1. Alfven Waves in Toroidal Plasmas
Summer School 2007, Chengdu
Alfven Waves in Toroidal Plasmas
S. Hu
College of Science, GZU
Supported by NSFC

2. Outline Introduction to Alfven waves Alfven waves in tokamaks
Toroidicity-induced Alfven Eigenmodes (TAE)
Energetic-particle modes (EPM)
Discrete Alfven eigenmodes ( TAE)
Summary

3. Introduction to Alfven Waves
Basic pictures of Alfven waves
Importance of Alfven waves
Alfven waves in nonuniform plasmas
Shear modes vs. compressional modes

4. Alfven Waves (Shear Modes)

5. Alfven Waves & Energetic Particles
Importance in Fusion Studies:
The Alfven frequencies are comparable to the characteristic frequencies of energetic / alpha particles in heating / ignition experiments.
Basic Waves in Space Investigations:
The Alfven waves widely exist in space, e.g., the Earth’s magnetosphere, the solar-terrestrial region, and so on. The interactions between the Alfven waves and the energetic particles also play important roles in physical understandings.

6. Alfven Waves

7. Alfven Waves (Compressional Modes)

8. Alfven Waves in Tokamaks
Basic equations
Ballooning formalism
Shear Alfven equation
The s- diagram
[ Lee and Van Dam, 1977
Connor, Hastie, Taylor, 1978 ]

9. Basic Equations

10. Ballooning Formalism

11. Shear Alfven Equation

12. The s- Diagram First ballooning-mode stable regime
(with the low pressure-gradient)
Ballooning-mode unstable regime
(with pressure-gradient inbetween)
Second ballooning-mode stable regime
(with the high pressure-gradient)

13. TAE Localized and extended potentials
Alfven continuum and frequency gap
Toroidicity-induced Alfven eigenmodes
TAE features
[ Cheng, Chen, Chance, AoP, 1985 ]

14. Localized and Extended Potentials

15. Alfven Frequency Spectrum

16. Toroidal Alfven Eigenmodes

17. TAE Features
Existence of the Alfven frequency gap due to the finite-toroidicity coupling between the neighboring poloidal harmonics.
Existence of eigenmodes with their frequencies located inside the Alfven frequency gap.
These modes experience negligible damping due to their frequencies decoupled from the continuum spectrum.

18. EPM Gyro-kinetic equation Vorticity equation Wave-particle resonances
EPM features
[ Chen, PoP, 1994 ]

19. Gyro-Kinetic Equation

20. Gyro-Kinetic Equation (cont.)

21. Vorticity Equation

22. Vorticity Equation (cont.)

23. Wave-Particle Resonances

24. EPM Features
The Alfven modes gain energy by resonant interactions between Alfven waves and energetic particles.
The mode frequencies are characterized by the typical frequencies of energetic particles via the wave-particle resonance conditions.
The gained energy can overcome the continuum damping.

25. TAE
Theoretical model
Bound states in the second ballooning-mode stable regime
Basic features
Kinetic excitations
[ Hu and Chen, PoP, 2004 ]

26. Theoretical Model

27. Basic Equations

28. Some Definitions

31. TAE Features
Existence of potential wells due to ballooning curvature drive.
Bound states of Alfven modes trapped in the MHD potential wells.
The trapped feature decouples the discrete Alfven eigenmodes from the continuum spectrum.

39. Summary
Introduction to shear Alfven waves in tokamaks and their interaction with energetic particles.
Discussions on the toroidicity-induced Alfven eigenmode (TAE), the energetic-particle continuum mode (EPM), as well as the discrete Alfven eigenmode ( TAE).

40. Alpha-TAE vs. EPM/TAE
alpha-TAE: Bound states in the potential wells due to the ballooning drive.
EPM: Frequencies determined by the wave-particle resonance conditions.
TAE: Frequencies located inside the toroidal Alfven frequency gap.