Proceedings of ITC/ISHW2007

Study of neoclassical transport of LHD plasmas
applying the DCOM/NNW neoclassical transport database
Arimitsu Wakasa, Sadayoshi Murakamia) and Shun-ichi Oikawa
Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japana)

Department of Nuclear Engineering, Kyoto University, Kyoto 606-8501, Japan
wakasa@fusion.qe.eng.hokudai.ac.jp

Abstract
A neoclassical transport database, DCOM/NNW, is developed including the configuration changes due tofinite beta effect in LHD. The mono-energetic diffusion coefficients are evaluated based on the MonteCarlo method by DCOM code and the mono-energetic diffusion coefficient database is constructed using aneural network technique. Input parameters for the database are the collision frequency, the radial electricfield, the minor radius and the plasma beta. The DCOM/NNW database is applied to finite beta plasma, andthe neoclassical transport coefficients and the ambipolar radial electric field are evaluated in LHD.
Keywords:
LHD, neoclassical transport, neural network, Monte Carlo method
1. Introduction
In helical systems, neoclassical transport isone of the important issues for sustaining
high-temperature plasma. In particular, in thelong-mean-free-path(LMFP)
regime,theneoclassical transport coefficient increases as
collision frequency decreases (1/ regime), andneoclassical transport plays an important role aswell as anomalous transport by plasma turbulence.
Many studies have been carried out toevaluate the neoclassical transport coefficient
analytically and numerically in helical systems.
Among the studies, the Drift Kinetic Equation
Solver (DKES)[1, 2] code has been commonlyused for experimental data analyses[3, 4] andtheoretical predictions.[5] However, in the LMFP
regime, especially with finite beta, a large
number of Fourier modes of the magnetic fieldmust be used for distribution function and aconvergence problem occurs.
On the other hand, the neoclassicaltransport coefficient has also been evaluatedusing the Monte Carlo method directly following
particle orbits, where the mono-energetic
diffusion coefficients are estimated by the radial
diffusion of test particles.[6-8] This method has agood property in the LMFP regime except for its
long calculation time. Thus, we have developed
a Monte Carlo simulation code, the Diffusion
Coefficient Calculator by Monte Carlo Method(DCOM) code,[9] which is optimized inperformance in the vector computer.
To evaluate the neoclassical diffusion
coefficient of thermal plasma, we must take energy
convolution into account. Therefore, it is necessary
to interpolate discrete data using the DCOM. In a
non-axisymmetric system, the diffusion coefficientshows complex behavior and strongly depends on
collision frequency and radial electric field (e.g.,
1/ ,and regimes). The interpolation basedon a traditional analytical theory has a problem
with connected regions between two regimes.
The neural network (NNW)[10], which has astrong nonlinearity and high fitting abilities fornonlinear phenomena, is applied in the study offusion plasma and used in constructing the NNW
database for neoclassical transport in TJ-II
plasma.[11] Therefore, the NNW method is applied
to the fitting of the diffusion coefficient of LHD,
which shows a complex behavior in several
collisional regimes, i.e., ,, 1/ , plateau, andP-S regimes. A multilayer perceptron NNW with
only one hidden layer, generally known as MLP1,is used. The neoclassical transport database,
DCOM/NNW[12], has been constructed with input
parameters r/a, *, and G and D* can be obtained
as an output of the NNW, where * is thenormalized collision frequency, G is the normlizedradial electric field and D* is the normalized
diffusion coefficient respectively. We haveconstructed six database each of six LHD
configurations with different values of theProceedings of ITC/ISHW2007( 1, 0)( 2, 10)( 1, 10)( 3, 20)( 3, 10)0.3

Bm,n/B0,00.2( 2, 20)( 0, 10)( 2,-10)( 1,-60)( 4, 20)

0.1
0-0.1
-0.2

0

0.2 0.4 0.6 0.8
r/a(b)( 1, 0)( 1, 10)( 2, 10)( 2, 0)( 0, 10)( 3, 10)0.3
0.2
Bm,n/B0,0

