Effects of drag and diffusion on nonlinear behavior of EP-driven instabilities.

Transcription

1 IAEA-TM EP 2011 / 09 / 07 Effects of drag and diffusion on nonlinear behavior of EP-driven instabilities. Maxime Lesur Y. Idomura, X. Garbet, P. Diamond, Y. Todo, K. Shinohara, F. Zonca, S. Pinches, M. Lilley, B. Breizman, L. Villard, V. Grandgirard, C. Nguyen, N. Dubuit, A. Fasoli, Z. Guimarães-Filho.

2 Motivation 2 / 20 Fast particles transport and loss depend on the nonlinear saturation and non-localities fluid nonlinearities (coupling with GAM, ZF, spatial avalanches ) kinetic nonlinearities (particle trapping, frequency sweeping, avalanches in velocityspace ) NSTX Podestà, et al., PoP (10) Linearly damped modes can be driven by phase-space structures (holes, clumps, blobs, granulations ) Dupree, et al., PF (82) Berman, et al., PF (85) Berk, et al., PoP (99) Kosuga, et al., PoP (10) Creation and evolution of structures depend strongly on the nature and frequency of collisions. Lilley, et al., PoP (10) Lesur, et al., PoP (10)

3 Key points BB model as a tractable model with key ingredients Observed quantitative similarities between BB model, and TAEs. Periodic chirping regime is recovered only when collisions include drag and diffusion. 3 / 20 Berk, Breizman, et al., PFB(90), PRL(96), PoP(99) MAST JET Lesur, et al., under review BB Simulation BB model Fasoli, et al., PRL(98)

4 Outline 4 / 20 I) Chirping II) BB model with drag & diffusion III) Periodic chirping regime IV) Teaser: phasestrophy growth Summary Perspectives

5 Chirping (frequency sweeping) 5 / 20 NL Chirping Chaotic behavior, Splitting into two spectral modes, up and down in general, Lifetime << equilib. evolution time. Hole/clumps Berk, et al., PLA(97) LHD Toi, et al. MAST Shot #5658 Gryaznevich, Sharapov, PPCF(04) Observations of chirping TAEs Kusama, et al., NF(99) Not shown: DIII-D, NSTX, CHS

6 Chirping (2) 6 / 20 Chirping GAMs Chirping e-fishbones HL-2A Berk, et al., NF(06) Nazikian, et al., PRL(08) Chen, et al., NF(10)

7 BB-model simulations TAE simulation Chirping in simulations 7 / 20 Pinches, et al., PPCF(04) Todo, et al., JPFR(03) Vann et al., PRL(07) Lilley et al., PoP(10)

8 Normalized velocity distribution Holes and Clumps 8 / 20 Shifting frequencies correspond to the evolution of holes and clumps. Berk, Breizman, Petviashvili, PLA(97) resonant velocity clump hole

9 Navigation 9 / 20 I) Chirping II) BB model with drag & diffusion III) Periodic chirping regime IV) Teaser: phasestrophy growth Summary Perspectives

10 The BB model 10 / 20 Berk, et al., PoP(95) Classic bump-on-tail instability, with collisions and external damping. 1D kinetic equation with a collision operator including dynamical friction (drag), and velocity-space diffusion, Displacement Current Equation with an external wave damping accounting for background dissipative mechanisms, Initial distribution function Single electrostatic wave, COBBLES COnservative Berk-Breizman semi-lagrangian Extended Solver Lesur, et al., JAEA-R(07) Lesur, et al., PoP(09)

11 The BB model 10 / 20 Berk, et al., PoP(95) Classic bump-on-tail instability, with collisions and external damping. 1D kinetic equation with a collision operator including dynamical friction (drag), and velocity-space diffusion, ν d /ν f 1.4 in ITER Lilley et al., PRL(10) maybe double or more with impurities Lesur et al., PoP(10) Displacement Current Equation with an external wave damping accounting for background dissipative mechanisms, Initial distribution function Single electrostatic wave, COBBLES COnservative Berk-Breizman semi-lagrangian Extended Solver Lesur, et al., JAEA-R(07) Lesur, et al., PoP(09)

12 Small drag (ν d /ν f = 5) Bounce frequency 11 / 20 Time Collision frequency (diffusion ) Similar behaviors with Krook operator. Vann, et al., PoP(03) Lesur, et al., PoP(09) External damping rate

