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Alpha-driven localized cyclotron modes in nonuniform magnetic field
as a challenging issue
in resonance, relativity, and ITER
K. R. Chen
Plasma and Space Science CenterPhysics Department of and Institute of Electro-Optics
National Cheng Kung University
Work is supported by National Science Council, Taiwan
May 15-19, 2006 Workshop on ITER Simulation, Peking Univ., Beijing, China
Outline
• Introduction
• Fundamental mechanics
• Applications in experiments
• Localized cyclotron modes in non-uniform magnetic field
• Summary
Introduction
• Fusion energy is essential for human’s future, if ITER is successful.The dynamics of alpha particle is important to burning fusion plasma.
• Resonance is a fundamental issue in science. It requires precise synchronization. For magnetized plasmas, the resonance condition is
n c ~ 0 , c = qmc
• For fusion-produced alpha, = 1.00094. Can relativity be important?
• Also, for relativistic cyclotron instabilities, the resonance condition is n c = r ii r > 0 |r| ,, i << n (As decided by the fundamental wave particle interaction mechanism,
the wave frequency is required to be larger than the harmonic cyclotron frequency.[Ref. K. R. Chu, Rev. Mod. Phys. 76, p.489 (2004)]
• Can these instabilities survive when the non-uniformity of the magnetic field is large
(i.e., the resonance condition is not satisfied over one gyro-radius)?
• If they can, what are the wave structure, the wave frequency, and the mismatch?
Fundamental mechanics
Two-gyro-streams in the gyro-phase of momentum space
Two streams in real space can cause a strong two-stream instability
Two-gyro-streams
In wave frame of real spaceV
x
V1
V2
Vph= k
V
xV1
V2
Vphdt
dxV
V decreases when decreases
c c
z eB
m c
wavel fcf
lscs
In wave frame of gyro-spaced
ωdt
c increases when decrease
s
• Two-gyro-streams can drive two-gyro-stream instabilities.• When slow ion is cold, single-stream can still drive beam-type instability.
vy
vx
• •lscs
lf cf
Xxx
kv2 < < kv1
lf cf < lswcsK. R. Chen, PLA, 1993.
A positive frequency mismatch lscs - lf cf is required to drive two-gyro-stream instability
.
Characteristics and consequences depend on relative ion rest masses
dielectric function
lf cf lscs
0
1
2
3
0 200 400 600
t=0 ; * 0.5t=800t=1000t=3200Maxwellian
dis
trib
utio
n
fun
ctio
n
P• Fast alphas in thermal deuterons can not satisfy. Beam-type instability
can be driven at high harmonics where thermal deuterons are cold.• Their perpendicular momentums are selectively gyro-broadened.
• Fast protons in thermal deuterons can satisfy.• Their perpendicular momentums are thermalized. [This is the first and only non-resistive mechanism.]
0
100
200
300
-300 -200 -100 0 100 200 300
P
Pz
Fig. 2. by Chen
K. R. Chen, PRL, 1994.K. R. Chen, PLA,1998; PoP, 2003.
K. R. Chen, PLA, 1993; PoP, 2000.
The history of field energy; energy extraction
• There is no instability when we use Newton equation instead of Lorentz equation.• So, the instabilities for high harmonic cyclotron waves are due to the relativistic mass variation effect.• Waves at high harmonics grow with rates approximately equal to theory.
10-6
10-5
10-4
10-3
0 500 1000 1500 2000
field
ene
rgy
/ ini
tial a
lpha
kin
etic
ene
rgy
time (cD
-1)
non-relativistic
relativistic
The growth rate peaks at
J13’(k) ~ 0
'J
J
icrelativistnon
icrelativist
n
n
Energy extraction
Fruchtman, Fisch, and Valeo,
PoP, 1997.
K. R. Chen, NF, 1995.
Alfvenic behavior and instability transition
Electromagnetic relativistic ion cyclotron instabilities
cubic
quadratic
Instability transition
Alfvenic behavior
K. R. Chen, et. al., PRE, 2005
Instability transits from cubic to quadratic without much change in spectral profile.
Applications in experiments
Theoretical prediction:1st harmonic =0.16 at =4.2p
2nd harmonic =0.08 at =1.4p
is consistent with the PIC simulation.
Consistent with JET’s observations.0
2
4
6
0 1 2 3pow
er s
pect
rum
(arb
itra
ry a
mp
litu
de)
frequency (/cf)
10-6
10-5
1010 1011
peak
fie
ld e
nerg
y
fast ion density
The straight line is the 0.84 power of the proton density while
Joint European Tokamak shows 0.9±0.1.The scaling is consistent with
the experimental measurements.
Cyclotron emission spectrum being consistent with JET
• Both the relative spectral amplitudes and the scaling with fast ion density are consistent with the JET’s experimental measurements.
• However, there are other mechanisms (Coppi, Dendy) proposed.
K. R. Chen, et. al., PoP, 1994.
Dominance of relativistic effect in magnetoacoustic cyclotron instability
0
0.001
0.002
0.003
0.004
4.35 4.4 4.45 4.5 4.55
kp0
i/
cf
relativity
classical
Classical result is the same as that in Fig. 1 and 2, respectively, of[R.O. Dendy, C.N. Lashmore-Davies, and K. F. Kam, PoP, 1992.]