2. Construction of neoclassical transport
database including finite beta effect
The Fourier spectrum of the magnetic fieldBm,n in the Boozer coordinate as functions ofnormalized minor radius r/a in the Raxis = 3.75 m
configuration with 0 = 0% and 0 = 3% is shownin Fig.1. Here, B0,0 is the amplitude of the (0, 0)
component of magnetic field strength at themagnetic axis, where (m, n) represents the Fouriercomponent of magnetic field with the poloidal
number, m, and the troidal number, n. It is showntaht the dominant magnetic field components inLHD with 0=0% plasma are (2,10) and (1,0);(2,10) corresponds to the main helical mode and(1,0) is the toroidicity. Figure 1(b) shows that theamplitude of (2,10) and (1,0) component decrease
and a lot of other components increase by theeffect of beta value, especially in the edge plasma,and the magnetic field configuration becomescomplex.
In
thisstudy,
wecalculateamono-energetic diffusion coefficient, D, byDCOM code. Using DCOM code, we can calculate
the diffusion coefficient without convergenceproblem even if we assume a lot of Fourier modesof the magnetic field. We use 50 Fourier modes toevaluate the magnetic field in this research. To
obtain mono-energetic diffusion coefficients, we
assume the electron energy of 1.0 10 3 eV. The
magnetic field is set to 3T at the magnetic axis.
The test particles are monitored for severalcollision times until the diffusion coefficient isconverged. Figure 2 shows the normalized
diffusion coefficient D* at r/a = 0.5 in Raxis =3.75m calculated using the DCOM code as afunction of the normalized collision frequency *(a)1

( 4, 10)( 5, 10)( 3, 0)( 6, 10)( 7, 10)( 1,-10)

0.1
0-0.1
-0.2
-0.2 0 0.2 0.4 0.6 0.8 1
r/a

Fig. 1. Amplitude of dominant Bm,ncomponent normalized by B0,0 (r/a) as
function of normalized minor radius r/awith (a) 0=0% and (b) 0=3%.

D*magnetic axis shift in the major radius direction
between Raxis = 3.45 m and Raxis = 3.90 m.
At construction of this database, weassumed only vacuum plasma and have not taken
into account finite beta plasma. However, arelatively large Shafranov shift occurs in finite beta
LHD plasmas and the magnetic field configuration
becomes complex leading to rapid increase of thenumber of Fourier modes in Boozer coordinates.
In order to analyze transport ofexperimental plasma with finite beta, we have totake account into finite beta. Therefore, we apply
effect of finite beta to the neoclassical database,
DCOM/NNW. We evaluate the neoclassicaltransport coefficients and the ambipolar radial
electric field using improved DCOM/NNW..

vacuumbeta 1%beta 2%
beta 3%
-6

10-4

10-2

100

*

102

10

104Fig. 2. Normalized monoenergetic diffusion
coefficients as function of normalized
collision frequency without radial electricfield at r/a = 0.5 in Raxis = 3.75m.

without radial electric field. Here, we normalize
the collision frequency by v / R , and thediffusion coefficient by the tokamak plateau value
inthemono-energetic
case,DP,as
D P ( /16)(v 3 / R axis c 2 ) , where v, and c areProceedings of ITC/ISHW2007

D*1/nuplateau
P-S

01

2

3[%]
0Fig. 3. Normalized diffusion coefficientas function of beta value, 0, without
radial electric field at r/a =0.5 in Raxis =3.75m.

1
The sum of square error

the minor radius, the velocity of test particles, therotational transform and cyclotron frequency oftest particle, respectively. It is found that thenormalized diffusion coefficients rise rapidly as
the central plasma beta values increases from 0%
to 3%.
In Fig. 3, the diffusion coefficient at r/a=0.5 are shown as function of 0 in each regime.
We can see that the diffusion coefficients areincreased exponentially with 0 in 1/ regime. It isfound that the plateau values are also increased by
the beta value. On the other hand, in the P-Sregime, the diffusion coefficients are almost
independent with the value of 0.Thus, it isnecessary to take into account the effect of the beta
value in order to accurately evaluate transport inhigh temperature plasma, in which the collision
frequency is in the 1/ or the plateau regime.
We add the beta values to the inputs ofDCOM/NNW which is the neoclassical transport
database.. We consider two LHD configurations,
Raxis =3.60m and Raxis =3.75m. We adjust theweights of NNW using the computational resultsof DCOM, which are called training data. As beta
value, we calculate the diffusion coefficient with
0=0%, 0=1%, 0=2% and 0 = 3% for training
data. In this research, we totally used 2688 training
data for Raxis = 3.75m configuration using theDCOM with various collision frequencies, radial
electric fields, minor radius and beta values. Also,
1777 training data are used for Raxis = 3.60m
configuration.
The accuracy of the NNW depends on thenumber of hidden units. The error between training
data and outputs of NNW decrease as the numberof hidden unit increases. However, if we increase
the number of hidden unit too much, overlearning
occurs, which gives poor prediction for a newparameter data although the error is very small.
In Fig.4, it is shown that the sum of relative
error as function of the number of hidden unit ofNNW. In this research, we assume the number ofhidden unit is set to 15 and mean relative error isabout 12.3 %.
We can obtain D* by using the improved
NNW database for arbitrary *, G, r/a and newly,*0. Figure 5 shows contour plot of D , obtained by
the newly NNW, at r/a = 0.5, G = 0.0 in Raxis =3.75m. The horizontal axis indicate * and thevertical axis is 0. We can found that D*dose notdepend on the beta value in P-S regime. And D* isincreased in 1/ and plateau regime as 0 increases.
It is also seen that the plateau regime has narrowed

