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.

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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

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

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 eigenmodes

  • Tokamak equilibrium

  • Shear Alfven wave equation

  • Alfven eignemodes

  • Summary


Shear alfven spectrum continuum damping and discrete modes

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.


Lecture series in energetic particle physics of fusion plasmas

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

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


Shear alfven equation

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

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


Gae can exist below shear alfven continuum due to magnetic shear

GAE can exist below shear Alfven continuum due to magnetic shear

wA(r)

U

wGAE

rmin

r

rmin

r


Mode coupling between m and m 1 induces a continuum gap

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

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 q min

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 alfven cascades were observed in jet plasmas

RSAE (Alfven cascades) were observed in JET plasmas


Rsae exists due to toroidicity pressure gradient or energetic particle effects

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

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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