Transport analysis of the LHD plasma using the integrated code TASK3D
A. Wakasa, A. Fukuyama, S. Murakami,
a)
C.D.
Beidler
,
a)
H.
Maassberg
,
b)
M. Yokoyama,
b)
M. Sato
Department of Nuclear Engineering, Kyoto University, Kyoto 606

8501, Japan
a)
Max

Planck

Institute
für
Plasmaphysik
, EURATOM Ass., Greifswald, Germany
b)
National Institute for Fusion Science,322

6
Oroshi

cho
, Toki 509

5292, Japan
The collision frequency
n
plateau regime
high temperature
The diffusion coefficient
D
classical transport
Neoclassical transport in
helical
device
anomalous
transport
banana
regime
P

S
regime
In helical devices, the
neoclassical transport is important issue.
n
regime
1/
n
regime
The helical trapped particles
increase the neoclassical diffusion
in the low collision frequency
regime.
As the temperature of plasma is raised in helical
device,
the neoclassical transport increases up to
the anomalous transport or more
.
Therefore, We have developed a Monte Carlo simulation code,
the
D
iffusion
Co
efficient Calculator by the
M
onte Carlo Method,
DCOM
.
accurate examination of the neoclassical transport is necessary in helical device.
DCOM
DCOM
can calculate
the mono

energetic
diffusion coefficient
without
convergence problem
even if in
LMFP
regime of
the
finite beta plasma where
a large number of Fourier modes of the magnetic field must be considered
.
Therefore, We apply the results of
GSRAKE
code to the neoclassical
transport database in the extremely low collision regime.
▼
the
problems of the computing time
.
The necessary CPU time increases rapidly in the LMFP regime
because we have to trace particle orbits for long time.
A neoclassical transport data base,
DCOM+GSRAKE/NNW for LHD (DGN/LHD)
has been constructed.
It
is necessary to interpolate
DCOM results when
we take the convolutions
of
the
mono

energetic
diffusion coefficient
because DCOM and GSRAKE results
are
discrete data.
We apply the
Neural Network technique
to the fitting of
DCOM
results.
Introduction
Monte Carlo calculation of diffusion coefficient
B
: the magnetic field
E
: the electric field
q
: charge of the particle
m
: the particle mass
v

: the velocity parallel to magnetic field.
: the velocity perpendicular to magnetic field.
Collision
Lorentz collision operator
n
d
: the deflection collision frequency
The pitch angle scattering
: time
=
cos
h
The drift velocity of the guiding center
DCOM
code evaluate the
monoenergetic
local
diffusion coefficient.
The particle orbits are directly traced.
r
0
:
initila
position of particles
r
j
: the position of
j
th
particle after
t
sec.
N
: the number of particles
●
initial radial position
r
0
are set uniformly.
initial toroidal position
initial poloidal position
N mono

energetic
particles
are released.
●
collision frequency
●
radial electric field
2
t
N
N
r
j
r
0
The diffusion coefficient is obtained
by calculating the dispersion as this
expression,
Connection of the results of
DCOM
and
GSRAKE
In
extremely low collision frequency regime
, we
combine the results of
GSRAKE
code with the results of
DCOM
to construct the neoclassical
transport database.
In the LMFP regime, a necessary computing time of DCOM code increases
in
inverse proportion to collision frequency. (e.g. at
n
*
=1
×
10

8
,
500 hours are
required
.
)
GSRAKE code is a general solution of the ripple

averaged kinetic equation.
0.000001
0.0001
0.01
1
100
10000
1000000
1E09
0.000001
0.001
1
1000
r
/
a
=0.75
G=0e0: DCOM
G=1e3: DCOM
G=3e3: DCOM
G=1e2: DCOM
G=3e2: DCOM
G=1e1: DCOM
G=0e0: GSRAKE
G=1e3: GSRAKE
G=3e3: GSRAKE
G=1e2: GSRAKE
G=3e2: GSRAKE
G=1e1: GSRAKE
DCOM
GSRAKE
R
axis
=
3.60m
b
0
=0.0%
Normalized collision frequency
Normalized radial electric field
Normalized diffusion coefficient
Normalized collision frequency,
n
*
Normalized diffusion coefficient,
D
*
Consequently, the
DCOM
results are insufficient in extremely low collision
frequency regime.
GSRAKE
can estimate the diffusion coefficients
with less computation time
but
not as detailed as in
DCOM
.
Neural network
(
NNW
) is
a technique that imitates
the
cranial nerve of
the living body
. Because NNW can have a strong nonlinear feature,
the NNW is
applicable to fitting of arbitrary, nonlinear function
.
Construction of
the neoclassical transport
database using Neural network in
LHD
:
DCOM+GSRAKE
/NNW for
LHD
,
DGN
/
LHD
synapse
axon
dendrite
cell body
output
neuron
input
electric
signal
Action of a
neuron
.
Engineering
model of a neuron.
inputs
y
=
f
(
z
)
weight
neuron
x
1
x
2
x
n
・
・
・
weighted sum
x
1
x
2
x
n
・
・
・
w
1
w
n
w
2
weighted sum
y
=
f
(
z
)
output
y
y = f
(
z
)
=
tanh
(
z
)
y
=
f
(
z
)
y
=
f
(
z
)
weighted sum
neural network
n
*
1
,
G
1
,
r/a
1
,
b
0,1
n
*
2
,
G
2
,
r/
a
2
,
b
0,2
n
*
N
,
G
N
,
r/
a
N
,
b
0,N
・
・
・
DCOM
results
and
GKASE
results
Training data
D
1, NNW
D
2, NNW
D
N, NNW
・
・
・
D
*
1
D
*
2
D
*
N
・
・
・
We modify weight values appropriately by
minimizing the root mean square error
between the training data and NNW
results.
inputs
outputs
The values which
should
output (
DCOM
and G
SRAKE
results).
bias
bias
w
1
w
3
w
0
w
2
w
1
w
2
・
・
・
・
・
・
・
R
axis
=
3.60m
:
(
n
*
,
G
,
r/a
,
D
*
,
b
0
)
・
R
axis
=
3.75m
: (
n
*
,
G
,
r/a
,
D
*
,
b
0
)
An arbitrary I/O relation can be given to NNW by
adjusting the weight to an
appropriate value.
We
adjust the weight values
of NNW using the results of
DCOM
and
GSRAKE
.
1960 DCOM
results and
200 GSRAKE
results are
precomputed
for training data in each
R
axis
.
1960
+
200
=
2160
First, weight values are given
randomly, so, NNW outputs wrong
diffusion coefficient.
The adjustable parameters of the NNW model are determined by a modified
quasi

