Transcript resonance - National Cheng Kung University

Alpha-driven localized cyclotron modes
in nonuniform magnetic field
K. R. Chen
Physics Department and
Plasma and Space Science Center
National Cheng Kung University
Collaborators: T. H. Tsai and L. Chen
20081107
FISFES at NCKU, Tainan, Taiwan
Outline
• Introduction
• Particle-in-cell simulation
• Analytical theory
• 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
w - n wc ~ 0 , wc = qB/gmc
• For fusion-produced alpha, g = 1.00094. Can relativity be important?
• Also, for relativistic cyclotron instabilities, the resonance condition is
w - n wc = dwr + i wi dwr > 0 |dwr| ,, wi << n (g-1) << 1
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?
Two-gyro-streams in the gyro-phase of momentum space
Two streams in real space can cause a strong two-stream instability
In wave frame of real space
V
V1
V2
V
V1
Vph= w / k
Vph
V2
x
kv2 < w < kv1
Two-gyro-streams
Wc
vy
wc 
•
g

lf wcf
•
x
X
B
vx
V decreases when g decreases
In wave frame of gyro-space
w
gmc
lsw cs
lswcs
x
z eB
x
dx
V
dt
l f w cf
lf wcf < w < lswcs
ω
wwave
f
df
dt
wc increases when g decreases
K. R. Chen, PLA, 1993.
• Two-gyro-streams can drive two-gyro-stream instabilities.
• When slow ion is cold, single-stream can still drive beam-type instability.
Characteristics and consequences depend on relative ion rest masses
A positive frequency mismatch D  lswcs - lf wcf
dielectric function
is required to drive two-gyro-stream instability.
K. R. Chen, PLA, 1993; PoP, 2000.
• Fast protons in thermal deuterons can satisfy.
lf wcf
• Their perpendicular momentums are thermalized.
[This is the first and only non-resistive mechanism.]
K. R. Chen, PRL, 1994.
3
w
lswcs
K. R. Chen, PLA,1998; PoP, 2003.
t=0 ; * 0.5
t=800
t=1000
t=3200
Maxwellian
300
distribution
function
2
P
^
1
0
0
200
100
200
400
600
0
-300
-200
-100
0
100
200
300
P
P^
z
• 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.
power spectrum
(arbitrary amplitude)
Cyclotron emission spectrum being consistent with JET
Theoretical prediction:
1st harmonic h=0.16 at l=4.2rp
4
2nd harmonic h=0.08 at l=1.4rp
is consistent with the PIC simulation
2
and JET’s observations.
00
3 e Landau damping is not important if
1
2
frequency (w/wcf)
poloidal m < qaRw/rve ~1000
finite k// due to shear B is not important if
poloidal m < qaRw/rc ~100
(linear thinking)
10-5
The straight line is the 0.84 power of
the proton density while
Joint European Tokamak shows 0.9±0.1.
10-6
The scaling is consistent with
11
10
10
10
the experimental measurements.
fast ion density
peak field energy
6
K. R. Chen, et. al., PoP, 1994.
• 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.
Explanation for TFTR experimental anomaly of alpha energy spectrum
birth distributions
calculated vs. measured spectrums
reduced chi-square
• Relativistic effect has led to good agreement. K. R. Chen, PLA, 2004;
• The reduced chi-square can be one. KR Chen & TH Tsai, PoP, 2005.
• Thus, it provides the sole explanation for the experimental anomaly.
Particle-in-cell simulation on
localized cyclotron modes
in non-uniform magnetic field
PIC and hybrid simulations with non-uniform B
• Physical parameters:
na = 2x109cm-3
Ea 3.5 MeV (g = 1.00094)
nD = 1x1013cm-3
TD = 10 KeV
B = 5T
harmonic > 12 unstable; for n = 13, wi,max/w = 0.00035 >> (w-13wca)r / w
• PIC parameters (uniform B):
periodic system length = 1024 dx, r0 =245dx
wave modes kept from 1 to 15
unit time to = wcD-1 dt = 0.025
total deuterons no. = 59,048
total alphas no.= 23,328
• Hybrid PIC parameters (non-uniform B):
periodic system length = 4096dx, r0 =125dx
wave modes kept from 1 to 2048
unit time to=wcao-1 , dt=0.025
fluid deuterons
particle alphas
relativistic
-5
10
10-6
classical
10-7
0
1000
time (w cD-1)
2000
dB/B = ±1%
Can wave grow while the resonance can not be maintained?
dB/B = ± 1%
1% in 1000 cells
Particle: uniform dw/w << g-10.00094 < 0.2% ~ 2ro=250 cells
Wave: non-uniform dw < damping < growth; but, << dw of width~4ro (shown later)
Thus, it is generally believed that the resonance excitation can not survive.
However,
• Relativistic ion cyclotron instability is robust against non-uniform magnetic field.
• This result challenges our understanding of resonance.
Electric field vs. X for localized modes in non-uniform B
t=1200
t=1400
t=2000
t=2400
t=1800
t=3000
• 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.
Structure of the localized wave modes
Ex vs. X
Field energy vs. k
t=1400
Mode 1
Mode 1
Mode 2
Mode 2
4 ro
Structure of wave modes vs. magnetic field non-uniformity
dB/B = 0
dB/B = ± 0.6%
dB/B = ± 0.2%
dB/B = ± 0.8%
dB/B = ± 0.4%
dB/B = ± 1%
Frequency of wave modes vs. magnetic field non-uniformity
13
13.1
13 w ca
13 w ca
w
w
13.05
12.99
13
dB/B = 0
dB/B = ± 0.6%
12.95
12.98
12.9
0
1000
2000
x
3000
4000
0
1000
2000
x
3000
4000
13.15
13.1
13 w ca
w
13.05
13 w ca
13.1
w
13.05
13
dB/B = ± 0.8%
13
dB/B = ± 1%
12.95
12.95
12.9
12.9
12.85
0
1000
2000
x
3000
4000
12.85
0
1000
2000
3000
4000
• The localized wave modes are coherent with
its frequency being able to be lower than the local harmonic cyclotron frequency.
Frequencies vs. magnetic field non-uniformity
At the vicinity of minimum of dB/B = ± 1%
dw/wcf = 3.5 x 10-2
damping 1.4×10-3
growth 4.7×10-3
• 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=2000
t=1400
t=2400
t=1800
t=3000
• The perturbation of alpha’s momentum Py grows anti-symmetrically and
then breaks from each respective center. Alphas have been transported.
t=3000
Py vs X
Pz vs P丄
Ex vs X
P丄vs X
fluid Px vs X
f(g)
• The localized perturbation on alphas’ perpendicular momentum has clear edges
and some alphas have been selectively slowed down (accelerated up) to 1 (6) MeV.
Perturbation theory for
localized cyclotron modes
in non-uniform magnetic field
Perturbation theory for dispersion relation
The dispersion relation and eigenfunction for nonuniform plasma
D(w,k,x)f (x)=0
Assumption: local homogeneity
Taking two-scale-length expansion
Perturbation
f ( x)  fˆ( x)eik x
*
w  w* + dw, k  k* - i x , x  x0 + d x
Nonuniform magnetic field
B ( X )  B0 (1+
1
 b x2 )
2
The dispersion relation for uniform plasma and magnetic field is
D(w* , k* , x0 )  0
Perturbed terms
(w* , k* ) is chosen for absolute instability
1 2 D
1 2 D 2
2 D
2
ˆ( x)  0
[Q(dw ) +
(
i

