# Alpha-driven localized cyclotron modes in nonuniform magnetic field

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```					   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                                                 V
dx
V1                                            V1                            V
Vph= w / k                                        Vph                 dt
V2                                                                      x
V2
x
kv2 < w < kv1                      V decreases when g decreases
Two-gyro-streams
vy     Wc                    z eB     In wave frame of gyro-space
wc 
g

gmc               w                         df
ω
lf wcf
•
lswcs•                         lsw cs
wwave            f
dt
xx
X                     vx          l f w cf
B
wc increases when g decreases
lf wcf < w < lswcs
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.                                             w
lf wcf                 lswcs
• Their perpendicular momentums are thermalized.
[This is the first and only non-resistive mechanism.]
K. R. Chen, PRL, 1994.
3                                       K. R. Chen, PLA,1998; PoP, 2003.
t=0 ; * 0.5
t=800
t=1000                        300
t=3200
Maxwellian
2
distribution

200
P
^
function

1                                     100

0
0                                       -300   -200   -100   0   100    200    300
0      200        400   600               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.
Cyclotron emission spectrum being consistent with JET
(arbitrary amplitude)
Theoretical prediction:
power spectrum

6
1st harmonic h=0.16 at l=4.2rp
4                                     2nd harmonic h=0.08 at l=1.4rp
2                          is consistent with the PIC simulation
and JET’s observations.
00         1       2
-
3 e Landau damping is not important if
frequency (w/wcf)               poloidal m < qaRw/rve ~1000
finite k// due to shear B is not important if
peak field energy

poloidal m < qaRw/rc ~100
-5
10                            (linear thinking)
The straight line is the 0.84 power of
the proton density while
-6
10
Joint European Tokamak shows 0.9±0.1.
The scaling is consistent with
10
10               1011
fast ion density                the experimental measurements.
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):
relativistic
-5
10

periodic system length = 1024 dx, r0 =245dx
wave modes kept from 1 to 15
unit time to = wcD-1 dt = 0.025                    10-6

total deuterons no. = 59,048                                                    classical

total alphas no.= 23,328                           10-7
0    1000            2000
time (w cD-1)

• 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                                                dB/B = ±1%
particle alphas
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=1800

t=2000                     t=2400                            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           t=1400                   Field energy vs. k

Mode 1

Mode 1        Mode 2

Mode 2

4 ro
Structure of wave modes vs. magnetic field non-uniformity

dB/B = 0               dB/B = ± 0.2%         dB/B = ± 0.4%

dB/B = ± 0.6%          dB/B = ± 0.8%         dB/B = ± 1%
Frequency of wave modes vs. magnetic field non-uniformity
13                                              13.1

13 w ca                                           13 w ca

w                13.05                            w

12.99                                              13
dB/B = ± 0.6%
dB/B = 0                         12.95

12.98                                             12.9
0    1000     2000   3000        4000            0   1000   2000   3000        4000
x                                            x
13.1                                           13.15
13 w ca                                 13 w ca
w            13.1
13.05                                                                        w
13.05
13
dB/B = ± 0.8%                  13
12.95                                                           dB/B = ± 1%
12.95
12.9
12.9

12.85                                            12.85
0   1000     2000    3000        4000           0   1000   2000   3000        4000
x

• 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=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.
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                       ˆ
f ( x )  f ( x )e ik        *x

Perturbation      w  w* + dw , k  k* - i x , x  x0 + d x
1
Nonuniform magnetic field            B( X )  B0 (1+  b x 2 )
2
The dispersion relation for uniform plasma and magnetic field is
D(w* , k* , x0 )  0         (w* , k* ) is chosen for absolute instability
Perturbed terms
1 2 D           1 2 D 2      2 D             ˆ
[Q(dw ) +        (-ix) +
2
dx +          dw(-ix)]f ( x)  0
2 k*2           2 x02
w*k*
D       1  D 2 1 3 D 3
2
where Q(dw )           dw +         dw +          dw + ...
w*       2 w* 2
3! w*3

ˆ
For further simplification f ( x )   ( x )eik1 x
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*

2 D           2 D
Choose        k1  -[                      ] dw       to eliminate the term of  x 
w*k*          k*2

1 2 D 2              1 2 D 2
Then,       (-         x + Q(dw ) +        x )  0
2 k*2
2 x02

The dispersion relation can be rewritten as a parabolic cylinder eq.
 2 1 2
- ( t -  )  0                     t        2 x
t 2
4
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
x 10               psi&ksi vs ksr [lbrunid=abs-k-b01a]
3.5                                                                                  0.7

3                                                                            0.6

2.5                                                                                  0.5

2                                                                            0.4
Imag(k)   The k with peak growth rate
Growth rate
ps (i)

ksi
1.5                                                                                  0.3
1                                                                            0.2

0.5                                                                                  0.1

0                                                                            0
10             15    20        25       30        35        40       45    50
ksr

Re(k)
-3
x 10               psr&ksi vs ksr [lbrunid=abs-k-b01a]
5                                                                            1

Frequency                                                                                                                    Imag(k)   The frequency mismatch is minus
ps (r)

ksi

0                                                                            0.5

mismatch                                                                                                                               at the k of peak growth rate.

-5                                                                            0
10             15    20         25       30       35           40    45    50
ksr

Re(k)
Eigenfunctions from the non-uniform theory

N=0                               k space
x space

N=1
Compare with the wave distribution in simulation
x space
Combined
Theoretical solution             Simulation for k=all modes
for N=1 mode                     (N=1 dominates)
Ex1 vs ig1
800

600

400

200

Ex1
0

-200

-400

-600

-800
2200 2400 2600 2800 3000 3200 3400 3600 3800 4000
ig1

k space
9
x 10                   |Ek1|2 vs k1
18

16

14

12

10
|Ek1|2

8

6

4

2

0
12         13   14    15       16         17   18   19    20
k1
Compare with the wave distribution in simulation
x space
Simulation for only keeping k=15.77~18.64
Theoretical solution for
(only N=0 can survive)
N=0 mode
Ex1 vs ig1
1000

800

600

400

200

Ex1
0

-200

-400

-600

-800

-1000
2200 2400 2600 2800 3000 3200 3400 3600 3800 4000
ig1

k space                                                                                                               N=1
10
x 10                   |Ek1|2 vs k1
4

3.5

3

2.5
|Ek1|2

2

1.5

1

0.5

0
15         15.5   16   16.5       17       17.5   18   18.5   19
k1
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.)

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