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 (Jan.3)
(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. Tokamak equilibrium, shear Alfven wave equation, Alfven eigenmodes
Alfven eignemodes
Summary

4. 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.

5. 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.

6. Shear Alfven Eigenmodes
Cylindrical limit  Global Alfven Eigenmodes
Toroidal coupling  TAE and Reversed shear Alfven eigenmodes
Elongation  EAE and Reversed shear Alfven eigenmodes
Triangularity  NAE
FLR effectsKTAE

7. Shear Alfven Equation
Assume low-beta, large aspect ratio, shear Alfven wave equation can be written as
G.Y. Fu and H.L. Berk, Phys. Plasmas 13, (2006)

8. Shear Alfven Equation: cylindrical limit (straight tokamak)
When Alfven continuum has a minimum, a discrete mode can exist
Below this minimum when This mode is called global
Alfven Eigenmode (GAE)
K. Appert, R. Gruber, F. Troyuon and J. Vaclavik 1982, Plasma Phys. 24, 1147

9. GAE can exist below shear Alfven continuum due to magnetic shear
wA(r) U
wGAE
rmin r
rmin r

10. Mode coupling between m and m+1 induces a continuum gap
Continuum spectrum is modified by toroidicity. at

11. Toroidal Alfven Eigenmode (TAE) can exist inside continuum gap
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

12. Reversed shear Alfven eigenmode (RSAE) can exist above maximum of Alfven continuum at q=qminwA
wRSAE
rmin r
rmin r
rmin r
w = (n-m/qmin)/R

13. RSAE (Alfven cascades) were observed in JET plasmas

14. RSAE exists due to toroidicity, pressure gradient or energetic particle effects
H.L. Berk, D.N. Borba, B.N. Breizman, S.D. Pinches and S.E. Sharapov 2001,
Phys. Rev. Lett
S.E. Sharapov, et al. 2001, Phys. Lett. A 289, 127
B.N. Breizman et al, Phys. Plasmas 10, 3649 (2003)
G.Y. Fu and H.L. Berk, Phys. Plasmas 13, (2006)

15. Summary Mode coupling induces gaps in shear Alfven continuum spectrum.
Discrete Alfven eigenmodes can usually exist near Alfven continuum accumulation point (inside gaps, near continuum minimum or maximum).
Existence of Alfven eigenmodes are due to “small” effects such as magnetic shear, toroidicity, elongation, and non-resonant energetic particle effects.