13 Small drag (ν d /ν f = 5) 12 / 20 Collision frequency (diffusion ) Time External damping rate

14 Small drag (ν d /ν f = 5) Krook case is qualitatively different. 12 / 20 Collision frequency (diffusion ) Time External damping rate

15 Small drag (ν d /ν f = 5) Krook case is qualitatively different. 12 / 20 Collision frequency (diffusion ) Subcritical instability Time External damping rate

16 Large drag (ν d /ν f = 1) 13 / 20 Collision frequency (diffusion ) External damping rate

17 Large drag (ν d /ν f = 1) 14 / 20 Agrees with Lilley et al., PoP(10) Collision frequency (diffusion ) External damping rate

18 Large drag (ν d /ν f = 1) 14 / 20 Agrees with Lilley et al., PoP(10) Collision frequency (diffusion ) Subcritical instability External damping rate

19 Navigation 15 / 20 I) Chirping II) BB model with drag & diffusion III) Periodic chirping regime IV) Teaser: phasestrophy growth Summary Perspectives

20 Frequency shift Frequency shift Chirping period 16 / 20 chirping period chirping period In experimentallyrelevant conditions, the time between 2 chirping bursts decreases with: Increasing because for these time-scales, drag is the main mechanism of unstable slope recovery. Decreasing because diffusion contradicts an effect of drag. Increasing because the initial slope to which the distribution relaxes is steeper. * general picture

21 h/c shift δf h/c amplitude Electric field amplitude h/c width Effect of drag on period 17 / 20 Time Frequency shift hole/clump model: Gaussian Time

22 h/c shift δf h/c amplitude Electric field amplitude h/c width Effect of drag on period 17 / 20 Time The growth of perturbation amplitude is not linear. It gives only a rough estimation of. Frequency shift hole/clump model: Gaussian Time

23 h/c shift δf h/c amplitude Electric field amplitude h/c width Effect of drag on period 17 / 20 Time The growth of perturbation amplitude is not linear. It gives only a rough estimation of. Frequency shift hole/clump model: Gaussian The linear growth rate increases with hole/clump drag-induced shift. The chirping period decreases with increasing drag. Time

24 Navigation 18 / 20 I) Chirping II) BB model with drag & diffusion III) Periodic chirping regime IV) Teaser: phasestrophy growth Summary Perspectives

25 Phasestrophy growth 19 / 20 In the spirit of Dupree s theory of phasespace density holes, Dupree, PF(82) - The growth is nonlinear - Growth compatible with negative linear growth wave

26 In a nutshell 20 / 20 1D BB model can reproduce complex nonlinear features of frequency sweeping TAEs. Taking into account collisional drag and diffusion changes the NL behavior bifurcations, and introduces new kinds of saturation, in particular qualitatively different chirping regimes. Dominant frequency in the spectrogram may be shifted up to 50%. Chirping bursts correspond to relaxation oscillation due to hole/clump dynamic. In the periodic chirping regime, bursts frequency decreases with decreasing drag, without significantly increasing bursts amplitude. Subcritical instabilities are driven by momentum exchange via. The growth rate of structures is proportional to. Take-out message Chirping bursts, which are detrimental to EP confinement, can be mitigated by increasing collisional diffusion or reducing drag. This effect should be investigated in experiments.

27 Backup slides 27 / 20

28 In our (rough) scan, one solution: Analysis of JT-60U 28 / 20 Lesur, et al. PoP(10) JT-60U 8.8% 4.7% 0.42% 1.7% 4.6% Simulation Noise level Quantitative agreement for growth and decay of chirping structures

29 Quantitative agreement for growth and decay of chirping structures In our (rough) scan, one solution: Analysis of MAST 29 / 20 (paper in preparation) MAST 9.9% 5.0% 0.55% 2.3% 5.5% Simulation

30 Effect of h/c shift on 30 / 20 no h/c limit After the plateau formation and a chirping phase, the growth rate is negative. The growth rate increases with hole/clump draginduced shift. no h/c limit The chirping period decreases with increasing drag.

31 From 3D TAE to 1D BB 31 / 20 3D TAE, expanded around one resonance Neglecting variations around the resonant phase-space surface, 1D bump-on-tail instability, In a frame moving with the wave. Magnetic shear Electrostatic potential P θ µ I Resonant phase-space surface P ζ Adding collisions and external damping processes Berk-Breizman (BB) model Lichtenberg (69) Berk, Breizman, Pekker, NF(95)