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New Insights into Energetic Ion Transport by Instabilities: The importance of phase

New Insights into Energetic Ion Transport by Instabilities: The importance of phase. W.W. (Bill) Heidbrink* UC Irvine. *in collaboration with the DIII-D & LAPD teams, especially:. Shu Zhou, Xi Chen, Liu Chen, Yubao Zhu University of California, Irvine

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New Insights into Energetic Ion Transport by Instabilities: The importance of phase

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  1. New Insights into Energetic Ion Transport by Instabilities: The importance of phase W.W. (Bill) Heidbrink* UC Irvine *in collaboration with the DIII-D & LAPD teams, especially: Shu Zhou, Xi Chen, Liu Chen, Yubao Zhu University of California, Irvine T. Carter, S. Vincena, S. K. P. Tripathi University of California, Los Angeles M. Van Zeeland, D. Pace, R. Fisher General Atomics G. Kramer, B. Grierson, R. White, K. Ghantous, N. Gorelenkov Princeton Plasma Physics Laboratory E. Bass University of California, San Diego

  2. Fast-ion orbits have large excursions from magnetic field lines Elevation (80 keV D+ ion in DIII-D) • Perp. velocity  gyromotion • Parallel velocity  follows flux surface • Curvature & Grad B drifts  excursion from flux surface Plan view Parallel ~ v Drift ~ (vll2 + v2/2)  Large excursions for large velocities

  3. Complex EP orbits are most simply described using constants of motion Projection of 80 keV D+ orbits in the DIII-D tokamak Constants of motion on orbital timescale: energy (W), magnetic moment (m), toroidal angular momentum (Pz) Distribution function: f(W,m,Pz) Roscoe White, Theory of toroidally confined plasmas

  4. The wave phase determines the sign of the force Resonance occurs when the orbit-averaged phase is constant in time, i.e., mathematically, resonance produces a secular term ~ t

  5. Outline FishbonesConvective resonant transport for kperpρ<<1 Energetic-particle GAMNonlinear sub-harmonic resonances at large amplitude (kperpρ<<1) Drift WavesOrbit-averaging for kperpρ>>1 Alfvén EigenmodesNon-resonant losses for kperpρ~1 Alfvén Eigenmodes“Stiff” transport for many small-amplitude modes with kperpρ~1

  6. Outline FishbonesConvective resonant transport for kperpρ<<1 Energetic-particle GAMNonlinear sub-harmonic resonances at large amplitude (kperpρ<<1) Drift WavesOrbit-averaging for kperpρ>>1 Alfvén EigenmodesNon-resonant losses for kperpρ~1 Alfvén Eigenmodes“Stiff” transport for many small-amplitude modes with kperpρ~1

  7. Resonant transport occurs when an aspect of the orbital motion matches the wave frequency (main energy exchange) Write vd as a Fourier expansion in terms of poloidal angle q: Wave mode #s Drift harmonic wq Parallel resonance condition: w = nwz + pwq Time to complete toroidal orbit  wz Time to complete poloidal orbit  wq vllEll0 (when Ell~0) wz Energy exchange resonance condition: w - nwz - (m+l)wq=0

  8. Fast-ion Loss Detector (FILD) measures lost trapped ions at off-axis fishbone burst • Scintillator acts as a magnetic spectrometer to measure energy & pitch of lost fast ions Projection of lost orbit DIII-D off-axis fishbone data • Bright spot for ~80 keV, trapped fast ions that satisfy resonance condition Heidbrink, Plasma Phys. Cont. Fusion 53 (2011) 085028

  9. Losses have a definite phase relative to the mode DIII-D off-axis fishbone data • Particles are expelled in a “beacon” that rotates with the mode • Caused by Eqx Bfconvective transport • Losses occur at the phase that pushes particles outward Heidbrink, Plasma Phys. Cont. Fusion 53 (2011) 085028

  10. Coherent convective transport occurs for modes that maintain resonance across the plasma Calculated Fishbone Loss Orbit • The fishbone was a globally extended, low-frequency mode (kperpρ<<1) • Low frequency  1st & 2nd adiabatic invariants are conserved • μ conservation  particles that move out (to lower B) lose Wperp • Main loss mechanism: convective E x B radial transport White, Phys. Fluids 26 (1983) 2958

