)
Here Vi  Vb . Finally, the normalized parallel velocity (V ) equation for ions is solved in a
form :
V
(1   )
 V ,    V ||V 
|| p  F||neo   || 2 (V  Vt 0 )
t
2ne
where the term F||neo   i
(1.9)
qR0 
T (r )
r
 p
(
V

  ki e ) is taken into account for the
r r
qR0 ne r
r
mean flow (harmonic n=0, m=0 [see 17] ) and is zero for the perturbations of the parallel
velocity. Normalized neoclassical viscosity is: i  i( ph ) A . Note that similar to [17] we
introduced the parallel collisional viscosity (//) and a source of the toroidal rotation which
compensates diffusive terms: SV   ||2Vt 0 . This provide stationary toroidal rotation profile
identical to the initial one (V(r)=V(r)t=0) in the absence of RMPs.
Vorticity equation with neoclassical parallel viscosity terms can be obtained by taking
ez   of equation (1.5). The detailed derivation of the vorticity equation without neoclassical
terms is given in literature [24]. So we give here only the derivation of the additional
“neoclassical” term in the vorticity equation, resulting from (1.6) :
 1

Fneo,( ph )  ez    i 2 (V ,i  kiV ,Ti )e  
 b

 (Te ne )
1
1 
r

 Te
 i 2 ez   (
V

 ki
)e 
b
BT r
qR0 mi ne ci r
mi ci r
i
b2

(1.9)
 (Te ne )
1     1 
r

 Te   
r

V


k



  ;
i
r  r   BT r
qR0 mi ne ci r
mi ci r   
Note that here we assume that
  i 
 r  is small compared to other terms, which was verified
r  b2 
numerically using for i the expressions from [28]. Finally after the normalization the
vorticity equation is solved in a following form:
W

1

 W ,    W , p   || J    p,      Fneo    2W
t
ne
ne
ne
(1.10)
As in (1.10) the neoclassical term was used only in the equation for the mean poloidal flow:
neo

F
 qR 
  i  0 
 r 
2

Te   
1    r  p
W

r
V


k




 
i
r r   qR0 ne r
r   

(1.11)
Here    108 is a numerical viscosity added for stability reasons, typically i /  ~ 103  104 .
All variables in the RMHD code [25] used here are represented in Fourier series, for example
poloidal flux     nm e
m , n 
im  inz / R0
, and the harmonic n=0,m=0 is the equilibrium value. The
boundary conditions at r=1 are zero for all perturbations except for the magnetic flux
vac
harmonic amplitudes,  nm r 1   nm
, sep , which are approximated by the vacuum amplitudes
calculated in the toroidal geometry using code ERGOS [9]. Here we take r   pol . Since
the most essential physics in interest here is the neoclassical equilibrium, similarly to [24] we
take in equations normalized n0,0 ~ 1 except in the expressions for the neoclassical terms.
Finally the normalized system of
non-linear RMHD equations with diamagnetic and
neoclassical effects are solved in the following form:

  ( J  J t 0 )  || (   p )
t
(1.10)
W
2
 V ||W  W ,    p   || J     p,     Fneo
,00    W
t
(1.11)
p
 V || p   p,    k 2 ( p  pt 0 )
t
(1.12)
V
1
neo
 V ||V  V ,   
|| p  F||,0,0
  || 2 (V  Vt 0 )
t
2
(1.13)
Where averaged neoclassical terms will be taken into account only in the equations for values
averaged over magnetic surface (here (0,0) is for harmonic n=0, m=0):

Te (r )  
1  
r
 p0,0
Fneo



W

r
V


k



 
,0,0
neo ,1  0,0
0,0
i
r r  
qR0 n0,0 r
r  


neo
F||,0,0
  neo,2 (
 0,0
 V0,0
r
T (r )
r
 p00

  ki e )
qR0 n0,0 r
r
2
where
(1.14)
qR
 qR0 
neo ,2  i 0 ,

r
 r 
neo ,1  i 
W0,0  20,0  
(1.15)
1  0,0
(r
),
r r
r
n0,0 
p0,0
Te (r )
.
Diffusion terms are compensated by volume sources terms adjusted to keep initial profiles
pressure, current and toroidal velocity profile without RMPs. Note that for the equilibrium
case without RMPs:
 0,0
r
 V0,0b 
 p0,0
n0,0 r
  ki
Te
0
r
(1.16)
That corresponds for the physical values to:
pi
1 
r
1
1 Ti
V

 ki
0
BT r
qR0 mi ne ci r
mi ci r
(1.17)
which is equivalent to the force balance equation for the radial electric field:
Er  

