ExB Workshop November 1, 2018 Princeton, NJ MORE ON 1D AZIMUTHAL AND 2D R-THETA ‘ - SIMULATIONS Ken Hara 1 , Ian DesJardin 2 , Rob Martin 3 1 Texas A&M University; 2 University at Buffalo (NSF-REU summer student at TAMU); 3 Air Force Research Laboratory, Edwards AFB 1
PIC simulation of electron cyclotron drift Instability (ECDI) • Kinetic simulations, e.g., particle-in-cell (PIC) simulations, are useful in understanding the electron transport due to electron cyclotron drift instability (ECDI). • Collisionless scattering of electrons from azimuthal plasma waves • Good numerical test cases to benchmark codes (LANDMARK) ‘ - • Such kinetic simulations can serve as a validation tool with advanced experimental measurements (e.g., laser diagnostics) • We are interested in the numerical effects of the simulations proposed by others (Lafleur, Janhunen, Boeuf, etc.) Lafleur, T., et al. “Theory for the Anomalous Electron Transport in Hall Effect Thrusters. I. Insights from Particle-in- Cell Simulations.” Physics of Plasmas, vol. 23, no. 5, 2016, p. 053502., doi:10.1063/1.4948495. 2
Numerical setup of the azimuthal PIC simulations • A few numerical setups have been suggested. • Periodic axial ( unbounded ) case [Janhunen et al. PoP 2018] UNBOUNDED BOUNDED • Artificial axial ( bounded ) case: particles are randomized (effectively adding collisionality) in position & velocity [Lafleur et al. PoP 2016] • Realistic 2D (z-theta) case ‘ - • 2D (r-theta) case • 3D (r-z-theta) case • Two mechanisms need to be investigated. 1. The source of azimuthal plasma wave (ECDI): Studied. 2. The effect of azimuthal plasma wave to the cross-field electron transport : a bit more to do? 3
The effect of azimuthal plasma wave to the cross-field electron transport • Single particle theory in the presence of Ey Uy=Ez/Bx, Uz=0 fluctuation: E y = E 0 cos (ky) + constant Ez and Bx 4 Ey = 0 10000 6 m/s 5000 • Equations of motion 2 Ey, V/m Z velocity, 10 0 𝑒𝑨 𝑒𝑤 𝑨 𝑟 0 • 𝑒𝑢 = 𝑤 𝑨 ; 𝑒𝑢 = 𝑛 𝐹 𝑨 − 𝑤 𝑧 𝐶 𝑦 -5000 -2 -10000 𝑒𝑤 𝑧 𝑒𝑧 𝑟 ‘ - • 𝑒𝑢 = 𝑤 𝑧 ; 𝑒𝑢 = 𝑛 𝐹 𝑧 𝑧 + 𝑤 𝑨 𝐶 𝑦 -4 1 • Observations Z position, mm 0 4 Ey fluctuation 6 m/s • Chaotic trajectory in phase space -> 2 -1 Z velocity, 10 electron heating 0 -2 • Guiding center motion (constant -2 -3 Ey = 0 drift)? Shift in guiding center because Ey fluctuation -4 -4 0 20 40 60 80 100 120 of heating? -2 0 2 4 6 Y position, mm 6 m/s Y velocity, 10 4
Benchmark against long-domain modulational instability due to ECDI [Janhunen et. al PoP 2018] ‘ - Our simulation results using MPI-PIC simulation [Janhunen et al. PoP 2018] 5
Verification test for the wave-induced electron transport • Lafleur reviewed the plasma-wave induced electron transport theory. • In the limit of collisionless ( 𝜉 𝑛 → 0 hence Ω → ∞ ), the effective cross-field mobility is given by… 𝑅 = 1 𝑈 න 𝑒𝑢 1 න𝑅𝑒𝑧 ‘ - 𝑀 𝑧 𝑒𝐹 𝑧 • Using the Poisson equation, 𝜗 0 𝑒𝑧 = 𝑓 𝑜 𝑗 − 𝑜 𝑓 , for 1D azimuthal PIC simulation 6