  11. Convective phase locked transport “marches” particle across the plasma Convective phase locked (~ Br, large %) EPs stay in phase with wave as they “walk” out of plasma • Leftward motion on graph implies outward radial motion

  12. Resonant transport drives instability • Ions that move out lose energy (μ conservation) • Ions that move in gain energy • Fast-ion profile is peaked  more ions move out than in  wave gains energy • Equivalent explanation: Heidbrink, Phys. Plasmas 15 (2008) 055501

  13. Outline FishbonesConvective resonant transport for kperpρ<<1 (Ions “see” constant phase) Energetic-particle GAMNonlinear sub-harmonic resonances at large amplitude (kperpρ<<1) Drift WavesOrbit-averaging for kperpρ>>1 Alfvén EigenmodesNon-resonant losses for kperpρ~1 Alfvén Eigenmodes“Stiff” transport for many small-amplitude modes with kperpρ~1

  14. Standard theory: resonances at frequency harmonics Energy exchange resonance condition: w - nwz - (m+l)wq=0 • For n=0 mode, expect resonances when w/wq = 1, 2, ....

  15. Find subharmonic resonances in simulation of large-amplitude EGAM! DIII-D Simulation • Simulate energetic-particle driven geodesic acoustic mode (EGAM) • Mode has large electric field • For small potential, find usual harmonic resonances • For large amplitudes, subharmonic resonances appear • Analytic theory explains results Kramer, PRL 109 (2012) 035003

  16. Experimental evidence of subharmonic losses exists DIII-D data • No evidence of subharmonics in instability spectra • Coherent losses at 1/2 resonance appear when EGAM amplitude is large Kramer, PRL 109 (2012) 035003

  17. Outline FishbonesConvective resonant transport for kperpρ<<1 (Ions “see” constant phase) Energetic-particle GAMNonlinear sub-harmonic resonances at large amplitude (kperpρ<<1) Drift WavesOrbit-averaging for kperpρ>>1 Alfvén EigenmodesNon-resonant losses for kperpρ~1 Alfvén Eigenmodes“Stiff” transport for many small-amplitude modes with kperpρ~1

  18. Large orbits spatially filter electrostatic turbulence Drift wave created by an obstacle in the LAPD • Potential fluctuations in plane perpendicular to B • Small-orbit ion stays in phase with wave  large E x B kick • Large-orbit ion sees rapid phase change  small E x B kick Fluctuation Amplitude

  19. Large orbits spatially filter electrostatic turbulence Fluctuation Amplitude • Temporal average over gyromotion  spatial filter of the potential • Gyro-phase averaging scales as: • First simulation in 1979*  <> <> <> • Other types of orbital motion also phase-average *Naitou, J. Phys. Soc. Japan 46 (1979) 258

  20. Launch a beam of particles. How do they spread in time? LAPD Data TORPEX Simulation Review paper on LAPD & TORPEX experiments: Heidbrink, PPCF 54 (2012) 124007

  21. Transport is characterized by an exponent g The spread in the particle position W is used to extract a transport exponent: TORPEX Simulation • For example, since there is no force in the parallel direction, Dz=(Dvz) t, so g=2 (called “ballistic” or “convective” transport) g=1 “diffusive” g<1 “sub-diffusive” g>1 “super-diffusive” Gustafson, PoP 19 (2012) 062306

  22. Three expected turbulent transport regimes • Initially Dr=vkickt  g=2 (convective) • Wave phase changes  some particles pushed back toward initial positions  g<1 (sub-diffusive) • Eventually many random kicks  random walk with W2 ~ t (normal diffusion) T. Hauff and F. Jenko, Phys. Pl. 15 (2008)112307

  23. Experimental Setup: Fast ions orbit through turbulence LAPD • Create plasma with electrostatic fluctuations • Pass Li+ beam through waves • Scan collector spatially to measure beam spreading • Measure properties of turbulence S. Zhou, PoP 17 (2010) 092103 23

  24. Beam spot provides information on radial and parallel transport Collector scans measure beam spreading

  25. Use obstacles to enhance turbulence (LAPD) Photograph from end of machine Cu Obstacle Fast-ion Orbit Li Source • Obstacle creates sharp density gradient • Large fluctuations at obstacle edge • Control turbulence by biasing obstacle & changing plasma species [Zhou, Phys. Pl. 19 (2012) 012116] Density Fluctuations 25

