Lecture Series in Energetic Particle Physics of Fusion Plasmas

Guoyong Fu

Princeton Plasma Physics LaboratoryPrinceton 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 is resonantly absorbed at

continuum damping• Phase mixing and resonant absorption has exact

analogy with Landau damping for Vlasov plasma.

)())(

(1 2

2

2

22

||

2

r

B

rq

mn

RVk

A

1))(

( rdr

rAd

))(exp( tri

A

)( rA

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 toroidicityK=2 coupling is induced by elongationK=3 coupling is induced by triangularity.

n

kmq

atq

kmn

q

mn

2

2

||||

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

02])1([)(

11

22

||

2

2

22

2

B

PBBUB

BB

J

UBB

BB

BUvA

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

Shear Alfven Equation: cylindrical limit (straight tokamak)

0])1([)(

11 ||

2

2

bUBBB

JUB

BB

BU

vA

0'2

)(2

||2

||2

2

UrRq

qmkUk

vA

When Alfven continuum has a minimum, a discrete mode can existBelow this minimum when This mode is called globalAlfven Eigenmode (GAE)

0'||

qmk

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

A(r)

rrminrrmin

U

GAE

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

Continuum spectrum is modified by toroidicity.

at 2

1||

2

||

2

||22

2

2

1||

2

||

222

1||

2

||

2

1||

2

||22

2

2

2

2

2

1||2

2

2

||2

2

2

2

1

2

||2

||2

2

2

2

2

||2

2

0

1

0

1

11

0

)1()1(

1

]4)([)1(2

1

)())((

'2)()(

1

0

0

mm

m

A

mmmmmm

A

A

m

A

m

A

A

mm

m

A

m

A

m

mmmm

mmmm

kk

kv

kkkkkkv

vk

vk

v

rvrrr

LL

rRq

qmkk

vr

m

rk

vrrr

L

ULUL

ULUL

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

rrmin rrmin rrmin

q U

= (n-m/qmin)/R

RSAE

RSAE (Alfven cascades) were observed in JET plasmas

RSAE exists due to toroidicity, pressure gradient or energetic particle effects

][4

2)1(3

0])()(1[

||

||

22

2

2

2

2

2

2

2

2

||2

2

2

2

2

||2

2

e

jkn

rr

m

cB

eQ

QdrB

rdP

r

mq

vr

mQ

UQkvr

m

rk

vrrr

henergetic

energetic

A

m

A

m

A

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. A 289, 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.