x
)
+
d
x
+
dw
(
i

x
)]
f
2 k*2
2 x02
w*k*
2
D
1  D 2 1 3 D 3
dw +
dw +
dw + ...
where Q(dw ) 
2
3
w*
2 w*
3! w*
For further simplification fˆ( x)  ( x)eik1x
Dispersion relation as a parabolic cylinder equation
By eliminating term of e ik x, the dispersion relation becomes
1
1 2 D 2
2 D
2 D
1 2 D 2
1 2 D 2
2 D
Q(dw)  x  - i x [ 2 k1 +
w] +
k1  +
dx +
dwk1  0
2 k*2
k*
w*k*
2 k*2
2 x02
w*k*
Choose
Then,
2 D
k1  -[
w*k*
2 D
] dw
2
k*
to eliminate the term of  x
1 2 D 2
1 2 D 2
( x + Q(dw ) +
x )  0
2
2
2 k*
2 x0
The dispersion relation can be rewritten as a parabolic cylinder eq.
 2 1 2
- ( t -  )  0
2
t
4
t
2 x
Absolute instability condition in uniform theory with complex w, k
For the localized wave, we consider the k satisfies the absolute instability
condition which implies there is no wave group velocity.
-3
3.5
0.7
3
0.6
2.5
0.5
2
0.4
1.5
0.3
1
0.2
0.5
0.1
ksi
ps (i)
Growth rate
psi&ksi vs ksr [lbrunid=abs-k-b01a]
x 10
0
10
15
20
25
30
ksr
35
40
45
Imag(k)
The k with peak growth rate
is about 17.
Imag(k)
The frequency mismatch is minus
at the k of peak growth rate.
0
50
Re(k)
-3
psr&ksi vs ksr [lbrunid=abs-k-b01a]
x 10
1
0
-5
10
0.5
15
20
25
30
ksr
Re(k)
35
40
45
0
50
ksi
Frequency
mismatch
ps (r)
5
Eigenfunctions from the non-uniform theory
N=0
N=1
x space
k space
Compare with the wave distribution in simulation
x space
Combined
Theoretical solution
for N=1 mode
Simulation for k=all modes
(N=1 dominates)
Ex1 vs ig1
800
600
400
Ex1
200
0
-200
-400
-600
-800
2200 2400 2600 2800 3000 3200 3400 3600 3800 4000
ig1
k space
|Ek1|2 vs k1
9
18
x 10
16
14
|Ek1|2
12
10
8
6
4
2
0
12
13
14
15
16
k1
17
18
19
20
Compare with the wave distribution in simulation
x space
Theoretical solution for
N=0 mode
Simulation for only keeping k=15.77~18.64
(only N=0 can survive)
Ex1 vs ig1
1000
800
600
400
Ex1
200
0
-200
-400
-600
-800
-1000
2200 2400 2600 2800 3000 3200 3400 3600 3800 4000
ig1
k space
N=1
|Ek1|2 vs k1
10
4
x 10
3.5
3
|Ek1|2
2.5
2
1.5
1
0.5
0
15
15.5
16
16.5
17
k1
17.5
18
18.5
19
Summary
• For fusion produced a with g=1.00094, relativity is still important.
• The relativistic ion cyclotron instability, the resonance, and the resultant
consequence on fast ions can survive the non-uniformity of magnetic field.
• Localized cyclotron waves like a wavelet consisting twin coupled sub-waves are
observed and alphas are transported in the hybrid simulation.
• The results of perturbation theory for nonuniform magnetic field is found to be
consistent with the simulation.
• Resonance is the consequence of the need of instability, even the resonance
condition can not be maintained within one gyro-motion 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.)