Cheryl M. Lam Advisor: Mark A. Cappelli Stanford Plasma Physics Laboratory

1. AXIAL-AZIMUTHAL HYBRID FLUID-PIC SIMULATIONS OF COHERENT FLUCTUATION-DRIVEN ELECTRON TRANSPORT IN A HALL THRUSTER Cheryl M. Lam Advisor: Mark A. Cappelli Stanford Plasma Physics Laboratory Mechanical Engineering Department Dissertation Proposal Meeting December 20, 2013

2. Dissertation Outline • Introduction • Hall Thruster Simulations (Background) • Model Description: Hybrid Fluid-PIC z-θ Model • Model Sensitivities • Simulation Results • Discussion • Conclusions and Future Work

3. Hall Thruster • Electric space propulsion device • Demonstrated high thrust efficiencies • Up to 60% (depending on operating power) • Deployed production technology • Design Improvements • Better physics understanding • Basic Premise: Accelerate heavy (positive) ions through electric potential to create thrust • E x B  azimuthal Hall current • Radial B field (r) • Axial E field (z) • Ionization zone (high electron density region) • Electrons “trapped” • Neutral propellant (e.g., Xe) ionized via collisions with electrons  Plasma • Ions accelerated across imposed axial potential (Ez / Φz) & ejected from thruster

4. Motivation • Hall thruster anomalous electron transport • Super-classical electron mobility observed in experiments1 • Theory: Correlated (azimuthal) fluctuations in ne and uez induce super-classical electron transport • 2D r-z models use tuned mobility to account for azimuthal effects2,3 • 3D model is computationally expensive • First fully-resolved 2D z-θ simulations of entire thruster ** Initial development by E. Fernandez • Predict azimuthal (ExB) fluctuations • Quantify impact on electron transport Channel Diameter = 9 cm Channel Length = 8 cm 1Meezan, N. B., Hargus, W.A., Jr., and Cappelli, M. A., Physical Review, Vol. 63, No. 2, 026410, 2001. 2Fife, J. M., Ph.D. Dissertation, Massachusetts Inst. of Technology, Cambridge, MA, 1999. 3Fernandez et al, “2D simulations of Hall thrusters,” CTR Annual Research Briefs, Stanford Univ.,1998.

5. Hall Thruster Simulations • 2D radial-axial (r-z) simulations • J. M. Fife, 1998 Ph.D. Dissertation – hybrid fluid-PIC • E. Fernandez, M. K. Scharfe, 2009 Ph.D. Dissertation – use experimental/semi-empirical mobility to account for azimuthal effects • E. Cha – ongoing – alternate propellants, entropy closure model • 2D axial-azimuthal (z-θ) • A. K. Knoll, 2010 Ph.D. Dissertation – fluid continuum, predicts high frequency fluctuations ~1-40 MHz, run length ~10 μs • L. Garrigues et al., IEPC 2013, PIC, partial azimuth, grid/timestep scaling, run length ~40 μs • C. M. Lam – ongoing – hybrid fluid-PIC, run length ~200 μs • 3D • F. Taccogna et al, IEPC 2013, PIC-MCC, geometric scaling and partial azimuth to reduce computational cost, run length ~5μs • K. Matyash, R. Schneider, S. Mazzoufre, Y. Raitses et al, IEPC 2013, PIC-MCC, partial azimuth, run length ~25 μs

6. Relevance of Hybrid z-θ Simulations • Thruster geometry • Full-size thruster: Dthruster ≈ 9 cm, Lthruster = 8 cm • Axial extent: entire thruster (anode to exit plane) plus near-field plume • Resolve full azimuth • No artificial introduction of periodicity • No geometric (or grid/timestep) scaling • Time scales of interest • Hybrid approach enables longer (~100s μs) simulations • Enables study of low- to mid-frequency waves (~10 kHz – 100 MHz)

