1. 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 Center Physics 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

2. Outline Introduction Fundamental mechanics Applications in experiments Localized cyclotron modes in non-uniform magnetic field Summary

3. 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 = q  mc For fusion-produced alpha,  = 1.00094. Can relativity be important? Also, for relativistic cyclotron instabilities, the resonance condition is  n  c =  r  i  i  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?

4. Fundamental mechanics

5. 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 space V x V1V1 V2V2 V ph =  k V x V1V1 V2V2 V ph V decreases when  decreases  c   c   zeB  mc    wave l f  cf l s  cs In wave frame of gyro-space  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. vyvy vxvx  l s  cs l f  cf X x x kv 2 <  < kv 1 l f  cf <  l s w cs K. R. Chen, PLA, 1993.

6. A positive frequency mismatch  l s  cs - l f  cf is required to drive two-gyro-stream instability. Characteristics and consequences depend on relative ion rest masses  dielectric function l f  cf l s  cs 0 1 2 3 0 200 400600 t=0 ; * 0.5 t=800 t=1000 t=3200 Maxwellian distribution function 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.] K. R. Chen, PRL, 1994. K. R. Chen, PLA,1998; PoP, 2003. K. R. Chen, PLA, 1993; PoP, 2000.

7. 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. The growth rate peaks at J 13 ’(k  ) ~ 0 Energy extraction Fruchtman, Fisch, and Valeo, PoP, 1997. K. R. Chen, NF, 1995.

8. 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.

9. Applications in experiments

10. Theoretical prediction: 1st harmonic  =0.16 at =4.2  p 2nd harmonic  =0.08 at =1.4  p is consistent with the PIC simulation. Consistent with JET’s observations. 0 2 4 6 012 3 power spectrum (arbitrary amplitude) frequency (  /  cf ) 10 -6 10 -5 10 11 peak field energy 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.

11. Dominance of relativistic effect in magnetoacoustic cyclotron instability 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 dominate those of the classical instability at every harmonics.

12. 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.

13. Localized cyclotron modes in non-uniform magnetic field

14. PIC and hybrid simulations with non-uniform B Physical parameters: n  = 2x10 9 cm -3 E   eV (  = 1.00094) n D = 1x10 13 cm -3 T D = 10 KeV B = 5T harmonic > 12 unstable; for n = 13,  i,max /  = 0.00035 >> (  -13  c  ) r /  PIC parameters (uniform B): periodic system length = 1024 dx,  0 =245dx wave modes kept from 1 to 15 unit time t o =  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 t o =  c  o -1, dt=0.025 fluid deuterons total alphas no.= 10,000,000 (from a PC cluster built by my lab)  B/B = ±1%

15. 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 cells  cells <  o =123 cells Thus, it is generally believed that the resonance excitation can not survive. This result challenges our understanding of resonance. However,

16. 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=1200t=1400 t=1800 t=2000 t=2400t=3000

17. t=1400 Ex vs. X Mode 1 Mode 2 Structure of the localized wave modes 4  o Field energy vs. k Mode 1 Mode 2

18.  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

19. Power spectrum of localized wave modes  B/B=±1% t=1400 Ex vs. X Mode 1 Mode 2 Mode 1 Mode 2  c  o   c      c  o   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

20.  B/B = 0  B/B = ± 0.6%  B/B = ± 0.8%  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.

21. 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.

22. Alpha’s momentum Py vs. X t=1200t=1400t=1800 t=2000t=2400t=3000 The perturbation of alpha’s momentum Py grows anti-symmetrically and then breaks from each respective center. Alphas have been transported.

23. Momentum Py of fluid deuteron vs X The deuteron momentum Py distribution of the localized waves develops from symmetric into anti-symmetric during its growth. t=1200t=1400 t=1800 t=2000t=2400 t=3000

24. t=1800 Two localized waves grow and each consists twin coupled sub-waves. The Py of both fast alpha and thermal deuteron is anti-symmetric while Px is symmetric, due to finite cyclotron orbit and system’s symmetry. E x vs Xfluid Px vs X fluid Py vs X Px vs XPy vs XP 丄 vs X

25. 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 E x vs X P 丄 vs X Pz vs P 丄

26. 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.)