Newton method
, the
BFGS method
.
we
newly
apply
DGN/LHD
as
a
neoclassical
transport
analysis
module
to
TASK/
3
D
,
which
is
the
integrated
simulation
code
in
helical
plasmas,
and
study
the
role
of
the
neoclassical
transport
in
several
typical
LHD
plasmas
.
This work is supported by Grant

in

Aid for Scientific Research (S) (20226017) from JSPS, Japan.
DGN
/
LHD
We incorporated
neoclassical transport
database DGN/LHD into
TASK3D.
Module structure of
TASK3D
DCOM/NNW
DGN/LHD
The outputs of each database:
DCOM/NNW
and
DGN/LHD
In extremely low collision frequency
regime, because the computational
results of DCOM don't exist,
the outputs
of the DCOM/NNW are inaccurate
.
r
/
a
=0.5
R
axis
=3.60 m
b
0
=0.0 %
At the plasma of standard profile
of temperature and of density, the
outputs of DCOM/NNW are
accurate enough.
R
axis
=3.60 m
b
0
=0.0%
Test analysis(1)
:
Low

collisional
plasma
(high
T
and low
n
)
0E+0
2E4
4E4
6E4
8E4
0
0.2
0.4
0.6
0.8
1
collision frequency in
thermal
velocity
,
n
*
vth
r
/
a
electron
ion
2
×
10

4
4
×
10

4
6
×
10

4
8
×
10

4
0.0
n
*
in the 5v
th
≈
2
×
10

7
r
/
a
=0.5
we must consider energy
convolution to
extremely
low collision regime
.
D
*
Because the neoclassical transport database,
DGN/LHD, contains the result of GSRAKE, the
outputs of DGN/LHD are appropriate
even in the
extremely low collision frequency regime.
The results of Neoclassical transport analysis
Ambipolar radial electric field
Thermal conductivity of electron
Radial electric field: almost the same
Thermal conductivity:
ion root
：
increases by a factor of about
2.
electron root
：
increases by a factor of about
1.5.
By using new neoclassical transport database, DGN/LHD, an accurate
evaluation of the neoclassical transport in
low

collisional
plasma
become possible.
By using DGN/LHD,
Test simulation of LHD plasma by using TASK3D
In non

axisymmetric
systems, the radial electric field
E
r
are determined from
ambipolar condition
,
where
G
are neoclassical particle fluxes.
Here, D1, D2, and D3 are calculated by using DGN/LHD.
Using this ambipolar radial electric field,
TR module solves particle and heat transport equations.
Summary
The GSRAKE results in the extremely low collision regime
have been included in Neural network database,
DGN/LHD
.
We incorporated
neoclassical transport database
DGN
/
LHD
into
TASK3D
.
We have been constructed the
neoclassical transport database
for
LHD
plasmas by
using neural network technique.
Future work
・
Neural network technique will be applied to the database of
calculated results of TASK.
The transport simulation with pinch velocities and more
factual model of
has to be done.
R
ax
=3.60[m]
B
=2.75 [T]
density profile is fixed.
b
0
=1.00[%]
:
proportional to
Bohm
particle pinch velocity
thermal pinch velocity
= 0.
R
ax
=3.75[m]
B
=1.5 [T]
b
0
=0.04[%]
・
single
helisity
model
(
Shaing
model)
・
neoclassical transport module, DGN/LHD
2 types of neoclassical modules
Comparison of single
helisity
model and DGN/LHD
Only neoclassical transport are assumed.
particle pinch velocity
thermal pinch velocity
= 0.
Use
neoclassical transport module, DGN/LHD
Comparison of anomalous transport model
:
current diffusive interchange mode
(
i
)
(ii)
[1]
[1]
Itoh
K.,
Itoh
S.

I. and Fukuyama A. 1992
Phys.
Rev.Lett
. 69 1050
(
i
)
Bohm
model
(ii) CDIM model
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