Fusion Res.
Res. SERIES,SERIES, Vol.
Vol. 6 (2004) 000-000
527–530
J. Plasma Fusion

Monte Carlo Simulation of Particle Transport inthe LID Configuration
KANNO Ryutaro1,2, JIMBO Shigeaki2, TAKAMARU Hisanori3 and OKAMOTO Masao1,21National Institute for Fusion Science, Toki 509-5292, Japan
Graduate University for Advanced Studies, Toki, 509-5292, Japan3
Department of Computer Science, Chubu University, Kasugai, 487-8501, Japan2(Received: 9 December 2003 / Accepted: 23 February 2004)

Abstract
Monte Carlo simulation based on the test particle representation is carried out in order to understand the transport
in the edge region of the Large Helical Device configuration with the Local Island Divertor (LID). It has the advantageof appropriately treating the collisionless edge plasma in the three-dimensional field line structure with the island andergodic zones. The strike point pattern on the LID head is numerically observed under the effect of the Coulomb
collisions, and the observations show that the pattern strongly depends on collisionless.
Keywords:edge plasma transport, stochastic approach, Monte Carlo technique, Local Island Divertor, Large Helical Device

1. Introduction

model is frequently employed [1]. The model is based on
simplified Braginskii's fluid equations [1,2] neglecting theelectric field and plasma current, i.e. a simple neutral plasma
is assumed. The fluid equation for a fluid property f is given
generally as

Analyses of the edge plasma transport are classified to
the fluid and particle representations. When collisionality isstrong, it is known that the fluid approximation based on
Braginskii’s fluid equations is good enough [1,2]. If the hightemperature divertor operation (HT-operation) is considered,
there is a possibility to change this understanding, however.
The Local Island Divertor (LID) in the Large HelicalDevice (LHD) configuration [3,4] is one of the divertor
concepts, and it utilizes an m/n=1/1 island formed at the edge
region of the LHD. Control of the edge plasma by means of
the LID is expected to realize the HT-operation.
In the HT-operation, the edge temperature is evaluatedto be raised up to + 5 keV by efficient pumping, thereby
leading to enhancement in the energy confinement [4]. In thiscase, the edge plasma is collisionless, and the neo-classicaleffect on the transport becomes important.
Then, at first, in order to select the appropriate method
solving the transport, we examine two stochastic approaches
to the edge plasma transport, i.e. the fluid and test particle
representations, in the three-dimensional (3-d) field line
structure with the island and ergodic zones. After theexamination of the approaches, the strike point pattern on theLID head is numerically observed by using a better way inthe approaches.

)
2 f (x, t)+ d $ 7 V (x, t) f (x, t) A - d $ < D $ df (x, t) F = S (x, t), (1)2t)

where S is a source/sink of f and D is a tensor of a diffusion))coefficient expressed as D = D bb + D = (I - bb) . Hereb = B/ | B | = B/B is a unit vector of a magnetic field B and)I is the identity tensor. In order to solve directly the fluid
equation (1), two kinds of differential operators along andacross a field line, i.e. d and d = , should be needed. It is not
so easy to appropriately realize the operators in the 3-d field
line structure with the island and ergodic zones.
Accordingly, one may consider that the Langevinequation can be used to solve the fluid equations, instead of
directly solving them. In this case, the differential operatorsin the fluid equation (1) is represented by using the Langevinequation of a stochastic process X(t):dX (t) = ( V (X (t), t) + d $ D (X (t), t) 2 dt + v $ d Wt , (2)))

and the Langevin equation can be solved easily even in the3-d field line structure, e.g. see refs. [5-7], where))v = D bb + D = d I - bb n satisfies the condition
D ij = v im v jn g mn , g mn is the metric coefficient, and Wt

2. Stochastic approach to edge plasmatransport
2.1 Fluid representation
In studies of the edge plasma transport, a simplified fluid©2004 by The Japan Society of Plasma
Science and Nuclear Fusion Research

Corresponding author’s e-mail: kanno@nifs.ac.jp1527

Kanno Ryutaro et al., Monte Carlo Simulation of Particle Transport in the LID configurationt

0 = d J [X] = d # ds E < 1 DX (s) F
20denotes a Wiener process. We will, however, find difficultyin the precision of the method based on the Langevinequation through the following discussion. For the simplicity)of the discussion, we assume hereafter that the tensor D is)expressed as (1/2) I and there is no source/sink, S = 0 .
When the fluid equation (1) is linear for f , the solution ofeq. (1) with the initial condition f (x, 0) = { (x) is given byusing the Feynman-Kac (F-K) formula [8] asf (x, t) = # dy 3 g (x, t; y) { ( y),where E [ $ ] denotes a mathematical expectation based on thepath integral (4), t = const is the density, p is the pressure,and the mean forward derivative is defined as

