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Energetic Particle Physics in Tokamak Plasmas
Guoyong Fu
Princeton Plasma Physics LaboratoryPrinceton University
Princeton, NJ 08543, USA
6th Workshop on Nonlinear Plasma Science and IFTS WorkshopSuzhou, China, Sept. 24
Example of EPM: fishbone instability
Mode structure is of (m,n)=(1,1) internal kink;Mode is destabilized by energetic trappedparticles;Mode frequency is comparable to trapped particles’ precessional drift frequency
K. McGuire, R. Goldston, M. Bell, et al. 1983, Phys. Rev. Lett. 50, 891 L. Chen, R.B. White and M.N. Rosenbluth 1984, Phys. Rev. Lett. 52, 1122
First observation of TAE in TFTR
K.L. Wong, R.J. Fonck, S.F. Paul, et al. 1991, Phys. Rev. Lett. 66, 1874
.
RSAE (Alfven cascades) were observed in JET plasmas
1% neutron rate decrease:5% neutron rate decrease:• TAE avalanches cause enhanced fast-ion losses.
• Potential to model island overlap condition with full diagnostic set.
E. Fredrickson, Phys. Plasmas 13, 056109 (2006)
NSTX observes that multi-mode TAE bursts can lead tolarger fast-ion losses than single-mode bursts
Outline
• Roles of energetic particles in fusion plasmas• Single particle confinement• MHD limit: Shear Alfven continuum and Alfven
eigenmodes• Linear Kinetic Stability: discrete AE and EPM• Nonlinear Physics: saturation mechanisms,
frequency chirping, multi-mode coupling• Prospect for ITER and DEMO• Important problems for future
Roles of energetic particles in fusion plasmas
• Heat plasmas via Coulomb collision, drive plasma rotation and plasma current (e.g., NBI injection)
• Stabilize MHD modes (e.g., internal kink, RWM)• Destabilize shear Alfven waves via wave-particle
resonance • Energetic particle loss can damage reactor wall• Energetic particle redistribution may affect thermal
plasma via plasma heating profile, plasma rotation and current drive.
• Energetic particle-driven instability can be beneficial (alpha channeling, diagnostic of q profile etc)
Single Particle Confinement
• For an axi-symmetric torus, energetic particles are well confined (conservation of toroidal angular momentum).
• Toroidal field ripple can induce stochastic diffusion (important in advanced plasma regime with high q)
• Symmetry-breaking MHD modes can also cause energetic particle anomalous transport.
Shear Alfven spectrum and continuum damping
• Shear Alfven wave dispersion relation and continuum spectrum
)()
)((
1 2
2
2
22
||
2
r
B
rq
mn
RVk
A
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
q
kmn
q
mn
2
2_||||
or EPM
----- RSAE
Discrete 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
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 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
vr
rr
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)
Linear Stability
• Energetic particle destabilization mechanism
• Kinetic/MHD hybrid model
• TAE stability: energetic particle drive and dampings
• EPM stability: fishbone mode
Destabilize shear Alfven waves via wave-particle resonance
• Destabilization mechanism (universal drive)
Wave particle resonance at
For the appropriate phase (n/ >0), particles will lose energy going outward and gain 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 damping (or drive)due to velocity space gradient
Kinetic/MHD Hybrid Model
bbPPIPP
BvE
Et
B
BJ
BJPPdt
dv
h
hb
)(
0
||
Quadratic form
hk
bf
kf
h
hb
PxdW
BJBJPxdW
xdK
WWKt
v
bbPPIPP
BJBJPPt
v
3
3
23
2
2
||
)(
||
)(
Drift-kinetic Equation for Energetic Particle Response
gEvi
vdxdeW
Evi
E
fiegvv
t
gff
fBvdP
fBEvdP
dk
d
h
d
h
)(
))(()(
)(
)2(
33
||
3
3
||
Perturbative Calculation of Energetic Particle Drive
)exp()221(2
)(
)]3
()()[1(
2
242
2
2
2
xxxxxF
v
vF
v
vFq
K
W
WWK
h
A
h
Ah
h
h
k
kf
G.Y.Fu and J.W. Van Dam, Phys. Fluids B1, 1949 (1989)R. Betti et al, Phys. Fluids B4, 1465 (1992).
