Alfven Waves in Toroidal Plasmas

1. Summer School 2007, Chengdu Alfven Wavesin 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.