• Both peak and spectral width of the relativistic instability dominatethose of the classical instability at every harmonics.
0
0.0002
0.0004
0.0006
4.35 4.4 4.45 4.5 4.55
kp0
i/
cf
relativity
classical
Explanation for TFTR experimental anomaly of alpha energy spectrum
birth distributions
reduced chi-square
calculated vs. measured spectrums
• Relativistic effect has led to good agreement.• The reduced chi-square can be one. • Thus, it provides the sole explanation for the experimental anomaly.
K. R. Chen, PLA, 2004; KR Chen & TH Tsai, PoP, 2005.
Localized cyclotron modes
in
non-uniform magnetic field
PIC and hybrid simulations with non-uniform B
10-7
10-6
10-5
0 1000 2000time (
cD-1)
classical
relativistic
• Physical parameters: n = 2x109cm-3 EeV (= 1.00094)
nD = 1x1013cm-3 TD = 10 KeV B = 5T harmonic > 12 unstable; for n = 13, i,max/ = 0.00035 >> (-13c)r /
• PIC parameters (uniform B): periodic system length = 1024 dx, 0 =245dx wave modes kept from 1 to 15 unit time to = cD
-1 dt = 0.025 total deuterons no. = 59,048 total alphas no.= 23,328
• Hybrid PIC parameters (non-uniform B): periodic system length = 4096dx, 0 =123dx wave modes kept from 1 to 2048 unit time to=co
-1 , dt=0.025 fluid deuterons total alphas no.= 10,000,000 (from a PC cluster built by my lab)
B/B = ±1%
Can wave grow while the resonance can not be maintained?
• Relativistic ion cyclotron instability is robust against non-uniform magnetic field.
B/B = ± 1%
1% in 1000 cellscells < o=123 cellsThus, it is generally believed that the resonance excitation can not survive.
• This result challenges our understanding of resonance.
However,
Electric field vs. X for localized modes in non-uniform B
• Localized cyclotron waves like wavelets are observed to grow from noise. • A special wave form is created for the need of instability and energy dissipation.• A gyrokinetic theory has been developed. A wavelet kinetic theory may be possible.
t=1200 t=1400 t=1800
t=2000 t=2400 t=3000
t=1400Ex vs. X
Mode 1 Mode 2
Structure of the localized wave modes
4 o
Field energy vs. k
Mode 1
Mode 2
B/B = ± 1%
B/B = 0 B/B = ± 0.2% B/B = ± 0.4%
B/B = ± 0.6% B/B = ± 0.8%
Structure of wave modes vs. magnetic field non-uniformity
Power spectrum of localized wave modes
B/B=±1%
t=1400Ex vs. X
Mode 1 Mode 2
Mode 1
Mode 2
co
c
co
c
• Resonance is a consequence of the need to drive instability for dissipating free energy and increasing the entropy.
• A wave eigen-frequency (even c) is collectively decided in a coherent means; a special wave form in real space is created for this purpose, even without boundary.
c at peak B c
peak=13.118
c at x>3232 or x<2896
12.98
12.99
13
0 1000 2000 3000 4000
13 c
x
B/B = 0
12.9
12.95
13
13.05
13.1
0 1000 2000 3000 4000
13 c
x
B/B = ± 0.6%
12.85
12.9
12.95
13
13.05
13.1
0 1000 2000 3000 4000
13 c
x
B/B = ± 0.8%
12.85
12.9
12.95
13
13.05
13.1
13.15
0 1000 2000 3000 4000
13 c
B/B = ± 1%
Frequency of wave modes vs. magnetic field non-uniformity
• The localized wave modes are coherent with its frequency being able to be lower than the local harmonic cyclotron frequency.
12.85
12.9
12.95
13
13.05
13.1
13.15
0 1000 2000 3000 4000x
B/B = 0, 0.006, 0.008, 0.01
13 c. ___
Frequencies vs. magnetic field non-uniformity
• The wave frequency can be lower then the local harmonic ion cyclotron frequency,
in contrast to what required for relativistic cyclotron instability.
Alpha’s momentum Py vs. X
t=1200 t=1400 t=1800
t=2000 t=2400 t=3000
• The perturbation of alpha’s momentum Py grows anti-symmetrically and then breaks from each respective center. Alphas have been transported.
Summary
• For fusion produced with =1.00094, relativity is still important.
• The effect on alpha dynamics is profound.
• The results can explain the experimentally measured ion cyclotron emission and alpha energy spectrum.
• The relativistic ion cyclotron instability and the resonance can survive the non-uniformity of magnetic field; thus, it should be an important issue in burning fusion devices, especially in ITER.
• Localized cyclotron waves like a wavelet consisting twin coupled sub-waves are observed and alphas are transported in hybrid simulation with our PC cluster.
• These results challenge our understanding of resonance.
• Resonance is the consequence of the need of instability, even the resonance condition can not be maintained within one gyro-radius and wave frequency is lower than local harmonic cyclotron frequency.
• This provides new theoretical opportunity (e.g., for kinetic theory) and a difficult problem for ITER simulation (because of the requirement of low noise and relativity.)