1. W.W. (Bill) Heidbrink* UC Irvine 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 *in collaboration with the DIII-D & LAPD teams, especially: New Insights into Energetic Ion Transport by Instabilities: The importance of phase

2. Fast-ion orbits have large excursions from magnetic field lines Plan view Elevation (80 keV D + ion in DIII-D) Perp. velocity  gyromotion Parallel velocity  follows flux surface Curvature & Grad B drifts  excursion from flux surface Parallel ~ v Drift ~ (v ll 2 + 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 (  ), toroidal angular momentum (P  ) Roscoe White, Theory of toroidally confined plasmas Distribution function: f(W, ,P  )

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 Fishbones Convective resonant transport for k perp ρ<<1 Energetic-particle GAM Nonlinear sub-harmonic resonances at large amplitude (k perp ρ<<1) Drift Waves Orbit-averaging for k perp ρ>>1 Alfvén Eigenmodes Non-resonant losses for k perp ρ~1 Alfvén Eigenmodes “Stiff” transport for many small- amplitude modes with k perp ρ~1

6. Outline Fishbones Convective resonant transport for k perp ρ<<1 Energetic-particle GAM Nonlinear sub-harmonic resonances at large amplitude (k perp ρ<<1) Drift Waves Orbit-averaging for k perp ρ>>1 Alfvén Eigenmodes Non-resonant losses for k perp ρ~1 Alfvén Eigenmodes “Stiff” transport for many small- amplitude modes with k perp ρ~1

7. Resonant transport occurs when an aspect of the orbital motion matches the wave frequency Time to complete poloidal orbit    Time to complete toroidal orbit      v ll E ll  0 (when E ll ~0) Parallel resonance condition:  n    p   Write v d as a Fourier expansion in terms of poloidal angle  : Energy exchange resonance condition:  n    (m+l)    (main energy exchange) Drift harmonic Wave mode #s

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

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

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

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

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 Fishbones Convective resonant transport for k perp ρ<<1 (Ions “see” constant phase) Energetic-particle GAM Nonlinear sub-harmonic resonances at large amplitude (k perp ρ<<1) Drift Waves Orbit-averaging for k perp ρ>>1 Alfvén Eigenmodes Non-resonant losses for k perp ρ~1 Alfvén Eigenmodes “Stiff” transport for many small- amplitude modes with k perp ρ~1

14. Standard theory: resonances at frequency harmonics For n=0 mode, expect resonances when    Energy exchange resonance condition:  n    (m+l)   

15. Find subharmonic resonances in simulation of large- amplitude EGAM! 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 DIII-D Simulation Kramer, PRL 109 (2012) 035003

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

17. Outline Fishbones Convective resonant transport for k perp ρ<<1 (Ions “see” constant phase) Energetic-particle GAM Nonlinear sub-harmonic resonances at large amplitude (k perp ρ<<1) Drift Waves Orbit-averaging for k perp ρ>>1 Alfvén Eigenmodes Non-resonant losses for k perp ρ~1 Alfvén Eigenmodes “Stiff” transport for many small- amplitude modes with k perp ρ~1

18. Large orbits spatially filter electrostatic turbulence Fluctuation Amplitude  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 Drift wave created by an obstacle in the LAPD

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

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

21. Transport is characterized by an exponent  Gustafson, PoP 19 (2012) 062306 The spread in the particle position W is used to extract a transport exponent: For example, since there is no force in the parallel direction,  z=(  v z ) t, so  =2 (called “ballistic” or “convective” transport)  “diffusive”  “sub-diffusive”  “super-diffusive” TORPEX Simulation

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

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

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

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

26. Model the fields with fluid codes (constrained by measurements) then compute orbits 26 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)

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 (  2) Classical transport is diffusive (  ~1) W2W2 Data S. Zhou, PoP 17 (2010) 092103

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

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

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

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

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

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

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 180 0  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: o 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: o Waves with larger spatial scale size cause more fast-ion transport o 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 o Transport is convective when fast ion time-of-flight << wave period o 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 Fishbones Convective resonant transport for k perp ρ<<1 (Ions “see” constant phase) Energetic-particle GAM Nonlinear sub-harmonic resonances at large amplitude (k perp ρ<<1) Drift Waves Orbit-averaging for k perp ρ>>1 Alfvén Eigenmodes Non-resonant losses for k perp ρ~1 Alfvén Eigenmodes “Stiff” transport for many small- amplitude modes with k perp ρ~1

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

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

40. Loss signal oscillates at the Alfvén eigenmode frequency Xi Chen, PRL 110 (2013) 065004 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

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. TAE RSAE Difference 2 nd RSAE 2 nd TAE Sum Nonlinear interactions for multiple Alfvén eigenmodes Xi Chen, (2013) in preparation Fluctuations Losses Each mode alters the phase of the ion at the other mode: This generates fluctuations in the losses at the sum (ω 1 +ω 2 ) & difference (ω 1 -ω 2 ) frequencies