Before looking into ECDI, what is the effect of ion density fluctuations (physical/numerical) to the electron transport? • Theory shows that 𝑜 𝑓 𝐹 𝑧 = 𝑜 𝑗 𝐹 𝑧 for a collisionless case. Hence, 𝜈 ⊥,𝑓𝑔𝑔 is dependent on the ion density modulation . • We propose a simple verification test case. • Turn off the ion dynamics (frozen ions) • Investigate effects of ion density modulation to the ‘ - electron transport • Two cases are compared. 𝑜 𝑗 𝑧 = 𝑑𝑝𝑜𝑡𝑢. = 𝑜 0 [smooth ion density] 1. • 𝑜 𝑗 𝐹 𝑧 = 𝑜 0 𝐹 𝑧 = 0 • Hence, 𝝂 ⊥,𝒇𝒈𝒈 = 𝟏 2. 𝑜 𝑗 𝑧 ≠ 𝑑𝑝𝑜𝑡𝑢. is initialized by macroparticles with randomized position [noisy ion density] Hara, K. (unpublished). 7
Ion modulation itself + “bounded” BCs can excite electron transport without ECDI • Electron energy equation (collisionless): 𝜖 2 − 𝑜 𝑓 𝑣 𝑓𝜄 𝐹 𝜄 𝜖𝑢 𝑜 𝑓 𝜗 𝑓 + ∇ ⋅ 𝑜 𝑓 𝜗 𝑓 𝒗 𝑓 + 𝑞 𝑓𝑊 𝒗 𝑓 = 𝑜 𝑓 𝜈 ⊥ 𝐹 𝑨 • For unbounded , ‘ - • 𝝂 ⊥ = 𝟏 (smooth n i ): Te = constant • 𝝂 ⊥ > 𝟏 (noisy n i ): Te linearly increases • For bounded , • 𝝂 ⊥ > 𝟏 (T eV saturates, because convective heat flux balance with cross-field transport) Hara, K. (unpublished). 8
Mobility Estimates: Ions Frozen at Point • Ions were Frozen and Electron Dynamics were Restarted • Many Reduced HET Models Assume: Scale Separation t e vs. t i ‘ - • Still anomalous electron transport is observed. • Also <niEy> = <neEy> is shown Equilibrium Rapidly Reestablished • Can we use this in a multifluid Even from Cold Electron Restarts! approach (coupling electron PIC with a multifluid ion/electron solver) 9
ECDI simulations with small domain (Ly=5 mm) and unbounded axial BC: Np (# or particles/cell) = 100 - 50,000 𝜖 5 𝜖𝑢 න𝑜 𝑓 𝜗 𝑓 𝑒𝑊 = න𝑜 𝑓 𝑣 𝑨 𝐹 𝑨 𝑒𝑊 − න𝑜 𝑓 𝑣 𝑓𝜄 𝐹 𝜄 𝑒𝑊 0 10 3 -5 -10 -15 ‘ - 10 2 -20 0 1 2 3 4 5 N p = 100 N p = 200 N p = 300 N p = 500 Nonlinear saturation at Te = 40 eV as N p = 600 N p = 700 Np increases (due to ion trapping) N p = 800 10 1 N p = 900 N p = 1000 N p = 2500 N p = 5000 N p = 7500 Growth rate decreases as Np increases N p = 10000 N p = 20000 (smaller numerical noise due to ion density) N p = 30000 N p = 40000 N p = 50000 10 0 0 0.5 1 1.5 2 2.5 3 3.5 4 Cf) typically in one cell, 10 8 -10 10 real particles 10
Increasing macroparticle count reduces numerical noise • The larger Np, the slower instability growth starts, and the smaller growth rate. • This is consistent with the “noisy ion density” case with the verification test case, ultimately approaching convergence (?) for 𝑂 𝑞 → ∞ . 10 2 10 3 Initial phase (t < 200 ns) Transition to nonlinear N p = 100 N p = 200 saturation (200 ns - 900 ns) ‘ - N p = 500 N p = 1000 10 2 N p = 10000 N p = 30000 10 1 N p = 100 N p = 200 10 1 N p = 500 N p = 1000 N p = 10000 N p = 30000 10 0 10 0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11
