1. Lecture Series in Energetic Particle Physics of Fusion Plasmas Guoyong Fu Princeton Plasma Physics Laboratory Princeton University Princeton, NJ 08543, USA IFTS, Zhejiang University, Hangzhou, China, Jan. 3-8, 2007

2. A series of 5 lectures (1) Overview of Energetic Particle Physics in Tokamaks (today) (2) Tokamak equilibrium, shear Alfven wave equation, Alfven eigenmodes (Jan. 4) (3) Linear stability of energetic particle-driven modes (Jan. 5) (4) Nonlinear dynamics of energetic particle-driven modes (Jan. 6) (5) Summary and future direction for research in energetic particle physics (Jan. 8)

3. Outline Saturation mechanism Single mode saturation: bump-on-tail problem Multi-mode problem Hybrid simulation of fishbone instability Summary

4. Destabilize shear Alfven waves via wave-particle resonance Destabilization mechanism (universal drive) Wave particle resonance at For the right phase, particle will lose energy going outward and gaining energy going inward. As a result, particles will lose energy to waves. Energetic particle drive Spatial gradient driveLandau damping Due to velocity space gradient

5. TAE Stability: energetic particle drive and background dampings Energetic particle drive Dampings Ion and electron Landau damping, collisional damping, continuum damping, “radiative damping” due to kinetic Alfven waves Drive > damping for instability G.Y.Fu and J.W. Van Dam, Phys. Fluids B1, 1949 (1989). M.N. Rosenbluth, H.L. Berk, J.W. Van Dam and D.M. Lindberg 1992, Phys. Rev. Lett. 68, 596 R.R. Mett and S.M. Mahajan 1992, Phys. Fluids B 4, 2885

6. First observation of TAE in TFTR K.L. Wong, R.J. Fonck, S.F. Paul, et al. 1991, Phys. Rev. Lett. 66, 1874.

7. Example of EPM: fishbone instability Mode structure is of (m,n)=(1,1) internal kink; Mode is destabilized by energetic trapped particles; Mode frequency is comparable to trapped particles’ precessional drift frequency K. McGuire, R. Goldston, M. Bell, et al. 1983, Phys. Rev. Lett. 50, 891 L. Chen, R.B. White and M.N. Rosenbluth 1984, Phys. Rev. Lett. 52, 1122

8. Bump-on-tail problem: definition H.L. Berk and B.N. Breizman 1990, Phys. Fluids B 2, 2235

9. Bump-on-tail problem: saturation mechanism We first consider case of no source/sink and no damping. The instability then saturates at The instability saturates when the distribution is flattened at the resonance region (width of flattened region is on order of The saturation is due to wave-particle trapping.

10. Bump-on-tail problem: saturation with damping, source and sink Collisions tend to restore the original unstable distribution. Balance of nonlinear flattening and collisional restoration leads to mode saturation. It can be shown that the linear growth rate is reduced by a factor of. Thus, the mode saturates at H.L. Berk and B.N. Breizman 1990, Phys. Fluids B 2, 2235

11. H.L. Berk et al, Phys. Plasmas 2, 3007 (1995).

12. Transition from steady state saturation to explosive nonlinear regime B.N. Breizman et al Phys. Plasmas 4, 1559 (1997).

13. Hole-clump creation and frequency chirping For near stability threshold and small collision frequency, hole-clump will be created due to steepening of distribution function near the boundary of flattening region. As hole and clump moves up and down in the phase space of distribution function, the mode frequency also moves up and down. H.L. Berk et al., Phys. Plasma 6, 3102 (1999).

16. Experimental observation of frequency chirping M.P. Gryaznevich et al, Plasma Phys. Control. Fusion 46 S15, 2004.

17. Saturation due to mode-mode coupling Fluid nonlinearity induces n=0 perturbations which lead to equilibrium modification, narrowing of continuum gaps and enhancement of mode damping. D.A. Spong, B.A. Carreras and C.L. Hedrick 1994, Phys. Plasmas 1, 1503 F. Zonca, F. Romanelli, G. Vlad and C. Kar 1995, Phys. Rev. Lett. 74, 698 L. Chen, F. Zonca, R.A. Santoro and G. Hu 1998, Plasma Phys. Control. Fusion 40, 1823 At high-n, mode-mode coupling leads to mode cascade to lower frequencies via ion Compton scattering. As a result, modes saturate due to larger effective damping. T.S. Hahm and L. Chen 1995, Phys. Rev. Lett. 74, 266

18. . Multiple unstable modes can lead to resonance overlap and stochastic diffusion of energetic particles H.L. Berk et al, Phys. Plasmas 2, 3007 (1995).

19. Nonlinear Hybrid Simulation of Fishbone instability Particle/MHD hybrid model Use M3D code Observed dynamic distribution flattening as mode frequency decreases. G.Y. Fu et al, Phys. Plasmas 13, 052517 (2006)

20. M3D Code M3D is a 3D extended nonlinear MHD code with multiple level of physics: resistive MHD two fluids Particle/MHD hyrid

21. M3D XMHD Model

23. Experimental observation of fishbone instability in PDX

24. Excitation of Fishbone at high  h

25. Mode Structure: Ideal Kink v.s. Fishbone

26. Nonlinear evolution of mode structure and mode amplitude

27. Saturation amplitude scale as square of linear growth rate

28. Simulation of fishbone shows distribution fattening and strong frequency chirping distribution

29. Summary Single mode saturates due to wave-particle trapping or distribution flattening. Collisions tend to restore original unstable distribution. Near stability threshold, nonlinear evolution can be explosive when collision is sufficiently weak. Mode-mode coupling can enhance damping and induce mode saturation. Multiple modes can cause resonance overlap and enhance particle loss.