Alpha-driven localized cyclotron modes in nonuniform magnetic field

1. Alpha-driven localized cyclotron modes in nonuniform magnetic field Physics Department and Plasma and Space Science Center National Cheng Kung University K. R. Chen 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 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 widwr > 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?

4. V V1 l w Vph x s cs X V2 l w f cf z eB W c w = = c g g m c vy • lf wcf • w lswcs vx x x wwave f B 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 V1 Vph= w / k V2 x V decreases when g decreases kv2 < w < kv1 Two-gyro-streams In wave frame of gyro-space wcincreases wheng decreases lfwcf < 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.

5. dielectric function 3 t=0 ; * 0.5 t=800 t=1000 w t=3200 lf wcf lswcs Maxwellian 2 1 distribution function 0 0 400 600 200 P ^ Characteristics and consequences depend on relative ion rest masses A positive frequency mismatchD = lswcs- lf wcf is required to drive two-gyro-stream instability. K. R. Chen, PLA, 1993; PoP, 2000. • 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. • 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.

6. 6 -5 (arbitrary amplitude) 10 power spectrum 4 2 0 3 0 1 2 -6 10 peak field energy 11 10 10 10 fast ion density Cyclotron emission spectrum being consistent with JET Theoretical prediction: 1st harmonic h=0.16 at l=4.2rp 2nd harmonic h=0.08 at l=1.4rp is consistent with the PIC simulation and JET’s observations. e- Landau damping is not important if poloidal m < qaRw/rve ~1000 finite k// due to shear B is not important if poloidal m < qaRw/rc ~100 (linear thinking) frequency (w/wcf) 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. 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.

7. 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. Explanation for TFTR experimental anomaly of alpha energy spectrum birth distributions calculated vs. measured spectrums reduced chi-square

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: • na = 2x109cm-3Ea= 3.5 MeV (g = 1.00094) • nD= 1x1013cm-3TD = 10 KeV B = 5T • harmonic > 12 unstable; for n = 13, wi,max/w = 0.00035 >> (w-13wca)r / w • PIC parameters (uniform B): • periodic system length = 1024 dx,r0 =245dx • wave modes kept from 1 to 15 • unit time to = wcD-1dt = 0.025 • total deuterons no. = 59,048 • total alphas no.= 23,328 • 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 • particle alphas dB/B = ±1%

10. dB/B = ± 1% Can wave grow while the resonance can not be maintained? 1% in 1000 cells Particle:uniform dw/w << g-1=0.00094 < 0.2% ~ 2ro=250 cells Wave: non-uniform dw< damping < growth; but, << dwofwidth~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.

11. Electric field vs. Xfor 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 instabilityand energy dissipation. • A gyrokinetic theory has been developed. A wavelet kinetic theory may be possible.

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

13. Structure of wave modes vs. magnetic field non-uniformity dB/B = ± 0.4% dB/B = 0 dB/B = ± 0.2% dB/B = ± 0.8% dB/B = ± 0.6% dB/B = ± 1%

14. dB/B = ± 0.6% dB/B = 0 dB/B = ± 0.8% dB/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.

15. 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.

16. 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.

17. t=3000 Pz vs P丄 Py vs X 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.

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

19. Perturbation theory for dispersion relation 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 Perturbed terms where For further simplification

20. 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.

21. 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. The k with peak growth rate is about 17. Imag(k) Growth rate Re(k) The frequency mismatch is minus at the k of peak growth rate. Frequency mismatch Imag(k) Re(k)

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

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

24. Compare with the wave distribution in simulation x space Simulation for only keeping k=15.77~18.64 (only N=0 can survive) Theoretical solution for N=0 mode k space N=1

25. 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.)