  26. Model the fields with fluid codes (constrained by measurements) then compute orbits • Magnetic fluctuations are small  Assume electrostatic • Long parallel wavelengths  Assume 2D fluctuating fields • Adjust amplitude of simulated turbulence to match experiment • Apply a Lorentz orbit code in simulated fields Floating Potential Cross-spectrum Use the resistive fluid code BOUT to simulate the microturbulence. Popovich et al, PoP 17, 122312 (2010) 26

  27. Fast-Ion Transport Decreases with Increasing Fast-Ion Energy • Axial speed held constant S. Zhou, PoP 17 (2010) 092103

  28. Turbulent spreading is super-diffusive (g~2) Data W2 • Classical transport is diffusive (g~1) S. Zhou, PoP 17 (2010) 092103

  29. Test-particle simulation in a BOUT simulated wave field agrees well with experiment Data Simulation Result S. Zhou, PoP 17 (2010) 092103

  30. Energy Scaling of Beam Transport Shows Gyro-Averaging Effect Experimental Data Averaged Fluctuating Amplitude Turbulent transport • Gyro-Averaging Effect: • The effective potential is phase-averaged over the fast ion gyro-orbit S. Zhou, PoP 17 (2010) 092103 30

  31. Use annular obstacle to vary the turbulence Cu Obstacle • Fixed gyroradius • Vary correlation length Lcorr & scale length of dominant modes Ls Fast-ion Orbit Li Source Density Fluctuations S. Zhou, PoP 18 (2011) 082104 31

  32. Different Transport Levels are Observed in 3 Typical Background Turbulence Cases A Helium Vbias=0V Lcorr=23cm Ls=2.6cm δn/n=0.55 (B) B Neon Vbias=75V Lcorr=19cm Ls=6.3cm δn/n=0.35 (C) C Helium Vbias=100V Lcorr=6cm Ls=2.6cm δn/n=0.53 (A) S. Zhou, PoP 18 (2011) 082104 32 Distance

  33. A simple model explains the dependence on Lcorr and Ls • Wave potential (amplitude) modeled by: • Gyro averaging is applied along an off-axis orbit: • Larger Ls: Gyro-averaged f increases with increasing potential scale length • Gyro-averaged f increases for waves with more modes S. Zhou, PoP 18 (2011) 082104

  34. Large scale size Ls reduces gyro-averaging; Short correlation length Lcorr reduces phase-averaging A Helium Vbias=0V Lcorr=23cm Ls=2.6cm (B) (C) B Neon Vbias=75V Lcorr=19cm Ls=6.3cm (A) Simple Model C (C) Helium Vbias=100V Lcorr=6cm Ls=2.6cm (A) (B) 34

  35. Sub-Diffusive Regime is Observed when Fast Ion Time-of-Flight Exceeds Wave Half Period Convective Sub-diffusive • Simulation uses measured time-dependent wave fields • Flat-part of curve occurs when dominant mode changes by 1800 pushing ions the opposite way S. Zhou, PoP 18 (2011) 082104

  36. Conclusion on Fast Ion Transport in Electrostatic Turbulent Waves in the LAPD • In experiment with plate obstacle: • Fast ion transport decreases with increasing fast ion energy (more phase averaging)S. Zhou et al., Phys. Plasmas 17, 092103 (2010) • In experiment with annulus obstacle: • Waves with larger spatial scale size cause more fast-ion transport • Turbulent waves cause more fast-ion transport than coherent waves (less phase averaging)S. Zhou et al., Phys. Plasmas 18, 082104 (2011) • Beam diffusivity versus time • Transport is convective when fast ion time-of-flight << wave period • Transport is sub-diffusive when fast ion time-of-flight exceeds half the wave period (phase reversal pushes ions back) • S. Zhou et al., Phys. Plasmas 18, 082104 (2011) 36

  37. Outline FishbonesConvective resonant transport for kperpρ<<1 (Ions “see” constant phase) Energetic-particle GAMNonlinear sub-harmonic resonances at large amplitude (kperpρ<<1) Drift WavesOrbit-averaging for kperpρ>>1 Alfvén EigenmodesNon-resonant losses for kperpρ~1 Alfvén Eigenmodes“Stiff” transport for many small-amplitude modes with kperpρ~1

