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Lecture Series in Energetic Particle Physics of Fusion Plasmas

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### 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

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)

Tokamak equilibrium, shear Alfven wave equation, Alfven eigenmodes

- Tokamak equilibrium
- Shear Alfven wave equation
- Alfven eignemodes
- Summary

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.

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.

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 effectsKTAE

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 13,052502 (2006)

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

Mode coupling between m and m+1 induces a continuum gap

Continuum spectrum is modified by toroidicity.

at

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

Reversed shear Alfven eigenmode (RSAE) can exist above maximum of Alfven continuum at q=qmin

U

wA

q

wRSAE

rmin

r

rmin

r

rmin

r

w = (n-m/qmin)/R

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.87 185002

S.E. Sharapov, et al. 2001, Phys. Lett. A289, 127

B.N. Breizman et al, Phys. Plasmas 10, 3649 (2003)

G.Y. Fu and H.L. Berk, Phys. Plasmas 13,052502 (2006)

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.

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