Energy growth: ions total energy, electron total energy, and maximum electric field (potential energy) • Numerical heating is large for small Np. Numerical heating is still present at larger Np. • Increase in energy is consistently present even at nonlinear saturation => Numerical heating? Electric field 10 6 Ion total energy Electron total energy 10 3 N p = 100 N p = 1000 70.6 N p = 100 N p = 10000 70.4 N p = 1000 ‘ - N p = 10000 70.2 10 5 N p = 50000 10 2 70 69.8 69.6 69.4 10 4 10 1 N p = 100 69.2 N p = 1000 69 N p = 10000 N p = 50000 68.8 10 0 10 3 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 12
A measure for benchmarking codes: numerical convergence • Macroscopic quantities can be used for benchmarking purposes of numerical codes and understanding numerical convergence? • A phase space of spatially-averaged u ex (t) and u ey (t) is constructed from the nonlinear saturation region. -800 N p = 100 N p = 100 3.6 N p = 500 -850 N p = 1000 ‘ - N p = 10000 3.4 Azimuthal velocity N p = 50000 -900 3.2 3 -950 2.8 -1000 2.6 -1050 2.4 -1100 2.2 0 0.5 1 1.5 2 2.5 3 3.5 4 -1150 -200 -100 0 100 200 Time Axial velocity Hara, K. (unpublished). 13
Statistical error vs Np: convergence? (1) • PDF of u ex (t) vs. u ey (t) is constructed; and investigate the variance and covariance. 10 5 𝜏 = ±320 m/s u x,e variance • Standard deviation: 𝜏 𝑘 = ± 𝑊𝑏𝑠 𝑤 𝑘 u y,e variance −1/2 convergence is shown. • A general trend of 𝑂 𝑞 −1/2 𝑊𝑏𝑠 ∝ 𝑂 𝑞 • Questions : ‘ - • 𝜏 = ±100 m/s Where can we claim that numerical 10 4 convergence is achieved? • It seems like the results are not fully converged (e.g., hp convergence & round- off error, in CFD) • What is the measure to use for −1 𝑊𝑏𝑠 ∝ 𝑂 𝑞 convergence? (e.g., variance, covariance) 𝜏 = ±32 m/s 10 3 10 2 10 3 10 4 10 5 Hara, K. (unpublished). 14
Statistical error vs Np: convergence? (2) 𝑑𝑝𝑤 𝑌,𝑍 R = 𝜏 𝑌 𝜏 𝑍 10 2 10 0 Covariance Correlation 10 1 10 -1 −1 |𝐷𝑝𝑤| ∝ 𝑂 𝑞 −1 |𝑆| ∝ 𝑂 𝑞 10 0 10 -2 ‘ - 10 -1 10 -3 Approaching zero “correlation” 10 -2 10 -4 between u ez and u ey is good? 10 -3 10 -5 −2 |𝑆| ∝ 𝑂 𝑞 −2 | 𝐷𝑝𝑤| ∝ 𝑂 𝑞 10 -6 10 -4 10 2 10 3 10 4 10 5 10 2 10 3 10 4 10 5 15 Hara, K. (unpublished).
2D simulations for cosine ion density profile (1/2) • Multidimensional effects (wave structures in sheath / ion front bowing out) • Note f = 0 V (ref. potential) is assumed at northeast corner of the domain. ExB ‘ - Potential ( f ) Ion azimuthal mean velocity (U yi ) Movie 16
2D simulations for cosine ion density profile (2/2) • Coupling between “azimuthal” plasma wave and “radial” wall sheath. • ExB Radial electron flux is no longer locally zero, leading to radial Joule heating/cooling. • A 3D PIC code by F. Taccogna shows similar results. ‘ - Plasma density Electron radial flux (n e U xe ) Movie 17
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