7. Channel Diameter = 9 cm Channel Length = 8 cm Anode Cathode Anode Exit Plane extends 4 cm past channel exit Geometry • 2D in z-θ • No radial dynamics • E x B  + θ • Br: purely radial (measured from SHT laboratory discharge) • Imposed operating voltage (based on operating condition) G G SAMPLE GRID: z: 40 points, non-uniform θ: 50 points, uniform

8. Hybrid Fluid-PIC Model • Ions: Particle-In-Cell approach (super-particles) • Non-magnetized • No particle-particle collisions; Wall collisions modeled in some cases • Neutrals: Particle-In-Cell approach (super-particles) • Injected at anode per mass flow rate • No particle-particle collisions; Wall collisions modeled in some cases • Ionized per local ionization rate • Electron impaction ionization rate based on fits to experimentally-measured collision cross-sections, assuming Maxwellian distribution for electrons • Electrons: Fluid continuum • Continuity (species & current) • Momentum • Drift-diffusion equation • Inertial terms neglected • Energy (1D in z) • Convective & diffusive fluxes • Joule heating, Ionization losses, Effective wall loss Quasineutrality: ni = ne

9. Interpolation: Particle  Grid rNW rNE FNW FNE rSE FSE rSW FSW Interpolation: Grid  Particle PIC Ions & Neutrals • Particle-In-Cell (PIC) Approach • Particles: arbitrary positions • Force  Particle acceleration Interpolate: Grid  Particle • Plasma properties evaluated at grid points (Coupled to electron fluid solution) • Interpolate: Particle  Grid • Bilinear Interpolation • Ions subject to electric field: ≈ 0 neglect • Discrete particles result in “noisy” plasma properties at grid

10. Neutral Injection & Particle Collisions • Particle collisions with thruster walls – included for some simulations • Neutral particles reflected upon collision with anode or inner/outer radial channel walls • Ions recombine (with donor electron) to form neutral upon collision with inner/outer radial channel walls • Particles still otherwise collisionless, i.e., we do not model particle-particle collisions • Neutral injection: • Injection velocity sampled from half-Maxwellian distribution • No wall collisions: mean speed based on centerline (channel radial midpoint) velocity from r-z simulations • With wall collisions: Tanode ~ 1000K

11. Electron Fluid Equations • Species Continuity • Current Continuity 0 ni = ne

12. Electron Fluid Equations Classical Mobility • Momentum: Drift-Diffusion • Neglect inertial terms Classical Diffusion

13. Electron Fluid Equations Momentum: Drift-Diffusion Neglect inertial terms Classical Mobility Classical Diffusion E x B classical diamagnetic classical E x B diamagnetic θ fluctuations/dynamics

14. Electron Fluid Equations Combine current continuity and electron momentum to get convection-diffusion equation for Φ: where (φ is electric potential)

15. Electron Fluid Equations Energy (Temperature) Equation 1D in z (average over θ, then time advance 1D equation in z) neglect shape factor variation in z, other simplifying assumptions where with ionization cost factor αi = 1 (simplest model)

16. Ionization Loss Simplest model: Use constant ionization cost factor Dugan model: temperature-dependent ionization cost factor αi = 1 – 2.5 (up to ~5)

17. LEAP FROG  Time Advance Particle Positions & Velocities Neutrals & Ions (subject to F=qE) EGRID  EPART Ionize Neutrals Inject Neutrals Calculate Plasma Properties ni-PART, vi-PART, nn-PART, vn-PART  ni-GRID, vi-GRID, nn-GRID, vn-GRID QUASINEUTRALITY: ne = ni = nplamsa  Spline  RK4 Time Advance Te=Te(ne, ve) DIRECT SOLVE  2nd-order F-D Iterative Solve Φ Calculate Φ=Φ(ne, vi-GRID) ↔ EGRID Calculate ve=ve(Φ, ne, Te) r < ε0 CONVERGED r = Φ – Φlast-iteration Calculate vi-GRID-TEST= vi-GRID(EGRID) Solution Algorithm Boundary Conditions: • Dirichlet in z (Φ,Te) • Periodic in θ