Dampings of TAE
• Ion Landau damping
• Electron Landau damping
• Continuum damping
• Collisional damping
• “radiative damping” due to thermal ion gyroradius
G.Y.Fu and J.W. Van Dam, Phys. Fluids B1, 1949 (1989)R. Betti et al, Phys. Fluids B4, 1465 (1992).F. Zonca and L. Chen 1992, Phys. Rev. Lett. 68, 592 M.N. Rosenbluth, H.L. Berk, J.W. Van Dam and D.M. Lindberg 1992, Phys. Rev. Lett. 68, 596 R.R. Mett and S.M. Mahajan 1992, Phys. Fluids B 4, 2885
Fishbone dispersion relation as an example of EPM
L. Chen, R.B. White and M.N. Rosenbluth 1984, Phys. Rev. Lett. 52, 1122
For EPM, the energetic particle effects are non-perturbative and the mode frequency is determined by particle orbit frequency.
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;
Are typical weakly damped due to thermal plasma kinetic effects;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 strong energetic particle drive to overcome continuum damping.
Nonlinear Physics
• Nonlinear dynamics of a single mode Bump-on-tail problem:
saturation due to wave particle trapping
frequency chirping
• Multiple mode effects: mode-mode coupling, resonance overlap
Bump-on-tail problem: definition
H.L. Berk and B.N. Breizman 1990, Phys. Fluids B 2, 2235
0|
)cos(
)()(
/
0
kv
a
v
f
tkx
vQfvv
f
xm
e
x
fv
t
f
It can be shown that bump-on-tail problem is nearly equivalent tothat of energetic particle-driven instability.
saturation due to wave particle trapping
We first consider case of no source/sink and no damping.
The instability then saturates at Lb ~
The instability saturates when the distribution is flattened at the resonance region
Width of flattened region is on order of
eb m
ek 02
2
kvL/~
Bump-on-tail problem: steady state 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).
Transition from steady state saturation to explosive nonlinear regime (near threshold)
B.N. Breizman et al, Phys. Plasmas 4, 1559(1997).
Hole-clump creation and frequency chirping
• For near stability threshold and small collision frequency, hole-clump will be created due to steepening of distribution function near the boundary of flattening region.
• As hole and clump moves up and down in the phase space of distribution function, the mode frequency also moves up and down.
H.L. Berk et al., Phys. Plasma 6, 3102 (1999).
Experimental observation of frequency chirping
M.P. Gryaznevich et al, Plasma Phys. Control. Fusion 46 S15, 2004.
Saturation due to mode-mode coupling
• Fluid nonlinearity induces n=0 perturbations which lead to equilibrium modification, narrowing of continuum gaps and enhancement of mode damping.
D.A. Spong, B.A. Carreras and C.L. Hedrick 1994, Phys. Plasmas 1, 1503 F. Zonca, F. Romanelli, G. Vlad and C. Kar 1995, Phys. Rev. Lett. 74, 698 L. Chen, F. Zonca, R.A. Santoro and G. Hu 1998, Plasma Phys. Control.Fusion 40, 1823
• At high-n, mode-mode coupling leads to mode cascade to lower frequencies via ion Compton scattering. As a result, modes saturate due to larger effective damping.
T.S. Hahm and L. Chen 1995, Phys. Rev. Lett. 74, 266
.Multiple unstable modes can lead to resonance overlap and stochastic diffusion of energetic particles
H.L. Berk et al, Phys. Plasmas 2, 3007 (1995).
Recent Nonlinear Simulation Results from M3D code
• Strong frequency chirping of fishbone due to kinetic nonlinearity alone;
• Simulations of multiple TAEs in NSTX show strong mode-mode coupling due to resonance overlap;
• Nonlinearly generated modes;
M3D XMHD Model
Recent M3D results:
(1) Alpha particle stabilizationof n=1 kink in ITER;
(2) Nonlinear frequency chirpingof fishbone;
(3) beam-driven TAEs in DIII-D;
(4) beam-driven TAEs in NSTX.
