3, 4
Simulation of ECCD and ECRH for SUNIST
Z. T. Wang1, Y. X. Long1, J.Q. Dong1， Z.X. He1, F. Zonca2,
G. Y. Fu3
1. Southwestern Institute of Physics, P.O. Box 432, Chengdu
610041, P. R. C.
2. Associazione EURATOM-ENEA, sulla Fusione, C.P. 65-00044
Frascati, Rome, Italy
3. Plasma physics laboratory, Princeton University, Princeton,
New Jersey 08543
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Abstract
Quasi-linear formalism is developed by using
canonical variables for the relativistic particles.
It is self-consistent including spatial diffusion.
The spatial diffusion coefficient obtained is
similar to the one obtained by Hazeltine.
The formalism is compatible with the
numerical code developed in Frascati.
An attempt is made to simulate the process
of electron cyclotron current drive (ECCD)
and electron cyclotron resonant heating
(ECRH) for SUNIST.
The special features in this paper are the
relativistic quasi-linear formalism and to see
resonance in the long time scale.
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Ⅰ Introduction
Interaction of radio-frequency wave with plasma in
magnetic confinement devices has been a very
important discipline of plasma physics.
To approach more realistic description of waveplasma interaction in a time scale longer than the
kinetic time scales, bounce-average is needed.
The long time evolution of the kinetic distribution can
be treated by Fokker-Planck equation.
The behavior of the plasma and the most interesting
macroscopic effects are obtained by balancing the
diffusion term with a collision term.
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For the relativistic particles the action and angle
variables initiated by Kaufman [1] are introduced.
“There has been a gradual evolution over the
years away from the averaging approach and
towards the transformation approach” said
Littlejohn [2].
The
technique
of
the
area-conserved
transformation proposed by Lichtenberg and
Lieberman [3] is employed.
A new invariant is formed by using bounce
average which actually is an implicit Hamiltonian
and from which the bounce frequency and
processional frequency can be calculated.
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Using new action and angle variables
quasi-linear
equation
is
derived
including spatial diffusion.
For the circulating particles, under the
conditions of small Larmor radius and
first harmonic resonance, the derived
diffusion coefficient is compatible with
the numerical code developed in
Frascati [4].
The distribution function is obtained
after the wave power is put in. The
driven current and the absorbed power
are calculated for SUNIST.
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In section Ⅱ Exact guiding center variables
fOR the relativistic particles are obtained. The
bounce-averaged quasi-linear equation is
derived in section Ⅲ. Numerical results of
electron cyclotron current drive and resonant
heating for SUNIST are given in section Ⅳ. In
the last section summary is presented.
Supported by National Natural Science
Foundations of China under Grant Nos.
10475043,
10535020,
10375019
and
10135020.
II . Exact guiding center
variablesII .

In
tokamak
configuration,
the
relativistic Hamiltonian of a charged
particle can be expressed as
e
e
e
H  [( PR  AR ) 2  ( PZ  AZ ) 2  ( P  RA ) 2 / R 2 ]c 2  m02 c 4  e
c (4)
c
c
PR  m0 u R 
e
AR
c
e
P  Rm 0 u + RA
c
PZ  m0 u Z 
e
AZ
c
We introduce a generating function for changing to
the guiding center variables,
m0  0 R02
X
R
X
F1  
exp(
)(ln

) 2 tg  ZX
2 (11) m0  0 R0
R0 m0  0 R0
PR  m0  c e

Rc
cos 
PZ   X
RC
X m 0  0 R0 ln
R0
sin 
 cos 
R  RC exp( 
)
RC
2
Z  PX   sin  
sin 2
4RC
1
P  m0  C  2
2
The Jacobian in the area-conserved transformation is unity [3],
that is,
d  JdP dPx dP ddXd
J 1
The exact Hamiltonian for the relativistic particles is
Rc 2 2
1
2
H  {2m0  c P [( ) sin   cos  ]  2 [ P  e]2 }c 2  m02 c 4  e
R
R
It is suitable for particle simulation from which
we can get equations of motion and Vlasov’s equation.
Ⅲ Quasi-linear equation
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For the gyro-kinetics the Hamiltonian
could be averaged;
H  (2m0  c P  m02 u2 )c 2  m02 c 4  e
To derive the quasi-linear equation we form a new invariant
which actually is an implicit Hamiltonian
1