44. The zeroth-order adiabatic invariant μ 0 =W perp /B is not conserved in this process Kramer (2013) in preparation For k perp ρ~1, there is a correction to μ even for and ω<<Ω i Ion gets “kick” on one side but not other Applies for v ll δB perp and v perp δB ll too The calculated shift in μ is ~ 5%

45. The zeroth-order adiabatic invariant μ 0 =W perp /B is not conserved in this process Kramer (2013) in preparation Full-orbit SPIRAL * simulation calculates a jump in μ 0 when ion traverses mode Calculated FILD oscillation in good agreement with experiment Analytical calculation: 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 Fishbones Convective resonant transport for k perp ρ<<1 (Ions “see” constant phase) Energetic-particle GAM Nonlinear sub-harmonic resonances at large amplitude (k perp ρ<<1) Drift Waves Orbit-averaging for k perp ρ>>1 Alfvén Eigenmodes Non-resonant losses for k perp ρ~1 Alfvén Eigenmodes “Stiff” transport for many small- amplitude modes with k perp ρ~1

47. Many small amplitude Alfven eigenmodes flatten the fast-ion profile Radial  T e 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 p   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 White, Plasma Phys. Cont. Fusion 52 (2010) 045012 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

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...

51. q RSAEs are typically weak or not observed during discharges with only off- axis beams Consistent with weaker fast ion gradient near q min On-Axis InjectionOff-Axis Injection ECE RSAEs Representative Profiles Use off-axis beams to alter the spatial gradient that drives Alfvén eigenmodes Heidbrink, Nucl. Fusion (2013) submitted

52. Different combinations of on- & off-axis beams at ~ constant power Amplitudes summed for channels near q min Time-averaged mode amplitude depends on fast-ion gradient Near qmin Stability trends consistently observed Heidbrink, Nucl. Fusion (2013) submitted Classical prediction

53. FIDA diagnostic measures profiles Profiles differ later in discharge (when AEs are weak) Strong fast-ion transport by AE instabilities makes profiles similar for all cases Suggests a “ critical- gradient ” model can describe transport in this regime Actual profiles are nearly identical for all beam combinations! Heidbrink, Nucl. Fusion (2013) submitted

54. *K. Ghantous et al, Phys. Plasmas 19 (2012) 092511 Infinitely “stiff” transport Ion redistribution expands unstable region A simple critical-gradient model explains some features of fast-ion transport in this regime Initial profile Relaxed profile Linear threshold Initial gradient

55. A simple critical-gradient model explains some features of fast-ion transport in this regime Application of this model to these plasmas gives qualitative agreement with experiment  can use linear physics to predict profiles in ITER Heidbrink, Nucl. Fusion (2013) submitted

56. Conclusions: The importance of phase Fishbones Convective resonant transport because ions “see” constant phase (k perp ρ<<1) Energetic-particle GAM Large amplitude modifies the phase and produces fractional resonances Drift Waves Phase-averaging reduces transport when k perp ρ>>1 Prompt Alfvén Eigenmode Losses Non-resonant particles are pushed across loss boundaries for the proper phase Nonlinear perturbations to the phase produce sum & difference frequencies in the loss spectrum Alfvén Eigenmodes Many resonances scramble phases, producing a diffusive regime with stiff fast-ion transport

57. is conserved for modes with ω<<Ω i and δB/B<<1 Resonant transport is more important than non-resonant transport The resonance condition is ω=nω pre +(m+l)ω bounce with [n,m,l] integers To predict the alpha transport in ITER you must be able to predict the amplitude of Alfvén eigenmodes Four “ truths ” that aren ’ t quite true (there is a O(10%) correction for k perp ρ~1) (not near a loss boundary!) (fractional resonances for large amplitude) (not if the transport is stiff)

58. Backup Slides

59. Cartoon of a field line that scatters the pitch angle In a static magnetic field, energy is conserved  a change in μ is a pitch-angle scattering event Simple Cartesian model for the gyro- averaged change in parallel energy in an Alfvén wave: To get an effect, the field must have k perp ρ~1 be asymmetric relative to the gyro-orbit the v x δb y term is non- zero for this field line

60. Cross-field correlation function for I sat 400eV 600eV 800eV 1000eV Broadband Drift Waves Induced at the Obstacle Edge S. Zhou, PoP 17 (2010) 092103

61. Resonance condition, Ω np = n ω  + p ω θ – ω = 0 n=4, p = 1 n=6, p = 2 n=3, p = 1 n=5, p = 2 n=6, p = 3 n=7, p = 3 Prompt losses E [MeV] 4.5 5.05.56.06.57.0 7.5 Calculated resonances with observed TAEs during RF ion heating in JET 0 -50 -100 -200 -150 -250 Log (  f E /Ω np ) -5 -6 -7 -8 -9 -10 P   ci [MeV] Draw curves in phase space to see resonances Pinches, Nucl. Fusion 46 (2006) S904