Alpha-driven localized cyclotron modes in nonuniform magnetic field

K. R. Chen

Physics Department andPlasma 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

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?

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 decreases

• 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 < lscsK. 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.

Theoretical prediction:1st harmonic =0.16 at =4.2p

2nd harmonic =0.08 at =1.4p

is consistent with the PIC simulationand 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.

e- Landau damping is not important ifpoloidal m < qaR/rve ~1000

finite k// due to shear B is not important ifpoloidal m < qaR/rc ~100

(linear thinking)

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.

Particle-in-cell simulation on

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 =125dx wave modes kept from 1 to 2048 unit time to=co

-1 , dt=0.025 fluid deuterons particle alphas

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 cellsParticle:uniform ~ 2o=250 cellsWave: non-uniform < damping < growth; but, << ofwidth~4o (shown later)

Thus, 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

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.

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.

At the vicinity of minimum of B/B = ± 1%

cf = 3.5 x 10-2

damping 1.4×10-3

growth 4.7×10-3

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

• The localized perturbation on alphas’ perpendicular momentum has clear edges and some alphas have been selectively slowed down (accelerated up) to 1 (6) MeV.

f()

Py vs X

fluid Px vs X

Ex vs X

P 丄 vs X

Pz vs P 丄

Perturbation theory for

localized cyclotron modes

in non-uniform magnetic field

*ˆ( ) ( ) ik xx x e

20

1( ) (1+ )

2 bB X B x

* * 0, ,xk k i x x x

2 2 22 2

2 2* 0 * *

1 1 ˆ[ ( ) ( ) ( )] ( ) 02 2

D D DQ i x x i x x

k x k

* * 0( , , ) 0D k x

1ˆ( ) ( ) ik xx x e

D(w,k,x) (x)=0The dispersion relation and eigenfunction for nonuniform plasma

Assumption: local homogeneity

Taking two-scale-length expansion

Perturbation

Nonuniform magnetic field

The dispersion relation for uniform plasma and magnetic field is

is chosen for absolute instability

For further simplification

* *( , )kPerturbed terms

Perturbation theory for dispersion relation

2 32 3

2 3* * *

1 1( ) ...

2 3!

D D DQ

where

2 2 2 2 2 22 2 2

1 1 12 2 2 2* * * * * 0 * *

1 1 1( ) [ ] 0

2 2 2x x

D D D D D DQ i k k x k

k k k k x k

2 2

1 2* * *

[ ]D D

kk k

2 22 2

2 2* 0

1 1( ( ) ) 0

2 2x

D DQ x

k x

2t x2

22

1( ) 04

tt

Dispersion relation as a parabolic cylinder equation

1ik xeBy eliminating term of , the dispersion relation becomes

Choose to eliminate the term of x

Then,

The dispersion relation can be rewritten as a parabolic cylinder eq.

Absolute instability condition in uniform theory with complex , k

10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3

3.5x 10

-3

ps (

i)

ksr

psi&ksi vs ksr [lbrunid=abs-k-b01a]

10 15 20 25 30 35 40 45 500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

ksi

Re(k)

Growth rate

For the localized wave, we consider the k satisfies the absolute instability condition which implies there is no wave group velocity.

Frequency mismatch

10 15 20 25 30 35 40 45 50-5

0

5x 10

-3

ps (

r)

ksr

psr&ksi vs ksr [lbrunid=abs-k-b01a]

10 15 20 25 30 35 40 45 500

0.5

1ks

i

Imag(k)

Imag(k)

Re(k)

The k with peak growth rate is about 17.

The frequency mismatch is minus at the k of peak growth rate.

N=0x space k space

N=1

Eigenfunctions from the non-uniform theory

Compare with the wave distribution in simulation

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

-600

-400

-200

0

200

400

600

800

ig1

Ex1

Ex1 vs ig1

12 13 14 15 16 17 18 19 200

2

4

6

8

10

12

14

16

18x 10

9

k1

|Ek1

|2

|Ek1|2 vs k1

Simulation for k=all modes (N=1 dominates)

Theoretical solution for N=1 mode

x space

k space

Combined

Compare with the wave distribution in simulation

Simulation for only keeping k=15.77~18.64 (only N=0 can survive)

x space

k space

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

-800

-600

-400

-200

0

200

400

600

800

1000

ig1E

x1

Ex1 vs ig1

15 15.5 16 16.5 17 17.5 18 18.5 190

0.5

1

1.5

2

2.5

3

3.5

4x 10

10

k1

|Ek1

|2

|Ek1|2 vs k1

Theoretical solution for N=0 mode

N=1

Summary

• For fusion produced with =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.)