1 pi

 (V B  Vneo B )
r ene r
(1.18)
Here in the cylindrical approximation V ~ V ; B ~ BT  const . In the presence of RMPs
radial electric field and poloidal rotation are modified and deviate from the neoclassical
values.
2.2. Role of poloidal rotation in screening of RMPs.
It is demonstrated theoretically [15, 38] and numerically [16-23] that the plasma
response to RMPs mainly consists in generation of current perturbations localized near the
rational surfaces, leading to the deformation of the magnetic perturbation compared to the
vacuum case. As it was analyzed in [15,38], depending on the plasma parameters (viscosity,
resistivity, rotation, current profiles etc) the plasma response can vary between more or less
strong screening or in some cases amplification of the externally applied static magnetic
perturbations. It will be useful for the following discussion of the modeling results to analyze
single harmonic perturbation penetration into the rotating plasma. Magnetic flux, current and
electron pressure correspondingly are represented in RMHD code
   00 (r )  nm (r )e
p(r )  p00 (r )  pnme
b  
im inz / R0
im inz / R0
 cc; J  J 00 (r )  J nm (r )e
 cc .
For
the
im inz / R0
as follows
 cc;
equilibrium
values:
|| ( 00   p00 )  0 ,
 00
r

. Considering that J 00 (r ) ~ J 00 (t  0) , the linearized Ohm’s law (1.10) for a
r
qR0
single harmonic perturbation amplitude in stationary conditions can be written as follows:
 J nm  
i
m im nm  ( oo   poo )
( nm   pnm )(n  ) 
R0
q
r
r
(1.19)
At the rational surface q=-m/n (n<0, m>0 according to the convention used in RMHD code
[25]) and hence:
 J nm  
im nm  ( oo   poo )
im nm
r

