A Kirk

V Rozhansky

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Possible mechanism for filament motion in the SOL of a tokamak 
 
V Rozhansky
1
, A Kirk
2 
 
1St.Petersburg State Polytechnical University, Polytechnicheskaya 29, 195251 St.Petersburg, Russia 
2EURATOM/UKAEA Fusion Association, Culham Science Centre, Abingdon, Oxon, OX14 3DB, UK 
 
1. Introduction 
Filaments have been observed both in L-mode and during ELMs [1]-[2]. The structures 
observed in L-mode are thought to result from the interchange instability and move towards 
the wall due to BE
EE
·  drifts [3]. In the case of ELMs the ballooning mode evolves into several 
filamentary structures that are highly elongated along B
E
 [4]. These filaments with density and 
temperatures significantly larger than those in the SOL detach from the LFS of the plasma 
edge and travel away from the separatrix [1]-[2]. Their size perpendicular to B
E
 is of the order 
of a few cm, while they are extended along B
E
 by several metres. We consider the evolution 
of such a filament once it has detached from the core and moves towards the wall.  
In the absence of the ambient plasma the filaments are expected to accelerate, similar 
to the case of a pellet cloud [3]. The vertical Bı  driven current is balanced by the 
polarization current, which flows inside the filament in the opposite direction. This current is 
proportional to the temporal derivative of the vertical electric field tE •• / , which causes an 
acceleration in the LFS direction : )/()(2 RmTTg iifef -? , (  are filament temperatures). 
The acceleration of filaments in the SOL during ELMs has been observed [1-2], however, it 
was an order of magnitude smaller than predicted by this expression. The  driven current 
might be short-circuited by currents in the divertor plates. Such models were considered for 
blobs [3], [6]-[7]. The change in the sheath potential drop at the plates produces parallel 
currents, which short-circuit the  driven current of the filament. However, this mechanism 
is not applicable to strongly non-linear filaments. For parameters associated with ELMs, the 
parallel currents, which are required to compensate the 
ifeT ,
Bı
Bı
Bı  driven current of the filament, are 
larger than the ion saturation current at the plates. Hence only the perpendicular polarization 
current of the ambient plasma can contribute to the short-circuiting. The effect is amplified by 
the fact that the field lines come close to each other, especially near to the X-point, and a 
strong perpendicular electric field is created. This effect has been discussed in [8] with the 
assumption that short-circuiting takes place at scales smaller than the ion gyroradius, so 
34th EPS Conference on Plasma Phys. Warsaw, 2 - 6 July 2007 ECA Vol.31F, P-4.035  (2007)
electron perpendicular conductivity has been invoked. However, flux expansion near the X-
point is not that strong for the case of ELMs. The short-circuiting of the  driven current of 
the strongly non-linear filament by the plasma of the SOL is discussed below in an MHD 
approximation. It is shown that the filament is accelerated towards the LFS but that the 
acceleration is order of magnitude smaller than that in the absence of the ambient plasma.  
Bı
 
2. Model  
The  driven current of the filament is (the subscript f corresponds to filament parameters):  Bı
 
RB
TTn
j
f
ifeff
B
)(2 -
?ı . (1) 
The integral along B
E
 (half of the net current) is defined as , where  is 
the filament parallel size. This current is partially balanced by the polarization current inside 
the filament flowing in the opposite direction: 
Ð
¢
ııı ??
0
||ljdsjI BBB ||l
 ||
0
2
l
t
E
B
mn
dsjI
f
f
iff
p
f
p •
•
?? Ð
¢
. (2) 
In the absence of parallel currents in the ambient plasma, from , one obtains 
: 
f
p
B II ?ı
;tgBE ff ? )/()(2 RmTTg iifef -? . This field corresponds to the acceleration in the LFS (x) 
direction in Fig.1. Taking into account the SOL, the parallel current, which is necessary to 
compensate the  driven current of the filament,  is  Bı
 
`
-
?
l
l
RB
TTn
j
f
ifefff ||
||
)(2
. (3) 
If the  driven current is short-circuited by the plates, then the parallel current Eq.(3) would 
flow through the sheath. On the other hand, the maximal parallel current is restricted by the 
ion saturation current: , (subscript pl corresponds to plate 
parameters). Hence the  driven current could be short-circuited by the plates only if:  
Bı
2/1
|| )]/)[( iipleplpl
sat mTTenj -?
Bı
 
