Nonlinear Ponderomotive Force by Low Frequency Waves and

1. 12th International Workshop on Spherical Torus 2006 Oct.11-13 Chengdu China Nonlinear Ponderomotive Force by Low Frequency Waves and the Possibility of Alfvén Wave Current Drive in Spherical Tori Zhe Gao Department of Engineering Physics, Tsinghua University, Beijing 100084, China Electronic mail: gaozhe@mail.tsinghua.edu.cn in collaboration with Prof. N. J. Fisch, Dr. Hong Qin Princeton Plasma Physics Laboratory, Princeton, NJ08543, USA and Prof. Yexi He Department of Engineering Physics, Tsinghua University, Beijing 100084, China

2. Low frequency wave current drive • RF resonant currentdrive V Vph= /k Resonance condition • Low-frequency waves, Alfvén waves or fast waves, were considered as an attractive mechanism of driving plasma current because of its potential high efficiency, no density limit and the convenience of high power RF generating and launching. • Electron trapping may dramatically reduce the current drive efficiency in the subthermal resonant regime

3. Helicity inject current drive • Alternatively, the possibility of increasing the current drive efficiency by helicity injection has been proposed (Ohkawa 1989) • This idea was later described as helicity balance between the input by wave and the dissipation by resistivity and viscosity,(Taylor 1988, Mett1989) and developed for arbitrary polarization waves. (Chan et al 1990) • Helicity conversation • current form and sustain during plasma relaxation in toroidal pinches, spheromaks and low aspect ratio tokomaks (Taylor 1986, Jarboe et al 1994, 2002) • Is it possible to use RF wave to drive nonresonant current in steady status? helicity flux: by DC, AC & RF

4. Ponderomotive force by RF • helicity injection current driven by low-f waves can be referred to as a “ponderomotive force” of the waves • Nonresonant force both in fluid(Klima 1980, Elfimov et al 1994, Tsypin et al 1995)and kinetic(Chan and Chiu 1993, Fukuyama et al 1993)model • If true, this nonresonant current drive in the toroidal geometry does not depend on trapped particle effects and the current drive efficiency is expected to be strongly increased. • Some computations for AWCD design in ST were based on the nonresonant current drive scheme. (Cuperman 1998, Elfimov 2001 … …) Chan and Chiu 1993

5. An apparent incongruity exists • The well-known ponderomotive forces only in the direction of the gradient of second-order field quantities. (Gaponov and Miller 1958) Under the condition of , • However, for pure propagating waves without dissipation, the RF field is fully symmetric in the propagating direction. Where is the nonresonant force from? Is it ture?

6. Single particle picture • cancellation between the electric force and the magnetic force in the symmetry directions • consider simplified fields: and cancellation

7. Lagrangian fluid element analysis • Lagrangian fluid element (at a fixed spatial location) • the force due to the displacement of single particle, disappears; • a new electric field force acting on the charged fluid element appears. • the total EM force on the fluid element = the nonresonant force

8. Lagrangian fluid element analysis (II) • But, the stress must be included in the fluid momentum equation. • Then, the total parallel force reduces to zero. The fluid picture becomes consistent with the single particle picture. • The fluid helicity of the fluctuations fully compensates the dissipation of the EM helicity For cold plasma, the stress reduces to the Reynolds stress

9. Lagrangian fluid element analysis (III) • For hot plasmas, the thermal pressure and viscosity should be included in the fluid picture. • The reason for the confusion among previous results is whether the stress force is considered completely(including thermal pressure, viscosity and Reynolds stress) • correct treatment: a model for the equation of state needs to be given, e. g. the CGL model was used and it is found the steady-state collisionless dynamo effect is absent in the double-adiabatic MHD theory. (Litwin 1994) But, many nonlinear modifications make things complicated and different closure schemes may give different conclusions

10. Lagrangian fluid element analysis (IV) • incomplete treatment: no stress contribution (Chan and Chiu 1993, Fukuyama et al 1993) or only the Reynolds stress (Elfimov et al 1994) or the Reynolds stress and viscosity (Tsypin et al 1995) consideration. the Reynolds stress gives a similar trend as the kinetic stress force, but it cannot cancel all the terms related to the EM force • A big difficulty in the fluid description is the evaluation of the thermal pressure • Instead of adopting a particular closure scheme, we will carry out a nonlinear kinetic analysis in the following for hot plasmas.

11. Nonlinear RF force in kinetic formalism • Expand the distribution function in powers of electric field • the second-order, time-averaged Vlasov Eq. the equilibrium distribution, usually assumed to be Maxwellian the linear response to the RF field the slowly varying in time, second-order response

12. Nonlinear RF force in kinetic formalism (II) • the time-averaged parallel momentum equation quasi-linear EM force Nonlinear stress force Except for the J0 terms, all the components in the quasi-linear EM force are cancelled by the kinetic stress gradient force.

13. Nonlinear RF force in kinetic formalism (III) • under the assumption of low frequency Retaining the first order of , we get for real , , and Same as in Chan and Chiu 1993 Nonresonant force: divergence of helicity flux

14. Nonlinear RF force in kinetic formalism (IV) • the total nonlinear RF force or Unless the wavenumber or is complex, all the driving terms are multiplied by that is, only the Landau resonant forces survive. • For steady state using RF driven current in toroidal systems, the frequency and the toroidal and poloidal wavenumbers are real. Therefore, the collisionless nonresonant force vanishes

15. Alfvén Wave Current Drive in Spherical Tori • In ST, most of electrons are trapped and more resonant electron for low frequency wave are trapped. For example, R/a=1.3, r/a=0.3 ft~68% ( fu~32%) For vp/vte=0.1,the fraction of untrapped resonant electrons is The fractional energy absorbed by the untrapped electrons exceeds their abundance almost by a factor of 1~3 (Elfimov NF 1990)

16. redistribution of the momentum • Considering the redistribution of the momentum from trapped electrons to the bulk plasma over a period . The final fraction of the original wave momentum to the untrapped electrons is (Puri and Wilhelm, Conf. Proc. No. 190 APS 1989) (for Zeff=1.5) (for conventional Tokamak, this fraction can up to above 50%) • Energy input produces a current of duration of low phase velocity increases the current-drive efficiency, while low untrapped (moment) fraction decreases it

17. Ware pinch and bootstrap current the moment absorbed by trapped electrons Ware pinch will be clear when Alfven wave is injected. Bootstrap current can completely recover the moment carried by trapped electrons? Strong peak of density due to the wave inject will change plasma equilibrium. (self-consistent equilibrium)

18. the electric field force due to the movement of single particle. cancellation the nonresonant force (the quasi-linear EM force) the nonlinear stress force (reduces to the Reynolds stress) cancellation Summary • In collisionless plasmas, the RF force by low f wave, in the picture of single particle, is shown to be consistent with that in the fluid and kinetic theory. • In the parallel direction, for a single particle, and for a Lagrangian fluid element, • Therefore, in collisionless plasmas, only the Landau resonant forces survive in the parallel direction, and none of the ponderomotive forces by low frequency waves can drive nonresonant current. Trapped electron effects must be faced. • In ST, low untrapped (moment) fraction may dramatically decrease the drive efficiency, however, the key of Alfven wave current drive in ST is whether/how the collisional and neoclassical effects be of real effect. the Lorentz force

19. ACKNOWLEDGEMENT • This work is supported by PRC-US Fusion Cooperation Program, National Science Foundation of China(Grant No. 10535020), and Foundation for the Author of National Excellent Doctoral Dissertation of PR China