(VE ,  Ve*, )  bnm
(VE ,  Ve*, )
r
r
r
(1.20)
The current perturbation is zero on the rational surface if (VE ,  Ve*, ) ~ 0 , meaning that with
respect to the static RMP imposed from the external coils electrons are at rest, and therefore
no screening of RMPs is expected for this particular harmonic in this region.
3. Generic features of the single RMP penetration into the plasma with flows.
In this chapter DIIII-D-like parameters [4] were used for modeling : RM  1.8m , a  0.6m ;
B ~ BT  1.9T , cylindrical q95 ~ 3.15 . Plasma parameters profiles used in modeling are
presented on Fig.1. Central density, temperature, toroidal rotation correspondingly were :
ne,0  8.1019 m3 ; Te,0  1.5keV V0  72km / s . These plasma parameters correspond typically to
the high collisionality regime [4, 5], note however that no specific DIII-D shots where
modeled here, that is why we call it here “DIII-D like” case. Density, rotation and temperature
profiles are approximated here by the cubic polynomials multiplied by a factor
f (r )  0.5(1  tanh((r  rbar ) /  ) with: rbar=0.98, and pedestal width ~=0.06. Resistivity
profile follows dependence:  (r ) ~ 0 (Te (r ) / Te (0)) 3/ 2 with typical experimental-like central
0  108 . Parallel viscosity is  ||  106 . Perpendicular numerical viscosity is taken    108 ,
and much larger perpendicular viscosity neo ,1,2   || , v is due to the neoclassical mechanism
[28]. The exact profiles of neo,1,2 (r ); ki (r ) for the parameters (Fig.1) will be used later on in
the paper. For the initial study of the generic features of RMP-plasma interaction, neoclassical
coefficients were taken constants: neo,1,2  5.105 ; ki  0.8 . At the boundary, normalized
perturbation amplitude was taken  nm (1) ~ 5.10 5 , which was estimated from vacuum
modeling with toroidal vacuum code ERGOS [9] for DIII-D I-coils at 4kAt, n=-3 even parity
configuration. As it was demonstrated in [17], screening of RMPs by plasma is negligible at
high resistivity, so we call here a “vacuum-like” case if RMHD modeling is done at high
resistivity: 0  104 . The “vacuum-like” m=8, n=-3 island without neoclassical viscosity
and without rotation ( neo ,1,2  0 ,=0. and V ,0  0 ) is presented on the Poincare plot on
Fig.2(a).
Corresponding
magnetic
flux
and
radial
magnetic
field
perturbation:
 (r ,  , z  0);  br (r , , z  0) are presented on Fig.2(b,c). Note the expected phase shift
between  ~ cos(m ) and  b r ~ sin( m ) , since  b r 
1  ( )
. The same kind of plots,
r 
but with plasma response with parameters V0/VA~0.022 (which corresponds to central rotation
~72km/s), 0  108 ,   0.03 , q95=3.15 are presented on Fig.3 at t=105A. As it was shown in
[17] the penetration time for RMPs scales approximately as ~1/, so in following analysis we
present amplitudes of perturbations after t=104-105A, when stationary conditions are reached
in modeling and the corresponding island has its final size. Note that current perturbation
(Fig.3(d)) is localized on the rational surface and is in phase with radial magnetic field
perturbation (Fig.3(c)) and strong screening of vacuum magnetic perturbation is observed :
 br r r ~ 0 . However, for example, at q95=3.44 RMP penetration is seen in modeling
res
(Fig.4). In this case the corresponding perturbations have “vacuum-like” structures and in
particular the current perturbation is zero at r=rres. The radial profiles of magnetic flux and
current perturbation harmonics amplitudes for q95=3.15 and q95=3.44 cases are presented on
Fig.5. The performed q95 scans (Fig.6) demonstrated, that RMP penetrate in a window situated
around a certain q95
for different values of edge RMP amplitude (here for
 nm (1) ~ 5.10 5 and nm (1) ~ 2.5105 ). The decrease of the absolute value of RMP amplitude
at the resonance for higher q95 values seen on Fig.6 is explained by the radial dependence
 nm (r ) ~ r m in vacuum. The resonant surface qres=8/3 moves toward the plasma centre when
q95 increases leading to a smaller “vacuum” island. On Fig.6 RMP penetration window q95
for (8/3)-island is q95= 1.-0.5 (smaller corresponds to smaller RMP amplitude). The reason
for no RMP screening seen at q95=3.44 is that (VE ,  Ve*, ) ~ 0 at the resonance surface q=8/3
(Fig.7(a)). On the contrary (VE ,  Ve*, )  0 for q95=3.15 (Fig.7(b)), so screening currents are
not zero at the rational surface (see discussion in Sec.2.2). As it was discussed already in [15,
17], at higher resistivity, screening currents are small even at non-zero poloidal rotation and
RMP can penetrate. To illustrate this, the radial profiles of the magnetic flux perturbation
harmonics are presented for different resistivities on Fig.8(a) for q95=3.15 case. One can see