2
 P dX
x
For the trapped particles in the large aspect ratio configuration
8qR0 m0 ( 0 P / m0 ) 0.5
2
t 
[ E (k1 )  (1  k1 ) K (k1 )]

which is the toroidal magnetic fluxen closed by drift
surface. The bounce frequency and the procession
frequency are obtained
 bt
 ( 0 P / m0 ) 0.5

2qR0 K (k1 )
2 0 P
4 0 P sˆ E (k1 )
E ( k1 ) 1
2
 t 
[

]

[

(
1

k
1 )]
2
2
 p m0 R0 K (k1 ) 2  p m0 R0 K (k1 )
For the circulating particles,
2qR0 m0 u 0
 0 m0 r
c 

E (k )
2

2
bc 
u 0
2qR0 K (k )
u20 sˆ E (k )
E (k )
k2
c  q bc 
[
 (1  )] 
2 p rR0 K (k )
2
 p R02 K (k )
u20
The bounce-averaged gyro-frequency for the trapped particles is

0
E (k ) 1
 [1  2 ( 1  )]

K (k1 ) 2
while for the circulating particles,
2
E (k )
k2

 {1  2
[
 (1  )]}

2
k (1   ) K (k )
0
New momenta
 , P , P
are conjugate to
In the extended phase space
the Hamiltonian is written as follows,
H ( p, q )  H ( p, q, t )  H
pn1   H , qn 1  t
, , 
According to Liouville’s theorem, the distribution function,
f, satisfies Vlasov’s equation
f
f
f
f  f  f  f
f
     
   P
 P
 H
0
t




P
P
H
where f can be divided in two parts, the
averaged part and oscillatory part,
~
f  f  f
The linear solution of Eq.(29)
~
f
f
f
f
f  (H 1
 nH 1
 mH1
 lH 1
)/
H
P

P
(  n   m  l)
The quasi-linear equation
f
ˆ DL
ˆf  C( f )
L

ˆ 
L

n

l



H
  
 P
  t
D  H12 (  n  m  l}
For one harmonic
J  J l 1 e
J  J l 1
e
e
H 1l  [(e k    Ak ) J l (k   )    AZk l 1
   ARk l 1
]
c
c
2
c
2i


i l exp( ik  rc  il  0 } exp( in   im  l  it )
P2
D  K (k )
D ( N res  N // )

P//
2
N res 
2 K (k )


( 
) /(  th P// )

which consistent with the code developed in Frascati
Ⅲ Numerical results for SUNIST
There is a magnetron for SUNIST. The frequency is 2.45GHz.
The power is about 100KW. For the experiment condition
0

is about 1.1,
R0 =0.3m, r=0.01m where r is
the resonant position.
N //=0.2
and ΔN =0.1, D  0.1
//
numerical results are given below,
.
Fig. 1 Distribution function
versus P and P//
.
Fig. 2 distribution function versus P//
Fig. 3 The driven current versus
time in ampere
.
Fig. 4 The temperature versus time
m0 c 2
normalized by
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Ⅳ. Summary
First the action and angle variables are used [1].
Secondary area-conserved transformation is
employed [2].
The bounce-averaged quasi-linear Fokker-Planck
equation for the relativistic particles is rigorously
obtained in canonical variables including spatial
diffusion.
The spatial diffusion coefficient obtained is
similar to the one obtained by Hazeltine [6].
For the SUNIST parameters the distribution
function, the driven current, the temperature are
calculated in Figs. 1-4.
The special features in this paper are the
relativistic quasi-linear formalism and to see
resonance in the long time scale.

For the SUNIST parameters the
distribution functions, the driven current,
the temperature, are calculated in Figs.
1-4. The special features in this paper
are relativistic quasiliear formalism and
to see resonance in the long time scale.
References
[1] Allan N. Kaufman, Phys. Fluids 15,
1063(1972).
[2] Robert G. Littlejohn, J. Plasma physics 29,
111(1983).
[3] J. Lichtenberg and Lieberman, Regular and
Stochastic Motion, Applied Sciences 38,
(Springer-Verlag New York Inc. 1983).
[4] A Cardinali, Report on Numerical solution
of the 2D relativistic Fokker-Planck equation
in presence of lower hybrid and
electron cyclotron waves.
[5]Zhongtian Wang, Plasma Phys. Control.
Fusion 41, A697(1999); Doe/ET-53088-593.
[6]R.D. Hazeltine, S.M. Mahajan, and
D.A. Hitchcock, Phys. Fluids, 24, 1164 (1972).
[7] Z. T. Wang, Y. X. Long, J.Q. Dong,
L. Wang, F. Zonca, Chin. Phys. Lett. 18,
158(2006).