introduction of wave-particle resonance in tokamaks j.q. dong southwestern institute of physics...
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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
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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
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• 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
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Tokamak magnetic configuration
• Equilibrium magnetic field: Toroidal field
Poloidal field
)/cos1( 00)/cos1( 0
0 RrBB RrB
),cos1(0
0 R
rBB
BB
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Charged particle motion in tokamaks
• Parallel (lognitudinal) motion:
• Rotation:
• Drifts of guiding center
i) Electric field drift:
thvv ||
thvv
B
bEVE
ˆ
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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
||
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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
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Diamagnetic drift of plasma fluids
2Bqn
BPcV
jj
idj
•It is in the vertical direction;
•It induces charge separation and then plasma outward motion.
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Wave-particle resonance due to parallel motion of particles
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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
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• 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
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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
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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
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Bump-on-tail problem with the presence of energetic particles
• Discrete Alfven eigenmodes • Energetic particle modes
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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
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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
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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
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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
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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 /
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H.L. Berk et al, Phys. Plasmas 2, 3007 (1995).
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.Multiple unstable modes can lead to resonance overlap and stochastic diffusion of energetic particles
H.L. Berk et al, Phys. Plasmas 2, 3007 (1995).
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First observation of TAE in TFTR
K.L. Wong, R.J. Fonck, S.F. Paul, et al. 1991, Phys. Rev. Lett. 66, 1874
.
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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.
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Wave-particle resonance due to drift motion of particles
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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
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Fishbone dispersion relation
L. Chen, R.B. White and M.N. Rosenbluth 1984, Phys. Rev. Lett. 52, 1122
dm
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Electron fishbone instability
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• HL-2A results need further explanation
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Wave-particle resonance due to rotation of particles
ECRH,
ICRH,
ECE
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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