0.8
0.6
0.4
0.2
0020406080 100

The number of hidden units

Fig. 4. The sum of relative error asfunction of number of hidden units ofNNW.

6105104103102101
100
10-110D* 610104010-2
104.0
3.0
2.0-610 10 -5 -4101.0-310 10 -2 -1
10 10 0110

*1020 [%]0.0Fig. 5. The normalized monoenergetic
diffusion coefficients, D* as a function
D*( *, G, r/a, 0) in Raxis = 3.75m, where G
and r/a are constant (G=0.0 and r/a = 0.5),
which are outputs of DCOM/NNW.

Proceedings of ITC/ISHW2007

as0 increases.
s48584_1.48sec

1
0.8
0.6
0.4

electron
ion

0.2
00

0.2

0.4 0.6 0.8
r/a

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Beasley, Jr., and E. C. Crume, Jr.: Phys. Fluids 29 (1986)
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21.510.501electron
ion

00.2

0.4 0.6 0.8
r/a

1Fig. 6. Plasma temperature and particle
density (#48584, t = 1.48sec, 0 = 0.72%).

radial electric field[kV/m](a)s48584 t=1.48sec

0-1-2
0-30-40= 0.00%= 0.72%0.2 0.4 0.6
r/a(b)

0.8

1s48584 t=1.48sec

0.2

000.15D1[m2/sec]

3. Summary
We have extended the neoclassicaltransport database, DCOM/NNW, to include theeffect of configuration changes due to the finite
beta effect. In the finite beta plasma, the magnetic
field becomes complex and the number ofnecessary Fourier mode increases. Using extended
DCOM/NNW we can estimate neoclassicaltransport more accurately than that using theprevious DCOM/NNW.
We have investigated the neoclassicaltransport and evaluate the ambipolar radial electricfield in LHD using DCOM/NNW.

s48584_1.48sec

2.5

1.2
density[10 19/m3]

Temperature[keV]

1.4

3. Neoclassical transport analysis
Next, we study the neoclassical transport byDCOM/NNW including the finite beta effect. In
this analysis, we consider hydrogen plasma andRaxis= 3.75m. The assumed temperature anddensity profiles of electrons and ions are shownin Fig .6, and the magnetic field is B=1.5T (#48584,t = 1.48sec).
Figure 7(a) shows the ambipolar radial
electric field as a function of r/a. There is only one
root (ion root) with 0= 0.0% and 0=0.72%. We
obtain similer profiles of the electric fields in bothcases. However, it is found that the diffusion
coefficient, D1, with b0=0.0% and with 0.72% aregreatly different as shown in Fig. 7(b). In the caseof 0=0.72%, D1 increases to about three times thatof 0 = 0.0%. The heat conductivity, D3, also
increases with an increase of 0. In the case of0=0.72%, D3 increases to about four times that of0 = 0.0%.
This results show that the inclusion offinite beta effect is necessary for the accurate
evaluation of neoclassical transport..

= 0.00%= 0.72%

0.1
0.05
000.2

0.4 0.6 0.8
r/a1Fig. 7. (a) Radial electric field and (b)Diffusion coefficient as function of r/a with
0 =0.0% and 0.72%.

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SERIES 4 (2001) 408[10] C. M. Bishop: Rev. Sci. Instrum. 65 (1994) 1803.
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