Transcript 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
• (1)
Overview of Energetic Particle Physics in Tokamaks (today)
• (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)
Overview of Energetic Particle Physics
in Tokamaks
•
•
•
•
•
Tokamak basics
Roles of energetic particles in fusion plasmas
Single particle confinement
Alfven continuum and shear Alfven eigenmodes
Energetic particle-driven collective instabilities:
discrete AE and EPM
• Nonlinear dynamics: single mode saturation and
multi-mode effects
Tokamak basics
Both fields are necessary to confine charged
particles or plasmas
• Safety factor q
• Particle orbits: trapped particles and circulating
particles, banana orbit, bounce frequency, drift
frequency
• Neutral beam heating, RF wave heating and
fusion alpha particles.
Roles of energetic particles in
fusion plasmas
• Heat plasmas via Coulomb collision
• Stabilize MHD modes
• Destabilize shear Alfven waves via waveparticle resonance
• Energetic particle redistribution/loss can
affect thermal plasma confinement,
degrade plasma heating, and damage
reactor wall
Single Particle Confinement
• For an axi-symmetric torus, particles are
confined as long as orbit width is not too large.
(conservation of toroidal angular momentum.)
• Energetic particles slow down due to collisions
with electrons and ions and heat thermal
particles. For typical parameters, energetic
particles mainly heat electrons.
• Toroidal field ripple (due to discrete coils) can
induce stochastic diffusion.
• Symmetry-breaking MHD modes can also cause
energetic particle anomalous transport.
Shear Alfven spectrum, continuum
damping, and discrete modes
• Shear Alfven wave dispersion relation
2
1
m
B
w  k V  (n 
)
R
q(r )  (r )
2
2
2
2
• Continuum spectrum
• Initial perturbation exp( iw A ( r ) t ) decays due to
phase mixing at time scale of ( dw Adr( r )r ) 1
• Driven perturbation at w is resonantly absorbed at
w  w A ( r )  continuum damping
• Phase mixing and resonant absorption has exact
analogy with Landau damping for Vlasov plasma.
||
A
2
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 :
m
mk
| n  || n 
| at
q
q
2m  k
q
2n
K=1 coupling is induced by toroidicity
K=2 coupling is induced by elongation
K=3 coupling is induced by triangularity.
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.
Shear Alfven Equation
• Assume low-beta, large aspect ratio, shear Alfven
wave equation can be written as
w
2
1
1
  U  B  B 
B  U
v
B
B
2

2
2

2
A
J
1
  B  P
  ( )  [ ( B  U )  B]  2
0
B
B
B
||
2
G.Y. Fu and H.L. Berk, Phys. Plasmas 13,052502 (2006)
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 effectsKTAE
GAE can exist below shear Alfven
continuum due to magnetic shear
wA(r)
U
wGAE
rmin
r
rmin
r
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
U
wA
q
wRSAE
rmin
r
w = (n-m/qmin)/R
rmin
r
rmin
r
Linear Stability
• Energetic particle destabilization
mechanism
• Kinetic/MHD hybrid model
• TAE stability: energetic particle drive and
dampings
• EPM stability: fishbone mode
• Summary
Destabilize shear Alfven waves via
wave-particle resonance
• Destabilization mechanism (universal drive)
Wave particle resonance at w  k|| v||
For the right phase, particle will lose energy going outward
and gaining energy going inward. As a result, particles will
lose energy to waves.
dP
n dE

dt
w dt
Energetic particle drive

n E df E df
  [

]
w
w fdP
fdE
h
h
h

Spatial gradient drive
Landau damping
Due to velocity space gradient
Kinetic/MHD Hybrid Model
dv
  P    P  J  B
dt
J  B
B
   E
t
E  vB  0
P  P I  ( P  P )bb
b
h

h
||

Quadratic form
v
  P    P  J  B  J  B
t
P  P I  (P  P )bb
b

h
h

||
v
 w 
t
w K  W  W
2
2
f
k
 K   d x |  |
W   d x  (  P  J  B  J  B )
3
2
3

f
b
 W   d x     P
3
k

h
G.Y. Fu et al. Phys. Fluids B5, 4040 (1993)
Drift-kinetic Equation for Energetic
Particle Response
P   d v (2 E  B )f
3
||
P   d v ( B )f
 f     f  g
3


h

f
i
(  v    v   ) g  ie
(w  w )( v  E )
t
E
w
i
W   e  d x  d v ( v  E ) g
h
||
3
k

d

3
w
d

d

Perturbative Calculation of
Energetic Particle Drive
w K   W  W
2
f
k
 w W

w 2w K

w
v
v
 q  (  1)[ F ( )  F ( )]
w
w
v
3v
k
2
h
h
2
A
A
h
h
F ( x) 

2
h
x (1  2 x  2 x ) exp(  x )
2
4
G.Y.Fu and J.W. Van Dam, Phys. Fluids B1, 1949 (1989)
R. Betti et al, Phys. Fluids B4, 1465 (1992).
2
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
w

w
dm
L. Chen, R.B. White and M.N. Rosenbluth 1984, Phys. Rev. Lett. 52, 1122
Nonlinear dynamics : single mode
saturation
• Saturation mechanism
Wave particle trapping leading to flattening of distribution function
and mode saturation
w ~
b
L
Collisions tend to restore the original unstable distribution. Balance
of nonlinear flattening and collisional restoration leads to mode
saturation.
w
b
 
eff
h
d
H.L. Berk and B.N. Breizman 1990, Phys. Fluids B 2, 2235
H.L. Berk et al, Phys. Plasmas 2, 3007 (1995).
Transition from steady state
saturation to explosive
nonlinear regime
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).
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).
Nonlinear Hybrid Simulation of
Fishbone instability
• Particle/MHD hybrid model
• Use M3D code
• Observed dynamic distribution flattening
as mode frequency decreases.
G.Y. Fu et al, Phys. Plasmas 13, 052517 (2006)
M3D XMHD Model
Simulation of fishbone shows distribution fattening
and strong frequency chirping
distribution
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).