the strong screening of the perturbation for r>rres at 0  108 and much smaller effect of
plasma on RMP at 0  107 compared to a “vacuum-like” case which corresponds in this
modeling to a resistivity 0  102 . Since resistivity increases towards the edge due to the
temperature dependence,
edge islands are typically less screened as it is illustrated on
Fig.8(b). The penetration time for RMP increases for lower resistivity (Fig.9) [17]. The
neoclassical viscosity neo,1,2    const (constant here) scan is presented on Fig.10 (a), where
maximum amplitude of a single harmonic
on the resonant surface q=8/3 is presented for
different q95 values. One can see that at larger  penetration window in terms of q95 is
narrower. In the case of larger neoclassical poloidal viscosity the initial radial electric field
(and hence, the poloidal component of ExB velocity) is less influenced by RMP. This is
illustrated on Fig.10 (b) where electric field profiles are presented for different values of 
Magnetic
topologies
resulting
from
application
of
RMP
spectrum
 n3;m6,7,..10 (1)  (8;7;...4) 105 at q95= 3.15 in vacuum and with plasma response (in stationary
conditions at t=6.104) are shown on Fig.11. Note strong screening of more central magnetic
islands except on q=7/3.
Screening factor on the resonant surface Snm 
pl
|  nm
|res
, is
vac
|  nm |res
presented for different values of resistivity ( 0 ) on Fig.12. The condition (VE ,  Ve*, ) ~ 0 in
vacuum and with plasma response (in stationary conditions at t=6.104, 0=10-8) are shown on
Fig.11. Note strong screening of more central magnetic island on q=6/3 and island formation
on q=7/3. Screening factor on the resonant surface Snm 
pl
|  nm
|res
, is presented for different
vac
|  nm |res
values of resistivity ( 0  108 ; 2.108 ; 8.108 ) on Fig.12. The condition (VE ,  Ve*, ) ~ 0
(Fig.13) is satisfied for the island (7/3) for all resistivities which explains why it is not
screened. Island q=(8/3) is screened since it is situated at maximum of poloidal velocity in all
cases (Fig.13). On the contrary, more central island on q=(6/3) forms for higher resistivity
values:
0  2.108 ; 8.108 The radial electric field Er structure with RMP spectrum at
t=6.104A is shown on Fig.14 in comparison with the initial equilibrium electric field. One can
see that, due to the non-linear interaction of RMPs with plasma, radial electric field is more
positive in the ergodic region and minimum of Er “well” moves inside the plasma (Fig.14),
which reminds the experimental trend [5].
The realistic neoclassical coefficients (Fig.15), calculated for profiles presented on Fig.1
according to [28] instead of constants, do not change the generic features of RMP interaction
with plasma described above. In particular, RMP screening by plasma is stronger for larger
rotation (including diamagnetic), smaller RMP amplitude, smaller resistivity and larger
viscosity. However, because of the increase of neo,1,2 towards the edge (Fig.15), smaller
penetration windowq95 ~0.3 (compared to Fig. 6 where q95 ~1) is observed for a single
island
(8/3) (Fig.16). However, edge resonant harmonics still penetrate (Fig.17), since
typically, the screening currents are smaller at higher resistivity.
4. RMP penetration on JET. In ELM mitigation experiments on JET, RMP are generated by
the external one-row EFCC coils [7] in n=1, n=2 configurations. Here we present an example
of RMHD modeling of plasma response in a shot JET#77329 with parameters RM  2.9m ,
a  0.89m ; B ~ BT  1.8T , toroidal rotation in the plasma centre
 ,0  38.6krad / s ,
parameter in diamagnetic terms was :   0.029 , cylindrical q95 ~ 3.8 , EFCC current
amplitude IEFCC=40kAt, n=-2. The realistic plasma parameters used in modelling are shown
on Fig.18. The corresponding normalized neoclassical viscosity coefficients calculated using
expressions from [28] are presented on Fig.19(a,b). The results of vacuum modeling by
ERGOS code [9] in toroidal geometry and equilibrium for the shot JET#77329 are presented
on Fig.20. At IEFCC=40kAt magnetic islands do not overlap even in vacuum modeling
(Fig.20(b)). The normalized absolute values of the magnetic flux perturbation harmonic
amplitudes as a function of radius ( r   pol ) are presented on Fig.21. The maximum values
of corresponding harmonics amplitudes are used in the cylindrical RMHD code as boundary
conditions:  n2;m3,4,..8 (1)  (9.9;6.5;4.4;2.9;2;1.4;) 105 . The poloidal rotation profile with
respect to the position of the resonances is a crucial factor in RMP screening by plasma.
However in order to have the same q-profile in cylindrical code as in toroidal geometry