1
DX (t) / lim
f E [X (t + f ) - X (t) X (t)] = V (X (t), t) (9)f. 0(3)

Here E [ $ | $ ] is a conditional expectation. According to thevariational principle, if we find the process X (t) giving theextremal value of the kinetic energy of fluid, then its trendterm V = DX (t) is the solution of the N-S equation. But atpresent the stochastic variational method is restricted bysevere conditions; e.g. an incompressible fluid should beconsidered, d $ V = 0 and t = const . Thus, the methodneeds the further extension. As a result of the above
discussions, we find difficulty in the fluid representation ofthe edge plasma transport in the 3-d field line structure with
the island and ergodic zones.

t2d X (s)g (x, t; y) = #D X exp )- 1 # ds <- V (X (s), s) F 3 . (4)2 0dsHere a stochastic process X (t) starts from X (0) = y andarrives at X (t) = x . If the trend term V is a mapping of thefluid property f itself, i.e. V = V (f) , then the fluid equation(1) becomes nonlinear, e.g. the Navier-Stokes (N-S)
equation: V = f . When the F-K formula is stretched ad hocin this case, the evolution of a fluid property f should berepresented by using eq. (3): for n = 0, 1, 2, ...,

2.2 Test particle representation
In the test particle representation, there is no difficultywhere we have seen in the previous subsection. We trace theorbits of the guiding centers in the fixed magnetic field underthe effects of the collisions and the anomalous diffusion, inorder to numerically observe the distribution of the guiding
centers in the configuration space. If the effects of thecollisions and the anomalous diffusion are neglected, theguiding center motion is expressed as [11](5)and the transition probability is expressed as

g (n) (x, t + f ; y, t)= #D X exp )- 1 #
2 t

t +f

ds <(6)2dX (s)- V _ f (n) (X (s), s)i F 3,dsRd X = v = 1 S v cB + ma v d # Bmea Bdt
B * SSTVn
ma 2+ e ea + e Bv o b # dB WW,
aWX
dvn
ma=c B + e B v d # B m $ dB ,dtama B *

where the time interval f is small enough compared with thetime scale of interesting evolution of f , i.e. O (f) % O (1) ,but is large enough to sufficiently diffuse random walkers
drawing the paths X (s) : its mean square displacement isestimated as (dx) 2 + O (f) . In the method, an error of theapproximation is required to be always less than O (f) . If anerror of f , h (x, t) ordered as | h | /| f | + O (f) , is caused, i.e.
f (n) = f + h = f (1 + fht ) , then we have the following
estimation in the next step of the scheme (5);Equation (5) &2f
1t
2t + d $ 7 V( f ) f A - 2 ∆ f - h f+ O (f ) + g = 0.

(8)

& 2V + (V $ d) V - 1 ∆ V = - t1 dp,2t2where the transition probability g (x, t; y) is expressed by thepath integral:f (n + 1) (x, t + f) = # dy 3 g (n) (x, t + f ; y, t) f (n) (y, t),2dn= 0,dt

(10)

(11)
(12)

where B * = B + (ma/eaB) v b : d # B , v = v : b is theparallel velocity, n is the magnetic moment, and a is thespecies of the particle. Here we neglect the effect of theelectric field. The Coulomb collision is expressed as [12](7)Here, expanding eq. (5) to first order in f , we have thedifferential equation (7). Therefore, in general it is not clearthat the method can accurately treat the fluid equation with
the nonlinear term.
In order to avoid this problem, one can use a stochastic
variational method [9,10]. The scheme of this method is
discussed in detail in ref. [10]. In the method, for example,
we see that the variational principle of a functional J [X] isequivalent to the N-S equation [9];

(a)

( m)

dm = - v d m dt + (1 - m 2 ) v (da) dW t ,RV(a)Jdv E NO W(a) SE3K
dE = - 2v E S E - K + (a)T dt
2 v E dE O WWSLPTX(E)+ 4Ta EvE(a) dW t ,(13)(14)

where m = v /| v | is the cosine of the pitch angle,
E = (1/2) m a | v | 2 is the energy of the particle, o (da) is thedeflection frequency, and o (Ea) is the frequency of the energy
scattering. The differential equations (10) - (14) are solved by2528

Kanno Ryutaro et al., Monte Carlo Simulation of Particle Transport in the LID configuration

