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

33 4,3 4,3 4,3

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.

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 wave-plasma 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.

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.

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.

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

ecmcRRAc

ePA

c

ePA

c

ePH ZZRR

420

22222 ]/)()()[(

0 RRR Ac

eumP R

c

e+0 AuRmP 0 ZZZ A

c

eumP

(4)

We introduce a generating function for changing to the guiding center variables,

))(lnexp(2

2

0000000

2000

1 ZXtgRm

X

R

R

Rm

XRmF

ln0

000 R

RRmX C

2

1 20 CmP

XPZ

)R

cosexp(RR

CC

24

2

sinR

sinPZC

X

(11)

sincos

0cRemP cR

The Jacobian in the area-conserved transformation is unity [3], that is,

dXdddPdPJdPd x

1J

The exact Hamiltonian for the relativistic particles is

ecmcePRR

RPmH c

c42

022

2222

0 }][1

]cossin)[(2{

It is suitable for particle simulation from which we can get equations of motion and Vlasov’s equation.

Ⅲ Quasi-linear equation For the gyro-kinetics the Hamiltonian

could be averaged;

ecmcumPmH c42

0222

00 )2(

To derive the quasi-linear equation we form a new invariant which actually is an implicit Hamiltonian

dXPx21

For the trapped particles in the large aspect ratio configuration

)]()1()([)/(8

12

11

5.00000 kKkkE

mPmqRt

For the circulating particles,

)(2

2000

200 kE

umqRrmc

which is the toroidal magnetic fluxen closed by drift surface. The bounce frequency and the procession frequency are obtained

)(2

)/(

10

5.000

kKqR

mPbt

)]1()(

)([

ˆ4]

2

1

)(

)([

2 21

1

1200

0

1

1200

0 kkK

kE

Rm

sP

kK

kE

Rm

P

ppt

)(2 0

0

kKqR

ubc

)(

)(ˆ)]

21(

)(

)([

2 20

20

2

0

20

kK

kE

R

suk

kK

kE

rR

uq

ppbcc

The bounce-averaged gyro-frequency for the trapped particles is

)]2

1

)(

)((21[

1

10

kK

kE

while for the circulating particles,

)]}2

1()(

)([

)1(

21{

2

20 k

kK

kE

k

PP ,, ,, New momenta are conjugate to

In the extended phase space the Hamiltonian is written as follows,

HtqpHqpH ),,(),(

tqHp nn 11 ,

According to Liouville’s theorem, the distribution function, f, satisfies Vlasov’s equation

0

H

fH

P

fP

P

fP

ffff

t

f

where f can be divided in two parts, the averaged part and oscillatory part, fff

~

The linear solution of Eq.(29)

)(

/)(~

1111

lmn

P

flH

fmH

P

fnH

H

fHf

The quasi-linear equation

)(ˆˆ fCfLDLf

t P

ln

HL

ˆ

}(21 lmnHD

For one harmonic

)exp(}exp(

]22

)()[(

0

11111

tiliminilrkii

i

JJA

c

eJJA

c

ekJA

c

eeH

cl

llRk

llZklkkl

)()(2

////

2

NNDP

PkKD res

)/()()(2

//PkK

N thres

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, 0R

//N //N 1.0D

=0.3m, r=0.01m where r is

=0.2 and Δ =0.1,

numerical results are given below,

the resonant position.

Fig. 2 distribution function versus //P

.

Fig. 1 Distribution function versus P

and //P

.

Fig. 3 The driven current versus time in ampere

Fig. 4 The temperature versus time normalized by

20cm

.

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).