Saturation amplitude scale as square of linear growth rate
Simulation of fishbone shows distribution fattening and strong frequency chirping
distribution
Evidence of a nonlinearly driven n=2 mode
n=1 n=2
Multi-mode simulations show strong mode-mode interaction.
n=2
n=3
Strong interaction between different modes is due to wave-particle resonance overlap
P
v v
=0.2 0.4
0.6 0.8
Prospect for ITER/DEMO
• Multiple high-n modes are expected to be unstable, especially in DEMO;
• Key question: (1) how strong is instability?
(2) can multiple modes induce global fast ion transport due to resonance overlap and/or avalanche?
(3) can fast ion instabilities affect thermal plasma significantly?
Linear Stability of TAE
• Alpha particle drive• Plasma dampings:
ion Landau dampingelectron collisional damping“radiative damping” due to FLR
stability is sensitive to plasma parameters and profiles !
),()1/(*
A
h
hreshh
h
V
vkF
i
A
ii
v
vx
xxq
3
)exp( 252
r
ms
T
T
gg
i
i
e
m
mrad
4
328
)1
exp()1(2
1
2/3
22
Alpha particle drive is maximized at
G.Y. Fu et al, Phys. FluidsB4, 3722 (1992)
k ~ 1
Parameters of Fusion Reactors
Device B(T) a(m) R(m) Ti0
(kev) ne(0)
(e20/m3)
c(0)
(%)
(0)
(%)
v
/vA
a/
nmax
TFTR-DT 5.0 0.87 2.5 28 0.76 4.6 0.2 1.6 18 5
JET-DT 3.8 0.94 2.9 23 0.45 5.7 0.4 1.7 15 4
ITER_steady 5.3 1.9 6.2 25 0.73 4.8 0.9 1.5 42 10
ARIES-AT 5.8 1.3 5.2 30 2.8 10 3.1 2.7 31 8
DEMO_EU 6.9 2.9 8.6 30 1.5 7.0 2.0 1.7 84 21
DEMO_Japan 6.8 2.1 6.5 45 1.0 7.0 4.0 1.4 60 15
Critical alpha parameters of TFTR/JET, ITER and DEMO
c
a/
JETTFTR
ITER
ARIES-AT
DEMO_Japan
ARIES-STDEMO_EU
Multiple high-n TAEs are expected be excited in ITER from NOVA-K
Instability is maximized at k ~ 1
N.N. Gorelenkov, Nucl. Fusion 2003
DEMO versus ITER
• Alpha drive is higher and mode number is larger in DEMO
• Expect stronger Alfven instability and more modes >> wave particle resonance overlap and alpha particle redistribution likely !
• Alpha beta is a significant fraction of thermal beta >> alpha particle effects on MHD modes and thermal plasma stronger !
Summary I: Discrete Alfven Eigenmodes
• 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.
Summary II: linear stability
• For discrete modes such as TAE, the stability can usually be calculated perturbatively. For EPM, a non-perturbative treatment is needed.
• For TAE, there are a variety of damping mechanisms. For instability, the energetic particle drive must overcome the sum of all dampings.
• For EPM to be unstable, the energetic particle drive must overcome continuum damping.
Summary III: nonlinear dynamics
• Single mode saturates due to wave-particle trapping or distribution flattening.
• Collisions tend to restore original unstable distribution.
• Near stability threshold, nonlinear evolution can be explosive when collision is sufficiently weak and result in hole-clump formation.
• Mode-mode coupling can enhance damping and induce mode saturation.
• Multiple modes can cause resonance overlap and enhance particle loss.
Important Energetic Particle Issues
• Linear Stability: basic mechanisms well understood, but lack of a comprehensive code which treats dampings and energetic particle drive non-perturbatively
• Nonlinear Physics: single mode saturation well understood, but lack of study for multi-mode dynamics
• Effects of energetic particles on thermal plasmas: needs a lot of work (integrated simulations).