q=qtor, it is necessary to adapt equilibrium poloidal field and current profile in RMHD code
accordingly and in particular:
B 
rB0
1
1 1  (rB )
; jz 
ez (  B) 
R0 qtor
0
0 r r
(4.1)
For obvious geometrical reasons, this current profile (4.1) is different from the toroidal one.
In particular, to provide the same strong magnetic shear in cylinder as in torus, current profile
tends to be negative at the edge in cylindrical code (Fig.22). The advantage of using this
current profile in RMHD cylindrical code is that q=qtor , but the drawback of this option is,
however, that MHD stability of the new current profile can be different and in particular more
tearing unstable at larger current gradient [39]. To avoid unrealistic negative currents in
RMHD code modelling, in some of the cases (for example DIII-D like parameters in Sec.3)
we have chosen a parabolic current profile which gives a different q-profile compared to
torus, but matching approximately the toroidal one at the edge and at the center (Fig.22).
Note, however, that the disadvantage of this solution is that resonance surfaces moves more
towards plasma center as it can be seen from vacuum-like RMHD code modelling without
rotation and at
0  102 (Fig.23 (a) compared to Fig.20(b)) Furthermore, the magnetic
islands have slightly smaller size compared to islands in toroidal geometry (closer to the edge)
and also different positions with respect to poloidal rotation profile. Magnetic topology with
rotating plasma response at 0  108 and t=6.103A, including diamagnetic effects and
realistic neoclassical coefficients [28], is presented on Fig.23(b). Note the strong screening of
all harmonics on JET compared to vacuum case. The poloidal velocity (Fig.24) at
corresponding r=rres is not zero for all harmonics (except for m=4, where island is formed on
q=2 surface), explaining strong screening of RMPs. The edge harmonic m=8 penetrate
because of higher resistivity as it can be concluded from the screening factors profiles for
harmonics (n=-2,m=3-8) presented on Fig.25. In the nearest future (in 2011) the increased
capability of the EFCC power supplies on JET will permit the operation at maximum EFCC
current 96kAt. Magnetic topology with the same plasma parameters but increased EFCC
current 96kAt instead of 40kAt (Fig.23(b)) is presented on Fig.26 – with (26(b)) and without
screening (26(a)). For comparison we presented the results of modeling with “unrealistic”
current profile (i.e., negative at the edge, Fig.22), but which provides q=qtor, IEFCC=96kAt.
One can see on Fig.27 (a) that all resonant surfaces are located on the same radial positions as
in torus (Fig.20(b)) The Poincare plot with plasma response is presented on Fig.27(b). Note
the larger ergodic region on Fig.27(b) compared to the cases of 40kAt (Fig. 23(b)) and to
96kAt case (Fig.26(b)- with different q-profile). Screening factors for the case q  qtor is
presented on Fig.28, showing strong screening at r> rres for all harmonics except for the very
edge resonant harmonic n=-2, m=8 . Now the most central islands at q=(3/2) (instead of
q=(4/2) on Fig.26(b) ) is located near (VE ,  Ve*, ) ~ 0 (Fig.24) and hence the corresponding
(n=-2, m=3) RMP harmonic can penetrate. It was tested numerically that the current profile
used in cylindrical code to provide: q=qtor is stable for the intrinsic n=2 mode for the
parameters used here.
5. RMP penetration on ITER. Finally let us consider ITER-like parameters: RM  6m ,
a  2m ; B ~ BT  5.3T . Plasma profiles for standard H-mode scenario presented on Fig.29,
toroidal rotation  ,0  6krad / s and, diamagnetic term parameter is   0.009 and central
resistivity is taken 0=0.510-8. The corresponding neoclassical coefficients are presented on
Fig.30. The neoclassical poloidal viscosity is smaller for ITER compared to JET (Fig.19) and
DIII-D (Fig.15) which is typical going to the lower collisionality plasmas. Vacuum modeling
in torus for the latest design of ITER in-vessel RMP coils was done with code ERGOS [9].
The 3 rows poloidally in each of 9 sectors are shown on sketch 31(a). Here coil coordinates
(R,Z,) are approximated as follows. Top coil coordinates are: upper conductor R1=7.735m
Z1=3.380m, lower conductor R2=8.262m
Z2=2.626m , centre = 30°, coil in width in
toroidal direction coil =30°, centre-centre =40°.
conductor -R1= 8.618m
Mid plane coil coordinates are: upper
Z1=1.790m, lower conductor- R2=8.661m
Z2= -0.550m , centre
= 26.7°, mid coil width in toroidal direction coil =20° , centre-centre =40°. Lower coil
coordinates are: upper conductor -R1= 8.230 m
Z1= -1.546m ,
lower conductor-
R2=7.771m, Z2= -2.381m , centre = 30°, coil in width in toroidal direction coil =30° ,