2.3 Strike point pattern on LID headusing the Monte Carlo method [12]. When an ad hoc
anomalous diffusion in the configuration space is considered,the motion in the configuration space is modified to the))Langevin equation (2) with V = v and D = Da , where))
Da = Da (I - bb) denotes the tensor of the anomalous
diffusion.
If we want to accurately treat the evolution of theparticle motions and the fields, of course we must select aparticle simulation [13-15]. But it is highly time-consuming
to solve all of the particle motions and the equations of thefields in the 3-d field line structure and in the interesting timescale. For the understanding of the edge plasma transport in
the 3-d field line structure, the test particle representation has
an advantage of relatively easily addressing it in spite of
fixing the fields. As a result of the discussions, we canconclude that a better way is the test particle representation.

By tracing the orbits of the guiding centers under theeffect of the Coulomb collisions, the strike point pattern onthe LID head is numerically observed. In this calculation, theshape of the head is simplified as a simple plate shown inFig. 1. We use the vacuum magnetic field to calculate theorbits. We consider the situation that the test particles are
mono-energetic protons with E = 1 keV, the distribution of
the pitch angles is uniform at the start points of the particles,
and the pitch angle scattering is dominant in collisions. We
also assume that all of the particles start from the magnetic
flux surface located at the edge of the core region, which isvery close to the island separatrix (see Fig.1), and theparticles are distributed uniformly on the surface. In this case,the width of the strike point pattern is evaluated as d + 6 cm,through the dimensional analysis: | v |/ L + 4D = /d 2 , whereL + 10 2 m is the connection length between the edge of thecore region and the LID head, and D = + 1 m 2 /s is aperpendicular diffusion coefficient guessed from the neoclassical coefficient in the core region for the collisionlessregime.
When the collision frequency of the edge plasma isestimated as o = 2 # 10 5 s - 1 , we find that the strike point
pattern is caused by the diffusion to the outside of the torus
and it is not symmetric, as shown in Fig. 2. Here the test
particles are assumed to be absorbed completely on thesurface of the head. On the other hand, when the collision
frequency is estimated as o = 1 # 10 3 s - 1 , the strike point
pattern is drastically changed as shown in Fig. 3. The widthof the pattern is measured as d + 6 cm. Comparison between
the strike point patterns in Fig. 2 and 3 shows that for thecollisional regime the orbits of the particles are not sensitiveFig. 1 (a) Poincaré plots of field lines of the vacuum
magnetic field and sketch of the simplified LID headat the horizontally elongated poloidal cross section.
The height of the head is 20 cm and the angle of thedepth is 5.2 degrees. The magnetic axis and the Opoint of the island are located at R ax = 3.54 m andR O = 4.09 m, respectively. (b) Expanded illustration ofFig. 1 (a). Field lines of the island are plotted by thedots, and the magnetic flux surface from which thetest particles start is shown by the solid square.

Fig. 2 Strike point pattern on the head for the collisional
regime. The number of the particles is 2 # 104, and 55% of them are absorbed into the head in this figure.
The remaining particles are moving in theconfiguration space. The pattern is not symmetric on
the surfaces of the head 2 and 4.

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Kanno Ryutaro et al., Monte Carlo Simulation of Particle Transport in the LID configuration

point patterns on the simplified LID head have been
numerically observed by tracing the orbits of the guiding
centers in the fixed magnetic field under the effect of theCoulomb collisions. We have seen that the shape of thepattern strongly depends on the collision frequency.
In this article, we have neglected effects of electric field,energy scattering, anomalous diffusion, charge exchange, etc.
These effects will be discussed in future work.

References[1] P.C. Stangeby, The Plasma Boundary of Magnetic
Fusion Devices (IOP, Bristol and Philadelphia, 2000).
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Fig. 3 Strike point pattern on the head for the collisionlessregime. The number of the particles is 2 # 104, and 60% of them are absorbed into the head in this figure.
The remaining particles are moving in theconfiguration space. The regions of the patternpointed by A and A’ are caused by the neo-classicaldiffusion.

to the field line structure of the island, however, for thecollisionless regime the orbits are affected by the island
structure and the pattern is caused by the neo-classicaldiffusion. The physical understanding of the strike point
patterns, especially the understanding of the broken
symmetry of the pattern in the collisional case, will bereported in detail in near future.

3. Conclusion
We have discussed the fluid and test particle
representations of the edge plasma transport in the 3-d field
line structure with the island and ergodic zones, and haveshown the advantage of the particle representation. The strike

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