1. 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

2. Outline Introduction Particle-in-cell simulation Analytical theory 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. 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  cs K. R. Chen, PLA, 1993.

5. 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.

6. Theoretical prediction: 1st harmonic  =0.16 at =4.2  p 2nd harmonic  =0.08 at =1.4  p is consistent with the PIC simulation and 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. e - Landau damping is not important if poloidal m < qaR  /rv e ~1000 finite k // due to shear B is not important if poloidal m < qaR  /rc ~100 (linear thinking)

7. 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.

8. Particle-in-cell simulation on localized cyclotron modes in non-uniform magnetic field

9. 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 =125dx wave modes kept from 1 to 2048 unit time t o =  c  o -1, dt=0.025 fluid deuterons particle alphas  B/B = ±1%

10. 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 Particle:  uniform  ~ 2  o =250 cells Wave: non-uniform  < damping < growth; but, <<  of  width~4  o (shown later) Thus, it is generally believed that the resonance excitation can not survive. This result challenges our understanding of resonance. However,

11. 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

12. 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

13.  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

14. 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

15.  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.

16. 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

17. 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.

18. 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

19. Perturbation theory for localized cyclotron modes in non-uniform magnetic field

20. The 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 Perturbed terms Perturbation theory for dispersion relation where

21. Dispersion relation as a parabolic cylinder equation By eliminating term of, the dispersion relation becomes Choose to eliminate the term of Then, The dispersion relation can be rewritten as a parabolic cylinder eq.

22. Absolute instability condition in uniform theory with complex , k 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 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.

23. N=0 x space k space N=1 Eigenfunctions from the non-uniform theory

24. Growth rate vs. nonuniformity Growth rate and eigenfunctions from the theory Frequency mismatch N=0 (the rank of eigenvalue) N=1 Frequency mismatch can be from negative to positive.

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

26. 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 Theoretical solution for N=0 mode N=1

27. 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.)