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. Overview of Energetic Particle Physics in Tokamaks
Tokamak basics
Roles of energetic particles in fusion plasmas
Single particle confinement
Alfven continuum and shear Alfven eigenmodes
Energetic particle-driven collective instabilities
Nonlinear dynamics
Summary

4. Tokamak basics
Tokamak is an axisymmetric torus with toroidal magnetic field produced by external coils and poloidal field produced by toroidal plasma current
Both fields are necessary to confine charged particles or plasmas
Tokamak equilibrium is achieved at
The plasma is heated externally to reach high temperature at high density in order to reach fusion ignition (Pfusion > Ploss)

5. What are energetic particles
Typical fusion plasmas have density nth~ 1020m-3and Tth~10kev
Typical energetic particle energy Eh>100kev >> Tth .
In current tokamak devices, energetic particles are usually introduced by neutral beam injection (NBI heating) or by Radio Frequency wave heating (RF heating).
In a fusion reactor, energetic alpha particles are produced by fusion reaction:
D+Ta(3.5Mev)+n(14Mev)
Energetic particles heat plasmas via Coulomb collisions (mainly heat electrons) and slow down to thermal energy.

6. Roles of energetic particles in fusion plasmas
Current fusion plasma research topics:
MHD stability (beta limit)
Transport (energy confinement)
Wave heating and current drive (equilibrium)
Edge physics (edge stability and confinement)
Energetic Particle Physics (collective instabilities)

7. Roles of energetic particles in fusion plasmas II
Heat plasmas via Coulomb collision
Stabilize MHD modes
Destabilize shear Alfven waves via wave-particle resonance
Energetic particle redistribution/loss can affect thermal plasma confinement, degrade plasma heating, and damage reactor wall

8. Typical fusion parameters

9. Single Particle Confinement
For an axi-symmetric torus, particles are confined as long as orbit width is not too large. (conservation of toroidal angular momentum.)
Energetic particles slow down due to collisions with electrons and ions and heat thermal particles. For typical parameters, energetic particles mainly heat electrons.
Toroidal field ripple (due to discrete coils) can induce stochastic diffusion.
Symmetry-breaking MHD modes can also cause energetic particle anomalous transport.

10. Shear Alfven spectrum, continuum damping, and discrete modes
Shear Alfven wave dispersion relation
Continuum spectrum
Initial perturbation decays due to phase mixing at time scale of
Driven perturbation at w is resonantly absorbed at  continuum damping
Phase mixing and resonant absorption has exact analogy with Landau damping for Vlasov plasma.

11. Discrete Alfven Eigenmodes can exist near continuum accumulation point due to small effects such as toroidicity, shaping, magnetic shear, and energetic particle effects.
Coupling of m and m+k modes breaks degeneracy of Alfven continuum :
K=1 coupling is induced by toroidicity
K=2 coupling is induced by elongation
K=3 coupling is induced by triangularity.

12. Discrete Alfven Eigenmodes versus Energetic Particle Modes
Discrete Alfven Eigenmodes (AE):
Mode frequencies located outside Alfven continuum (e.g., inside gaps);
Modes exist in the MHD limit;
energetic particle effects are often perturbative.
Energetic Particle Modes (EPM):
Mode frequencies located inside Alfven continuum and determined by energetic particle dynamics;
Energetic effects are non-perturbative;
Requires sufficient energetic particle drive to overcome continuum damping.

13. Example of Discrete AE: Toroidal Alfven Eigenmode (TAE)
TAE mode frequencies are located inside the toroidcity-induced Alfven gaps;
TAE modes peak at the gaps with two dominating poloidal harmonics.
C.Z. Cheng, L. Chen and M.S. Chance 1985, Ann. Phys. (N.Y.) 161, 21

14. 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 drive
Landau damping
Due to velocity space gradient

15. TAE Stability: energetic particle drive and background 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

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

17. 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 , Phys. Rev. Lett. 50, 891
L. Chen, R.B. White and M.N. Rosenbluth 1984, Phys. Rev. Lett. 52, 1122

18. Nonlinear dynamics I: single mode saturation
Saturation mechanism
Wave particle trapping leading to flattening of distribution function and mode saturation
Collisions tend to restore the original unstable distribution. Balance of nonlinear flattening and collisional restoration leads to mode saturation.
H.L. Berk and B.N. Breizman 1990, Phys. Fluids B 2, 2235

19. Nonlinear dynamics II: 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).

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

23. Nonlinear dynamics III: other saturation mechanisms
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

24. Nonlinear Dynamics: multiple modes
Multiple unstable modes can lead to resonance overlap and stochastic diffusion of energetic particles.

25. Summary
In absence of MHD perturbations, energetic particles are well confined in tokamak due to toroidal symmetry.
Wave particle resonances lead to a variety of energetic particle instabilities: discrete AEs and EPMs.
Rich nonlinear behavior: steady state saturation, hole-clump creations, and multi-mode coupling.