centre-centre =40°. DC currents in the coils are distributed in a way to generate mainly n=3
toroidal symmetry (in upper coils in [A]: 23220. -86940. 63630. 23310. -86940. 63630.
23310. -86940.
63630; in mid-plane coils: 0.
77940. -77940.
0. 77940. -77940.
0. 77940. -77940; in low coils: -27810. -60210. 88020. -27810. -60210. 88020. -27810.
-60210. 88020). The corresponding profiles of magnetic flux perturbation harmonics (n=-3,
m=4-11),
used
for
boundary
conditions
 n3;m 4,..11(1)  (6.89; 6.66; 6.31; 5.97; 5.64; 5.35; 4.98; 4.52)  105 ,
in
RMHD
are
presented
code:
on
Fig.31(b). As it was discussed already in Sec.4, the main difficulty of toroidal q profile
approximation in cylindrical code consists is the choice between cylindrical and toroidal q
profile since both options have their disadvantages and advantages, but neither can correctly
reproduce toroidal geometry (See discussion in Sec.4). Cylindrical and toroidal q profiles
used here are presented on Fig.32. One can see how far resonances at the same m- poloidal
number move deeper towards the plasma center if q  qcyl  qtor . Vacuum and with plasma
response Poincare plots for q  qcyl  qtor are presented on Fig.33. Only m=11, n=-3 island
chain forms on q=11/3 surface, but other harmonics are strongly screened.
For q  qtor
(Fig.34), q=m/n resonances have the same position compared to the torus, i.e. closer to the
edge. On Fig.35 one can see that the predictions for RMP screening factors for (n=-3, m)
harmonics in ITER are very different depending on the choice ( q  qcyl  qtor or q  qtor ),
since corresponding positions of rres and hence RMP amplitudes are different (larger towards
the edge). In particular, q  qtor gives more optimistic predictions for RMP penetration in
ITER. Screening factor is bigger then one in the case of amplification [5], which is seen for
example for m=11 and m=8 harmonics on Fig.35. Poloidal velocity profiles in stationary
conditions after t=104A are presented on Fig.36 for two approximations: (a)- q  qcyl  qtor
and (b)- q  qtor . Note that due to RMPs penetration, poloidal velocity is decreased almost to
zero only in the narrow region near q=(11/3) for (a)-case and in much larger region near
q=(7/3)-(11/3) resonances for the option-(b), where RMP amplitudes are closer to the edge
and hence larger on the resonant surfaces. Even within limitations of the cylindrical code
RMHD mentioned above, plasma response modeling indicated already that RMPs in plasma
could be different compared to vacuum modeling. It is illustrated even more clearly in the
following example of modeling for ITER. Here we used maximum possible current in each
coil 90kAt with -phase shift between coils in toroidal direction. Upper and lower coils in
each of 9 sectors have the same sign of current and mid-plane coil has the opposite signs
compared to upper and lower coils [40]. Vacuum modeling [40] predicted strong ergodisation
in larger edge region in this case compared to n=3 case discussed above. The main toroidal
number with these currents in RMP coils is n=4, but side harmonics amplitudes n=1,2,3,5 are
large. As a result islands overlapping is increased [40]. The boundary conditions for RMP
spectrum (here we used n=1:4 and corresponding to them the main resonant m - harmonics)
for RMHD code were obtained using ERGOS code [9]:
 n1;m14 (1)  (2.5;1.4;1.1;0.9) 105
 n2;m27 (1)  (1.;0.9;0.8;0.7;0.7;0.68) 105 ,
(5.1)
 n3;m611 (1)  (1.34;1.28;1.22;1.17;1.13;1.07) 105 ,
 n4;m412 (1)  (4.56;4.38;4.18;4.01;3.77;3.61;3.41;3.28;3.08) 105 .
Here toroidal q-profile q  qtor was used.
Note the strong screening of all harmonics for
this case (except the very edge ones) seen on Fig.37, leading to the conclusion that vacuum
predictions of plasma edge ergodisation by RMPs are totally irrelevant in this case.
6. Conclusions and discussion.
The non-linear reduced four-field RMHD model in cylinder with plasma rotation,
neoclassical poloidal viscosity and two fluid diamagnetic effects was applied to the problem
of RMP interaction with plasma. Due to the steep pressure gradients in the pedestal region the
poloidal components of flows including E  B and electron diamagnetic components:
(V , EB  V*,e ) play essential role in RMP screening by plasma.
Current perturbation located near resonance surface generated in rotating plasmas is
the underlying physics of RMP screening. The screening effect increases for lower resistivity,
stronger rotation and smaller RMPs amplitude. Typically RMPs penetration occurs in the
narrow region near the separatrix due to a higher resistivity. However, at certain plasma