  38. Perform an analogous experiment on DIII-D Plan view of DIII-D • Neutral beams are the fast-ion source • FILD is the detector • Alfvén waves with kperpρ~1 are the fluctuations Arrange the orbit so it passes close to FILD Xi Chen, Phys. Rev. Lett. 110 (2013) 065004

  39. Alfvén eigenmodes deflect fast ions to the scintillator after one bounce orbit Elevation • The contours show a calculated mode structure • Unperturbed and perturbed orbits are shown Xi Chen, Phys. Rev. Lett. 110 (2013) 065004

  40. Loss signal oscillates at the Alfvén eigenmode frequency • Enhanced losses only occur when unperturbed orbit passes close to the detector • Can infer the radial “kick” from the size of the coherent FILD fluctuations Xi Chen, PRL 110 (2013) 065004

  41. Displacement is linearly proportional to mode amplitude • Ions with correct phase are pushed out • Consistent with ballistic transport • Non-resonant ions are lost Xi Chen, Phys. Rev. Lett. 110 (2013) 065004

  42. Enhanced prompt losses are an important new effect • Powerful diagnostic technique  quantifies transport in well-defined orbit • Losses are concentrated spatially  possibility of wall damage • Non-resonant lost ions do not recover their energy  additional instability drive? Xi Chen, Phys. Rev. Lett. 110 (2013) 065004

  43. Nonlinear interactions for multiple Alfvén eigenmodes • Each mode alters the phase of the ion at the other mode: Fluctuations RSAE TAE • This generates fluctuations in the losses at the sum (ω1+ω2) & difference (ω1-ω2) frequencies Losses Difference Sum 2ndRSAE 2nd TAE Xi Chen, (2013) in preparation

  44. The zeroth-order adiabatic invariant μ0=Wperp/B is not conserved in this process • For kperpρ~1, there is a correction to μ even for and ω<<Ωi • Ion gets “kick” on one side but not other • Applies for vllδBperp and vperpδBll too • The calculated shift in μ is ~ 5% Kramer (2013) in preparation

  45. The zeroth-order adiabatic invariant μ0=Wperp/B is not conserved in this process • Full-orbit SPIRAL* simulation calculates a jump in μ0 when ion traverses mode • Calculated FILD oscillation in good agreement with experiment • Analytical calculation: Kramer (2013) in preparation • Similar deviations found for kinetic Alfvén waves in full-orbit simulations of astrophysical turbulence [Chandran, Ap. J. 720 (2010) 503] *Kramer, PPCF 55 (2013) 025013

  46. Outline FishbonesConvective resonant transport for kperpρ<<1 (Ions “see” constant phase) Energetic-particle GAMNonlinear sub-harmonic resonances at large amplitude (kperpρ<<1) Drift WavesOrbit-averaging for kperpρ>>1 Alfvén EigenmodesNon-resonant losses for kperpρ~1 Alfvén Eigenmodes“Stiff” transport for many small-amplitude modes with kperpρ~1

  47. Many small amplitude Alfven eigenmodes flatten the fast-ion profile Radial dTe profile during beam injection into DIII-D Radial fast-ion profile Heidbrink, PRL 99 (2007) 245002 Van Zeeland, PRL 97 (2006) 135001

  48. These plasmas have an enormous number of resonances Calculated energy change due to a single harmonic in a DIII-D plasma • Colors indicate energy exchange • Each pair is from one pwq of the resonance condition • Each toroidal mode is composed of multiple poloidal harmonics  hundreds of important resonances White, Plasma Phys. Cont. Fusion 52 (2010) 045012

  49. Many small-amplitude resonances  appreciable transport Partial island overlap of some of the resonances • Although the individual island widths are small, stochastic transport still occurs  flattened profile consistent with experiment • Recent work: efficient algorithm to calculate profile for situations with numerous small-amplitude modes White, Comm. Nonlinear Science Numerical Simulation 17 (2012) 2200 White, Plasma Phys. Cont. Fusion 52 (2010) 045012

  50. What I thought (until recently) ... Major goal of Energetic Particle research: Predict fast-ion transport in ITER (and other future devices) • Given the fields, we can calculate fast-ion transport but we have to know the mode amplitude & spectra • The mode spectra is very hard to predict (extremely complicated nonlinear physics) Our recent results with off-axis beam injection suggests there may be an easier way ...

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