1. Simulation of ECCD and ECRH for SUNIST Z. T. Wang 1, Y. X. Long 1, J.Q. Dong 1 ， Z.X. He 1, F. Zonca 2, G. Y. Fu 3 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

2. 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.

3. 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.

4. For the relativistic particles t he 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.

5. 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.

6. In section Ⅱ Exact guiding center variables f OR 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.

7. II. Exact guiding center variablesII. In tokamak configuration, the relativistic Hamiltonian of a charged particle can be expressed as (4)

8. We introduce a generating function for changing to the guiding center variables, (11)

9. The Jacobian in the area-conserved transformation is unity [3], that is, The exact Hamiltonian for the relativistic particles is It is suitable for particle simulation from which we can get equations of motion and Vlasov ’ s equation.

10. Ⅲ Quasi-linear equation For the gyro-kinetics the Hamiltonian could be averaged; To derive the quasi-linear equation we form a new invariant which actually is an implicit Hamiltonian For the trapped particles in the large aspect ratio configuration

11. For the circulating particles, which is the toroidal magnetic fluxen closed by drift surface. The bounce frequency and the procession frequency are obtained

12. The bounce-averaged gyro-frequency for the trapped particles is while for the circulating particles, New momenta are conjugate to In the extended phase space the Hamiltonian is written as follows,

13. According to Liouville ’ s theorem, the distribution function, f, satisfies Vlasov ’ s equation where f can be divided in two parts, the averaged part and oscillatory part, The linear solution of Eq.(29) The quasi-linear equation

14. For one harmonic which consistent with the code developed in Frascati

15. 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 is about 1.1, =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. Fig. 1 Distribution function versus and.

16. Fig. 3 The driven current versus time in ampere Fig. 4 The temperature versus time normalized by.

17. 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.

18. 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.

19. 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).

20. [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).