parameters or/and due to non-linear evolution of radial electric field due to RMPs the poloidal
E  B rotation can be compensated by the electron diamagnetic rotation (V , EB  V*,e ) ~ 0 . In
this case RMP harmonic (n,m) penetrate locally and form islands on the corresponding
resonance surface q=m/n.
For DIII-D-like parameters and RMPs with n=3 toroidal number islands are mainly
screened in plasma core and ergodic zone is much narrower (rerg>0.95) compared to vacuum
modeling (rerg>0.9).
Depending on the q-profile, islands (7/3), (8/3) can penetrate if
condition (V , EB  V*,e ) ~ 0 is satisfied on the corresponding resonant surfaces. The q95 scan
performed for (n=-3,m=8) harmonic demonstrated rather narrow q95~0.3 penetration
window for the island (8/3), situated approximately on the pedestal top.
For realistic JET parameters (shot #77329) and EFCC coils at n=2, IEFCC=40kAt, a
strong RMP screening was observed in modeling except for the very edge (r>0.98).
Depending on q-profile approximation in cylindrical code ( q  qcyl  qtor or q  qtor ), the
formation of central islands on q=(4/2) or q=(3/2) was seen in modeling near r~0.7 where
(V , EB  V*,e ) ~ 0 . The edge RMP penetration was slightly increased (r>0.96) at higher
current in EFCC coils 96kAt, compared to 40kAt.
ITER standard H-mode scenario and RMP in-vessel coils parameters were used in
modeling. For the case n=-3, m=4:11 spectrum and q  qcyl  qtor , strong screening of all
harmonics except for m=11 was demonstrated. On the contrary for more realistic positions of
resonances at
q  qtor edge harmonics m=7:11 can penetrate and form the ergodic region in
ITER. Poloidal plasma velocity is strongly modified by RMP and is close to zero in this case,
which is favorable for RMP penetration. In the case of n=1: 4 spectrum and maximum current
90kAt in each coil strong screening of RMPs is observed limiting ergodic region to the very
edge. With this respect n=-3 option remains more favorable for ITER.
The cylindrical RMHD code results presented here are obviously limited in exact
predictions of RMPs screening in ITER. Note also that, higher RMP amplitudes, higher
resistivity and lower neoclassical poloidal viscosity are favorable factors for RMP
penetration. ELM suppression physics with respect to screening of RMPs was not treated in
the paper and still remains an open question. The further step in understanding of RMP
penetration in ITER should be based on the non-linear MHD code JOREK in realistic toroidal
geometry [14]. This study is under way at present and will be presented in the dedicated
paper.
Acknowledgments. This work, supported by the European Communities under the contract
of Association between EURATOM and CEA, was carried out within the framework of the
European Fusion Development Agreement. The views and opinions expressed herein do not
necessarily reflect those of the European Commission. The Sec.5 of the paper was supported
within the framework of the Fusion for Energy grant F4E-GRT-055 (PMS-PE). This work
was also part-funded by the Grant Agency of the Czech Republic under grant P205/11/2341.
References:
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Figures:
Fig.1. DIII-D-like plasma parameters used in modelling.
Fig.2. (a)-Poincare plot for vacuum-like island on resonant surface q=8/3; (b) – corresponding
to (a) magnetic flux perturbation as a function of radial and poloidal coordinates at z=0 ; (c)corresponding to (a) radial magnetic field perturbation.
Fig.3 (a)-Poincare plot for (8/3) island with plasma response at q95=3.15; t=105A. (b)magnetic flux perturbation; (c)-radial magnetic field; (d)- toroidal current perturbation at z=0.
Fig.4. (a)-Poincare plot for (8/3) island with plasma response at q95=3.44, t=105A.; (b)magnetic flux perturbation; (c)-radial magnetic field; (d)- toroidal current perturbation at z=0.
Fig.5. Radial profiles of the magnetic flux perturbation (a) and toroidal current (b) harmonic
amplitudes (n=-3,m=8) at q95=3.15 (screening of RMP ) and q95=3.44 (no screening) in
stationary conditions at t=105A.
Fig.6. Maximum RMP amplitude on the resonant surface q=8/3 for single harmonic in q95
and RMP edge amplitude scans.
Fig.7. Poloidal components of the perpendicular velocity: VE , (dashed-dot line); electron
diamagnetic Ve*, (thin); the sum (VE ,  Ve*, ) (bold), q 103 profile (dashed). The position of
q=8/3 resonance is indicated by diamond for a case of screened island at q95=3.15 -(a) and no
screening at q95=3.44-(b).
Fig.8. (a) Radial profiles of the magnetic flux perturbationharmonic amplitudes (n=-3,m=8)
at q95=3.15 for different resistivities.(b)- The same as (a) for more external island (n=3,m=10).