18. Numerical Solution • Single numerical grid used for PIC and fluid solution • Cubic spline applied to PIC-derived grid properties (prior to use in fluid equations) • Electron energy (Te) equation • Central difference scheme for spatial derivatives • Calculate RHS for 2D grid, then average over θ to obtain 1D Te(z) • Time advance via 4th-order Runge-Kutta • Electric potential (Φ) equation • 2nd-order finite difference w/ upwind • Direct solve: block tridiagonal solver • Single timestep used for PIC and fluid (typically dt = 1 ns)

19. Model Sensitivities • Grid spacing • Current non-conservation • Predicted waves (azimuthal modes) • Effect of spline / PIC “noise” – required number of particles? • Initial Conditions • Boundary Conditions • Numerical stability/sensitivity of energy (Te) equation • Ionization cost factor • Constant factor • Dugan model • Energy loss to wall

20. 40 points non-uniform in z 50 points uniform in θ 61 points uniform in z 25 points uniform in θ Sample Numerical Grid ~400,000 (initial) particles per species dt = 1ns 6 days to run ~200 μs (single 64-bit Xeon x5355 2.66GHz processor core)

21. Sample ResultsIEPC 2009

22. Plasma Density Electron Temperature Axial Ion Velocity Potential Time-Averaged Plasma Properties100 V Simulation – IEPC 2009

23. E x B Axial Electron Velocity Distinct wave behavior observed: • Throughout channel (upstream) • Tilted: - z, + E x B • Lower frequency, slow moving, longer wavelength • Near exit plane • Peak Br, High shear (∂ueθ/∂z) • Tilted: + z, - ExB • Higher frequency, faster moving, shorter wavelength • Outside exit plane (downstream) • Purely axial: + z • Same structure (in θ) as exit plane waves

24. E x B Anode Cathode E x B E x B Fluctuations in θ f = 40 KHz λθ = 5 cm vph = 4000 m/s f = 700 KHz λθ = 4 cm vph = 40,000 m/s

25. Electron Transport • Simulation predicts super-classical electron mobility Axial Electron Mobility:

26. Stability Challenges • 100V simulation (2013) • Ionization cost factor = 2.1 • Wall collisions modeled • ICs: smooth neutral and ion density profiles, experimental Te(z) • Strong instability develops after ~100 μs (dt = 1 ns)

27. Stability Challenges • 100V simulation (2013) • Ionization cost factor = 2.1 • Wall collisions modeled • ICs: smooth neutral and ion density profiles, experimental Te(z) • Strong instability develops after ~100 μs (dt = 1 ns)

28. Runaway Ionization 100 V (2013)

29. Current Non-Conservation100V simulation (2013)

30. ξ t Theory:Azimuthal Fluctuations induce Axial Transport Eθ= E0cos(ωt) ne = n0cos(ωt + ξ) Consider Induced Current Induced current depends on phase shift ξ

31. Electron Fluid Equations Momentum: Drift-Diffusion Neglect inertial terms Correlated azimuthal fluctuations induce axial transport: Classical Mobility Classical Diffusion E x B classical diamagnetic classical E x B diamagnetic Previous models under-predict Jez=qneuez θ fluctuations/dynamics

32. Correlated ne and uez fluctuations generate axial electron current Uncorrelated Correlated fluctuations generate axial current 100 V (2013)

33. Characterize fluctuations Compare to experimental data Consider including dispersion analysis/maps Compare to theory or linearized dispersion relations? Role of fluctuations in enhanced (anomalous) electron transport Effect of shear, gradients, etc. on anomalous transport Effect of operating conditions Anomalous Electron Transport 100 V (2009)

34. Sample Experiments

35. Take-Aways • Simulations predict fluctuations • Complementary to other simulation efforts • Similar to those observed in experiment? • Consistent with theory? • Anomalous electron transport • Role of fluctuations • Effect of Hall thruster geometry, operating conditions, etc. • Suggestions for future work • Finite volume (in process) • Fully kinetic simulations

