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K. R. Chen

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.

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K. R. Chen

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  1. Alpha-driven localized cyclotron modes in nonuniform magnetic fieldas 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 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. Fundamental mechanics

  5. 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.

  6. 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.

  7. The history of field energy; energy extraction K. R. Chen, NF, 1995. The growth rate peaks at J13’(kr) ~ 0 Energy extraction Fruchtman, Fisch, and Valeo, PoP, 1997. • 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.

  8. cubic quadratic Alfvenic behavior and instability transition Electromagnetic relativistic ion cyclotron instabilities K. R. Chen, et. al., PRE, 2005 Instability transition Alfvenic behavior Instability transits from cubic to quadratic without much change in spectral profile.

  9. Applications in experiments

  10. 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. Consistent with JET’s observations. 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.

  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 instabilitydominate • those of the classical instability at every harmonics.

  12. 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

  13. Localized cyclotron modes in non-uniform magnetic field

  14. 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 =123dx • wave modes kept from 1 to 2048 • unit time to=wcao-1 , dt=0.025 • fluid deuterons • total alphas no.= 10,000,000 • (from a PC cluster built by my lab) dB/B = ±1%

  15. dB/B = ± 1% Can wave grow while the resonance can not be maintained? 1% in 1000 cells dw/w << g-1=0.00094 < 0.1% <- 100 cells < ro=123 cells 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.

  16. 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.

  17. 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

  18. 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%

  19. Mode 1 dB/B=±1% Mode 2 Power spectrum of localized wave modes w = 13.115 wcao =13 (wcao/g) (1.01-0.00021) w<13wca at peak B wcapeak=13.118 Ex vs. X t=1400 w = 12.862 wcao =13 (wcao/g) (0.99+0.00031) • < 13wca at x>3232 or x<2896 Mode 1 Mode 2 • Resonance is a consequence of the need to drive instability • for dissipating free energy and increasing the entropy. • A wave eigen-frequency (even w < wca) is collectively decided in a coherent means; • a special wave form in real space is created for this purpose, even without boundary.

  20. 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.

  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=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.

  23. Summary • For fusion produced a with g=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.)

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