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.)