36. Recent Progress & Challenges • Addition of particle collisions with thruster walls • Finer axial (z) grid resolution near anode • Stability challenges • Sensitivity to Initial Conditions and Boundary Conditions • Strong fluctuation in Te and Φ • Current conservation • Finite Difference – present model • Finite Volume – parallel effort (E. Fernandez)

37. Additional Simulations – 100V • Establish stable long-running simulation (~600 μs – 1 ms) for low voltage (100V) case • Start (continue) from IEPC 2009 simulation (run length = 200 μs) • Ionization cost factor = 1 • No wall collisions; Slow neutral injection velocity • Zero-slope BC for Te • Increase number of particles (ionizspc) to enable longer simulation • Grid refinement study • Finer grid in z: current non-conservation, wave structure near anode • Finer, varied grid in θ: impact periodicity, azimuthal wavelength/modes • Initial Conditions • Increased neutral density  resulting spoke at anode? • Shape of neutral density profile (flat vs. sloped, magnitude/gradient) • More realistic (experiment-like) plasma density profile and/or magnitude

38. Additional Simulations • Higher voltage • Incrementally increase operating voltage • Look for trends (in frequency/wavelength/direction of fluctuations, electron transport, anomalous contribution to transport) • Initial Conditions  Waves • Smooth initial profiles (based on prescribed profile or experiment)  allow fluctuations to evolve • Consider “seeding” simulation with particular waves (spatial modes) to study wave growth/dissipation and energy coupling (e.g., into other modes)

39. Model Improvements • Te stability and BC/IC impacts • Stability and sensitivity analysis – contribution of source/sink terms, esp. wall loss and ionization cost • Enforcement of experimental profile (as IC and/or prescribed profile at regular time interval) and/or experimental-based limits • Prescribed (fixed value) vs. zero-slope condition at axial domain boundaries • Ionization cost factor • Dugan model – exponential vs. algebraic form • Tuned constant factor? • Improved/tunable wall loss model • Introduction of diffusive damping term? • Effect of spline smoothing • Implicit solve • 2D Te equation • Improve stability – consider more global changes to model • “External” power supply circuit model (potential BC) • Hyperviscous damping (for potential equation)

40. Model Improvements • Incremental changes to PIC model – additional physics • Introduction of wall collisions (w/ higher neutral injection velocity) • Revisit ionization rate implementation • Enhance electron transport via prescribed electron mobility – sustain/generate waves (may also improve stability) • Additive “baseline” μ┴ or νen • Experimental mobility μexp(z) (in lieu of or in addition to μ┴) • Experimental or additional (or Bohm-like) mobility for electron fluid equations only

41. Publications • Conference Papers • C. M. Lam, A. K. Knoll, E. Fernandez, and M. A. Cappelli, “Two-Dimensional (z-θ) Simulations of Hall Thruster Anomalous Transport,” International Electric Propulsion Conference, 2009. • C. M. Lam, E. Fernandez, and M. A. Cappelli, “Two-Dimensional (z-θ) Hybrid Fluid-PIC Simulation of Enhanced Cross-field Electron Transport in an Annular E x B Discharge,” Gaseous Electronics Conference, 2012. • C. M. Lam, E. Fernandez, and M. A. Cappelli, “Two-Dimensional Simulations of Coherent Fluctuation-Driven Transport in a Hall Thruster,” International Electric Propulsion Conference, 2013. • Journal Papers • C. M. Lam, E. Fernadez, and M. A. Cappelli, “A Two-Dimensional Hybrid Hall Thruster Simulation that resolves the E × B Electron Drift Direction,” IEEE Transactions on Plasma Science, Special Edition – submitted for review, expect publication Dec 2014 • Planned additional publications • Journal paper: waves and transport

42. Research Progress Timeline

43. Questions?