`
-
?@
l
l
RB
TTn
jj
f
ifefffsat ||
||||
)(2
. (4) 
In the following we assume that this condition is not satisfied, which as will be shown later is 
the typical case for ELMs. Therefore we will neglect the current through the plates. In this 
case the parallel current in Eq. (4) should be equal to the divergence of the polarization 
current in the ambient plasma integrated along the magnetic field . The electric fbp jIdiv ||?`
34th EPS 2007; V.Rozhansky et al. : Possible mechanism for filament motion in the SOL of a tokamak 2 of 4
field in the filament is determined by current balance . Due to the distortion of 
the magnetic flux tube near the X-point, at a given distance s, a strong electric field arises 
almost in the radial direction, Fig.1. This electric field 
b
p
f
pB III -?ı
`fl? /bE  is significantly larger than 
the electric field in the filament `l? lE f / . Here )(s`f  is the smaller size of the distorted 
cross section of a flux tube which at 0?s  has a circular cross section with radius . The 
polarization current is flowing in the ambient plasma across the magnetic field mainly in the 
direction of the maximal potential gradient (
`l
x|  direction in Fig.1). The direction of the 
ambient polarization current projected along B
E
 to the filament is slightly tilted inside the 
filament forming an angle  with the y-axis. The closing of the circuit might be 
completed by a small additional electric field in the x-direction inside the filament. The 
current density at a given distance s is . This current density projected 
to the plane  should be  times larger due to the divergence of the magnetic field 
lines. The net polarization current of the ambient plasma ( ) is  
``f l/
tBmnj ibp •l•f?
/
`
/ /12
0?s `f lpol /
2
`` ?ff lpol
 ds
l
B
n
t
E
mI
plates
bf
i
b
p 2
2
0
2
`
`
f•
•
? Ð . (5) 
Finally, the acceleration of the filament is ( 1>c ).  
 )/()(2 RmTTg iifef -c? , 
1
2
2
0
2
||
2
1
/
`
`
Õ
Õ
Ö
Ô
Ä
Ä
Å
Ã
f
-?c Ð ds
l
B
n
ln
B plate
s
b
f
f
 (6) 
3. Comparison with experiment 
Typical parameters for filaments observed during ELMs in MAST [1]-[2] are: 
, , ,319102 /©? mn f eVTT ifef 80?? ml 8|| ? cml 4?` , TB 25.0? , mR 5.1? , and at the plates 
,  and fpl nmn 1.0102
318 ?©? / eVTepl 20? eplipl TT ? . This corresponds to . 
For the parameters chosen  and  , so the current could not be 
short circuited through the plates. On MAST the acceleration of the filaments was reported to 
be  [2], (in vacuum ). To calculate 
2
|| /15 mkAj
sat ?
2
|| /600 mkAj
f … 40/ |||| …
satf jj
28 /108.1 smg ©? 210 /10 smg ? c  we take parameters near 
the X-point: the ratio  and 16/ ?f``l 67.2/int ?fXpo BB . Assuming that the effective parallel 
length of the SOL plasma coincides with , we obtain . From Eq. (6) the 
predicted value of the filament acceleration is , which is in good agreement 
with the measured acceleration of  [2], Fig.2. During MAST – AUG similarity 
||l
2105.2 /©?c
28 /105.2 smg ©?
28 /108.1 sm©
34th EPS 2007; V.Rozhansky et al. : Possible mechanism for filament motion in the SOL of a tokamak 3 of 4
experiments [9] it was reported that the acceleration of the filaments is lower in AUG than in 
MAST in spite of the higher filament temperature in AUG. From the above analysis it can be 
shown that the current inside the filament can be more effectively short-circuited in AUG than 
in MAST. Indeed, for AUG the ratio 43.1/int ?fXpo BB  is smaller than for MAST and 
according to Eq. (6) the impact of the ambient plasma is determined by . The 
distortion of the magnetic tube for AUG is also slightly larger than for MAST: . 
As a result for AUG , which is smaller than in MAST and hence the 
corresponding acceleration will be also smaller in spite of the larger filament pressure.  
2
int )/( fXpo BB
6.26/ ?f``l
3105.2 /©?c
 
 4. Conclusions 
It is demonstrated that high density and temperature detached filaments, typical for ELMs, in 
the SOL are accelerated outwards. The acceleration is determined by the balance between 
the  drift current inside the filament and the polarization current in the SOL plasma. The 
impact of the polarization current is amplified by the distortion of the magnetic flux tube, 
especially near the X-point. The short-circuiting through the divertor plates does not play a 
significant role for such strongly non-linear filaments. The predicted acceleration values are in 
reasonable agreement with the filament accelerations observed on MAST and AUG.  
Bı
Acknowledgements 
This work was funded jointly by the United Kingdom Engineering and Physical Sciences Research Council and by the 
European Communities under the contract of Association between EURATOM and UKAEA. The views and opinions 
expressed herein do not necessarily reflect those of the European Commission. 
References 
[1] Kirk A et al 2006 Phys. Rev. Lett. 96 185001 
[2] Kirk A et al 2006 PPCF 48 B433 
[3] Krasheninnikov S I 2001 Phys. Lett. A 283 368 
[4] Wilson H et al 2006 PPCF 48 A71 
[5] Rozhansky V et al 1995 PPCF 37 399 
 
[6] Garcia O E et al 2004 Phys. Rev. Lett. 92 165003-4 
[7] Sarazin Y et al 2003 J. Nucl. Material 311-316 796 
[8] Cohen R.H., Ryutov D. D. 2006 CPP 46 475 
[9] Kirk A et al 2005 PPCF 47 995 
 
y'
x
x'
j
B
2f
2 l
2f
po
l
+
+
  
Fig.1. Scheme of magnetic tube distortion.          Fig.2. The change in toroidal and radial location of one filament 
                                                                                         as a function of time during a) an ELM and b) in L-mode. 
34th EPS 2007; V.Rozhansky et al. : Possible mechanism for filament motion in the SOL of a tokamak 4 of 4


Possible mechanism for filament motion in the sol of a tokamak

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Possible mechanism for filament motion in the sol of a tokamak

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