INTRODUCTION OF WAVE-PARTICLE RESONANCE IN TOKAMAKS J.Q. Dong Southwestern Institute of Physics Chengdu, China International School on Plasma Turbulence.

INTRODUCTION OF WAVE-PARTICLE RESONANCE IN TOKAMAKS

J.Q. Dong

Southwestern Institute of Physics

Chengdu, China

International School on Plasma Turbulence and Transport

August 16 – 18, 2007, Chengdu, China

Outline• Introduction• Tokamak magnetic configuration• Charged particle motion in tokamaks• Wave-particle resonance due to parallel motion of

particles• Wave-particle resonance due to drift motion of part

icles• Wave-particle resonance due to rotation of particle

s• Summary

• Plasmas are affluent in collective oscillations and waves

• Wave-particle interaction is an important part of magnetic fusion plasma science:

Excitation of turbulent flows and fluctuations leads turbulent mass, momentum and energy transport

Effects of external waves on plasma particles include trapping of particles in waves, chaotic behavior in particle orbits, particle acceleration,

plasma heating and current drive• Resonance is an efficient way for collisionless energy

transfer between particles and waves

Introduction

Tokamak magnetic configuration

• Equilibrium magnetic field： Toroidal field

Poloidal field

)/cos1( 00)/cos1( 0

0 RrBB RrB

),cos1(0

0 R

rBB

BB

Charged particle motion in tokamaks

• Parallel (lognitudinal) motion:

• Rotation:

• Drifts of guiding center

i) Electric field drift:

thvv ||

thvv

B

bEVE

ˆ

ii) magnetic gradient ( ) drift：B

bnR

VB

m

bV mg

ˆˆ2

ˆ 2

iii) magnetic curvature drift：

||

ˆˆ ,n

bR

R

bnV

l

bV

bVd

ˆˆˆˆ 2||2

||

iv)trapping, bounce and toroidal drifta) Particle trapping

b) Bounce period of the trapped particles

c) Toroidal drift of trapped particles

2

|| 2rR

12

),(24 2

2||2

5.0

v

v

r

RK

v

Rq

),2

1

)(

)((24

2

K

E

RqB

mvvd

Diamagnetic drift of plasma fluids

2Bqn

BPcV

jj

idj

•It is in the vertical direction;

•It induces charge separation and then plasma outward motion.

Wave-particle resonance due to parallel motion of particles

Landau damping & bump on tail instability

• Vlasov equation:

• Linearization:

• Langmuir wave:

• Consider the parallel motion of the electrons only

01

fBVc

Em

qfV

t

fV

~1 E

,10 fff

0011011

fEm

qfBV

m

qfV

t

fVV

tiikxtiikx evffe )(~,~~11

11

fk qf

kv m v

• Poison equation

• Dispersion equation• Landau damping: for Maxwellian distribution

214k q f dv

204 1

( , ) 1 0fq

k dvkm kv v

,3 222 km

T

epe

).2

3

)(2

1exp(

)(

1

8 23

kdkd

Instability: for bump on tail distribution

]}.2

)(exp[]

2

)({exp[

2

1

2)(

2

20

2

20

2

21

10 T

Vvm

T

Vvm

T

m

n

nTF

n

nf eee

em

ee

]}.)1()(2

/exp[)1()()

2

3

)(2

1{exp(

)(

1

820

22105.1

2

1

1

223

rr

kV

kd

TTkV

T

T

n

n

kdkd

Lower hybrid current drive

-10 -8 -6 -4 -2 0 2 4 6 8 10

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

______=0.0.............=0.2- - - - - - =0.4

f

v//

Electron velocity distribution functions with different trapping effects under LHCD

Bump-on-tail problem with the presence of energetic particles

• Discrete Alfven eigenmodes • Energetic particle modes

Destabilization of shear Alfven waves via

wave-particle resonance • Dispersion relation of shear Alfven wave

• Destabilization mechanism (universal drive)

Wave particle resonance at

For the right phase, particle will lose energy going outward and gaining energy going inward. As a result, particles will lose energy to waves.

Energetic particle drive

||||vk

dt

dEn

dt

dP

][fdE

dfE

fdP

dfEnhh

h

Spatial gradient drive Landau dampingDue to velocity space gradient

|| ,Ak v / 4Av B

Shear Alfven spectrum, continuum damping, and discrete modes

• Shear Alfven wave dispersion relation in tokamaks

• Continuum spectrum• Initial perturbation decays due to phase mix

ing at time scale of • Driven perturbation at is resonantly absorbed at

continuum damping• Phase mixing and resonant absorption has exact analog

y 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

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

rvr

rrLL

rRq

qmkk

vr

m

rk

vr

rrL

ULUL

ULUL

Example of Discrete AE: Toroidal Alfven Eigenmode (TAE)

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

Bump-on-tail problem: saturation with damping, source and sink

Collisions tend to restore the original unstable distribution. Balance of nonlinear flattening and collisional restoration leads to mode saturation. It can be shown that the linear growth rate is reduced by a factor of . Thus, the mode saturates at

d

h

effb

H.L. Berk and B.N. Breizman 1990, Phys. Fluids B 2, 2235

beff /

H.L. Berk et al, Phys. Plasmas 2, 3007 (1995).

.Multiple unstable modes can lead to resonance overlap and stochastic diffusion of energetic particles

H.L. Berk et al, Phys. Plasmas 2, 3007 (1995).

First observation of TAE in TFTR

K.L. Wong, R.J. Fonck, S.F. Paul, et al. 1991, Phys. Rev. Lett. 66, 1874

.

Discrete Alfven Eigenmodes versus Energetic Particle Modes

• Discrete Alfven Eigenmodes (AE): Mode frequencies located outside Alfven continuum (e.g., inside gaps);Modes exist in the MHD limit;energetic particle effects are often perturbative.

• Energetic Particle Modes (EPM):Mode frequencies located inside Alfven continuum and determined by energetic particle dynamics;Energetic effects are non-perturbative;Requires sufficient energetic particle drive to overcome continuum damping.

Wave-particle resonance due to drift motion of particles

Fishbone Instability• Induce by injection of hi

gh energy neutral beam

• Due to interaction between the injected particles and the m=1,n=1 MHD mode

• Resonance between the toroidal wave velocity of the mode and toroidal drift of the trapped particles

Fishbone dispersion relation

L. Chen, R.B. White and M.N. Rosenbluth 1984, Phys. Rev. Lett. 52, 1122

dm

Electron fishbone instability

• HL-2A results need further explanation

Wave-particle resonance due to rotation of particles

ECRH,

ICRH,

ECE

Summary• Wave-particle resonance is a basic and important

mechanism for wave-particle interaction in tokamak plasmas

• Externally launched waves may be absorbed and heat plasma or drive current in plasma by wave-particle resonance

• Waves may be driven by particle motion through wave-particle resonance in plasmas

• There are quite a few observations on wave excitation in plasmas need explanation