Alpha-driven localized cyclotron modes in nonuniform magnetic fieldas a challenging issue in resonance, relativity, and ITER K. R. ChenPlasma and Space Science CenterPhysics Department of and Institute of Electro-OpticsNational Cheng Kung UniversityWork is supported by National Science Council, TaiwanMay 15-19, 2006 Workshop on ITER Simulation, Peking Univ., Beijing, China

OutlineIntroductionFundamental mechanicsApplications in experimentsLocalized cyclotron modes in non-uniform magnetic fieldSummary

IntroductionFusion energy is essential for humans 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 isw - 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 wi dwr > 0 |dwr| ,, wi

Fundamental mechanics

Two-gyro-streams in the gyro-phase of momentum spaceTwo streams in real space can cause a strong two-stream instabilityTwo-gyro-streamsIn wave frame of real spaceVxV1V2Vph= w / kV decreases when g decreasesIn wave frame of gyro-spacewc increases when g decreases Two-gyro-streams can drive two-gyro-stream instabilities. When slow ion is cold, single-stream can still drive beam-type instability.kv2 < w < kv1lf wcf < w < lswcsK. R. Chen, PLA, 1993.

A positive frequency mismatch D = lswcs - lf wcf is required to drive two-gyro-stream instability.Characteristics and consequences depend on relative ion rest masses 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.

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 atJ13(kr) ~ 0Energy extractionFruchtman, Fisch, and Valeo, PoP, 1997.K. R. Chen, NF, 1995.

Alfvenic behavior and instability transitionElectromagnetic relativistic ion cyclotron instabilitiesInstability transitionAlfvenic behaviorK. R. Chen, et. al., PRE, 2005Instability transits from cubic to quadratic without much change in spectral profile.

Applications in experiments

The straight line is the 0.84 power of the proton density while Joint European Tokamak shows 0.90.1.The scaling is consistent with the experimental measurements.Theoretical prediction:1st harmonic h=0.16 at l=4.2rp2nd harmonic h=0.08 at l=1.4rpis consistent with the PIC simulation.

Consistent with JETs observations.frequency (w/wcf)Cyclotron emission spectrum being consistent with JET Both the relative spectral amplitudes and the scaling with fast ion density are consistent with the JETs experimental measurements. However, there are other mechanisms (Coppi, Dendy) proposed. K. R. Chen, et. al., PoP, 1994.

Dominance of relativistic effect in magnetoacoustic cyclotron instabilityClassical 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.

Explanation for TFTR experimental anomaly of alpha energy spectrum birth distributionsreduced chi-squarecalculated vs. measured spectrums

Localized cyclotron modes in non-uniform magnetic field

PIC and hybrid simulations with non-uniform B Physical parameters: na = 2x109cm-3 Ea= 3.5 MeV (g = 1.00094) nD = 1x1013cm-3 TD = 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-1 dt = 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%

Can wave grow while the resonance can not be maintained? Relativistic ion cyclotron instability is robust against non-uniform magnetic field. 1% in 1000 cellsdw/w

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=1400t=1800t=2000t=2400t=3000

Structure of the localized wave modest=1400Ex vs. XMode 1Mode 24 roField energy vs. kMode 1Mode 2

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

Power spectrum of localized wave modest=1400Ex vs. XMode 1Mode 2w = 13.115 wcao = 13 (wcao/g) (1.01-0.00021)w = 12.862 wcao = 13 (wcao/g) (0.99+0.00031) 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. w3232 or x

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.

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.

Alphas momentum Py vs. Xt=1200t=1400t=1800t=2000t=2400t=3000 The perturbation of alphas momentum Py grows anti-symmetrically and then breaks from each respective center. Alphas have been transported.

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=1400t=1800t=2000t=2400t=3000

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 systems symmetry.Ex vs Xfluid Px vs Xfluid Py vs XPx vs XPy vs XPvs X

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(g)Py vs Xfluid Px vs XEx vs XPvs XPz vs P

SummaryFor 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.)