Fig.9. Absolute value of (n=-3,m=10) harmonic amplitude on the resonant surface as a
function of time for different central resistivity values 0.
Fig.10. (a) – absolute value of the magnetic flux perturbation harmonic amplitude on the
resonant surface at t=105A as a function of q95 for different neoclassical viscosity constant
coefficient neo. (b)- stationary
Fig.11. (a)- Vacuum magnetic topology with RMP spectrum for DIII-D –like parameters. (b)the same as (a) but with neoclassical and diamagnetic effects at t=6.104A; q95=3.15.
Fig.12. Screening factor ( Snm 
pl
|  nm
|res
) profile for RMP spectrum in resistivity scan at
vac
|  nm |res
t=6.104A; q95=3.15.
Fig.13. Poloidal components of the perpendicular velocity for DIII-D like parameters and
RMP spectrum 
(1)  (8;7;...4) 105 with neoclassical and diamagnetic effects at
n 3;m6,7,..10
t=6.10 A; q95=3.15. Here: VE , (dashed-dot line); electron diamagnetic Ve*, (thin); the sum
4
(VE ,  Ve*, ) (bold), q 103 profile (dashed). The positions of resonances are indicated by
vertical lines.
Fig.14. Radial electric field profile for DIII-D like parameters and RMP spectrum
 n3;m6,7,..10 (1)  (8;7;...4) 105 with neoclassical and diamagnetic effects at t=6.104A;
q95=3.15. The case without RMP (or at t=0) is indicated in dashed line.
Fig.15. Neoclassical coefficients for DIII-D like parameters (Fig.1): RM  1.8m , a  0.6m ;
B ~ BT  1.9T , cylindrical q95 ~ 3.15 .
Fig.16. Screening factor ( Snm
pl
|  nm
|
 vac res ) on the resonant surface q=8/3 in q95 scan for a
|  nm |res
single harmonic (m=8,n=-3) with edge amplitude  nm (1) ~ 5.10 5 , 0=10-8, with realistic
neoclassical coefficients (presented on Fig.15) [28] .
.
Fig.17. Magnetic topology in the case of RMP spectrum  n3;m6,7,..12 (1)  5 105 at 0=10-8,
q95= 3.87 with realistic neoclassical coefficients (presented on Fig.15) [28]. Note penetration
of (8/3) island and very edge harmonics m=10-12.
Fig.18. JET parameters (#77329) used in modelling.
Fig.19. Corresponding to JET shot #77329 parameters normalized neoclassical poloidal
viscosity- (a) and ki -(b) coefficients.
Fig.20. Vacuum modelling in torus with code ERGOS [9] . (a)- Normalized Re al ( n )  0 ,
(b)-Poincare plot in magnetic flux coordinates
  ;  ;  0 in vacuum ( I
EFCC=40kAt,
n=-
2)
Fig.21.Normalized harmonics (n=-2,m=3:10) amplitudes profiles in vacuum resulting from
ERGOS code for IEFCC=40kAt. Edge values were used as input for RMHD code in JET case.
Fig.22. Current and corresponding q profiles used in following RMHD modelling.
Fig.23. Magnetic topology resulting from cylindrical code RMHD for JET case at
IEFCC=40kAt, n=-2. (a)-vacuum(0=10-2) without rotation, (b)- with rotating plasma response
at t=6.103A, q  qtor .
Fig.24. Poloidal components of the perpendicular velocity for JET parameters at IEFCC=40kAt,
n=-2, m=3-8 with neoclassical and diamagnetic effects at t=6.103A; q95=3.8. Here:
VE , (dashed line); electron diamagnetic Ve*, (dashed); the sum (VE ,  Ve*, ) (bold),
q 103 profile (dashed with diamonds indicating resonance surfaces q=(3/2); (4/2) etc). The
positions of resonances are indicated by vertical dot-lines. q  qtor .
pl
|  nm
|res
) profiles for JET parameters at IEFCC=40kAt,
vac
|  nm |res
resonant harmonics n=-2, m=3-8 with neoclassical and diamagnetic effects at t=6.103A;
q95=3.8; 0=10-8, q  qtor .
Fig.25. Screening factor ( Snm 
Fig.26. Magnetic topology resulting from cylindrical code RMHD for JET case at IEFCC=96
kAt, n=-2. (a)-vacuum (0=10-2) without rotation, (b)- with rotating plasma response at
t=6.103A, q  qtor .
Fig.27. The same as Fig.26, but with q  qtor .
Fig.28. The same as Fig.25, but with q  qtor in cylindrical code.
Fig.29. ITER H-mode scenario parameters used in modeling.
Fig.30.Neoclassican coefficients corresponding to parameters presented on Fig.29.
Fig.31. (a)- ITER RMP coils representation in ERGOS vacuum code in toroidal geometry;
(b)-magnetic flux perturbation profiles for n=3,m=4:11 harmonics.
Fig.32. q-profiles used in RMHD modeling for ITER.
Fig.33. Magnetic topology in ITER due to RMP coils at n=-3 with cylindrical q-profile:
q  qcyl  qtor in vacuum- (a), in plasma-(b).
Fig.34. Magnetic topology in ITER due to RMP coils at n=-3 with toroidal q-profile: q  qtor
in vacuum- (a), in plasma-(b).
Fig.35. Screening factor for different poloidal harmonics m at n=-3 for q  qtor (in white) ,
q  qcyl  qtor (in black).
Fig.36. Similar to Fig.24, but for ITER RMP modeling in ITER at n=-3 for (a)-with
q  qcyl  qtor , (b)-with q  qtor .
Fig.37. Poincare plots for vacuum (top) and with plasma response (bottom) for RMPs
modeling in ITER for n=4,3,2,1 spectrum (n=4 is the main harmonic) with maximum 90kAt
current in each coil.