Transport analysis of the LHD plasma using the integrated code TASK3D

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20 Οκτ 2013 (πριν από 3 χρόνια και 9 μήνες)

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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
1E-09
0.000001
0.001
1
1000
r
/
a
=0.75

G=0e0: DCOM
G=1e-3: DCOM
G=3e-3: DCOM
G=1e-2: DCOM
G=3e-2: DCOM
G=1e-1: DCOM
G=0e0: GSRAKE
G=1e-3: GSRAKE
G=3e-3: GSRAKE
G=1e-2: GSRAKE
G=3e-2: GSRAKE
G=1e-1: 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
2E-4
4E-4
6E-4
8E-4
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