MORE ON 1D AZIMUTHAL AND 2D R-THETA - SIMULATIONS Ken Hara 1 , Ian - - PowerPoint PPT Presentation

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Ken Hara1, Ian DesJardin2, Rob Martin3

1 Texas A&M University; 2 University at Buffalo (NSF-REU summer student at TAMU); 3 Air Force Research Laboratory, Edwards AFB

MORE ON 1D AZIMUTHAL AND 2D R-THETA SIMULATIONS

ExB Workshop November 1, 2018 Princeton, NJ

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• 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.)

PIC simulation of electron cyclotron drift Instability (ECDI)

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.

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• A few numerical setups have been suggested.
• Periodic axial (unbounded) case [Janhunen et al. PoP

2018]

• 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?

Numerical setup of the azimuthal PIC simulations

UNBOUNDED BOUNDED

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• Single particle theory in the presence of Ey

fluctuation: Ey = E0 cos (ky) + constant Ez and Bx

• Equations of motion
• 𝑒𝑨

𝑒𝑢 = 𝑤𝑨 ; 𝑒𝑤𝑨 𝑒𝑢 = 𝑟 𝑛 𝐹𝑨 − 𝑤𝑧𝐶𝑦

• 𝑒𝑧

𝑒𝑢 = 𝑤𝑧 ; 𝑒𝑤𝑧 𝑒𝑢 = 𝑟 𝑛 𝐹𝑧 𝑧 + 𝑤𝑨𝐶𝑦

• Observations
• Chaotic trajectory in phase space ->

electron heating

• Guiding center motion (constant

drift)? Shift in guiding center because

• f heating?

The effect of azimuthal plasma wave to the cross-field electron transport

Y position, mm Z position, mm

20 40 60 80 100 120

• 4
• 3
• 2
• 1

1 Ey = 0 Ey fluctuation

Ey, V/m

• 10000
• 5000

5000 10000

Y velocity, 10

6 m/s

Z velocity, 10

6 m/s

• 2

2 4 6

• 4
• 2

2 4

Z velocity, 10

6 m/s

• 4
• 2

2 4

Ey fluctuation Ey = 0

Uy=Ez/Bx, Uz=0

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Benchmark against long-domain modulational instability due to ECDI [Janhunen et. al PoP 2018]

[Janhunen et al. PoP 2018] Our simulation results using MPI-PIC simulation

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• Lafleur reviewed the plasma-wave induced electron transport theory.
• In the limit of collisionless (𝜉𝑛 → 0 hence Ω → ∞), the effective cross-field mobility is given

by…

• Using the Poisson equation, 𝜗0

𝑒𝐹𝑧 𝑒𝑧 = 𝑓 𝑜𝑗 − 𝑜𝑓 , for 1D azimuthal PIC simulation

Verification test for the wave-induced electron transport

𝑅 = 1 𝑈 න 𝑒𝑢 1 𝑀𝑧 න𝑅𝑒𝑧

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• Theory shows that 𝑜𝑓𝐹𝑧 = 𝑜𝑗𝐹𝑧 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.

1. 𝑜𝑗 𝑧 = 𝑑𝑝𝑜𝑡𝑢. = 𝑜0 [smooth ion density]

• 𝑜𝑗𝐹𝑧 = 𝑜0 𝐹𝑧 = 0
• Hence, 𝝂⊥,𝒇𝒈𝒈 = 𝟏

2. 𝑜𝑗 𝑧 ≠ 𝑑𝑝𝑜𝑡𝑢. is initialized by macroparticles with randomized position [noisy ion density]

Before looking into ECDI, what is the effect of ion density fluctuations (physical/numerical) to the electron transport?

Hara, K. (unpublished).

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• Electron energy equation (collisionless):

𝜖 𝜖𝑢 𝑜𝑓𝜗𝑓 + ∇ ⋅ 𝑜𝑓𝜗𝑓𝒗𝑓 + 𝑞𝑓𝑊𝒗𝑓 = 𝑜𝑓𝜈⊥𝐹𝑨

2 − 𝑜𝑓𝑣𝑓𝜄𝐹𝜄

• For unbounded,
• 𝝂⊥ = 𝟏 (smooth ni): Te = constant
• 𝝂⊥ > 𝟏 (noisy ni): Te linearly increases
• For bounded,
• 𝝂⊥ > 𝟏 (T

eV saturates, because convective heat

flux balance with cross-field transport)

Ion modulation itself + “bounded” BCs can excite electron transport without ECDI

Hara, K. (unpublished).

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Mobility Estimates: Ions Frozen at Point

Equilibrium Rapidly Reestablished Even from Cold Electron Restarts!

• Ions were Frozen and Electron Dynamics

were Restarted

• Many Reduced HET Models Assume:

Scale Separation te vs. ti

• Still anomalous electron transport is
• bserved.
• Also <niEy> = <neEy> is shown
• Can we use this in a multifluid

approach (coupling electron PIC with a multifluid ion/electron solver)

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0.5 1 1.5 2 2.5 3 3.5 4 100 101 102 103

Np = 100 Np = 200 Np = 300 Np = 500 Np = 600 Np = 700 Np = 800 Np = 900 Np = 1000 Np = 2500 Np = 5000 Np = 7500 Np = 10000 Np = 20000 Np = 30000 Np = 40000 Np = 50000

𝜖 𝜖𝑢 න𝑜𝑓𝜗𝑓𝑒𝑊 = න𝑜𝑓𝑣𝑨𝐹𝑨𝑒𝑊 − න𝑜𝑓𝑣𝑓𝜄𝐹𝜄 𝑒𝑊

ECDI simulations with small domain (Ly=5 mm) and unbounded axial BC: Np (# or particles/cell) = 100 - 50,000

Growth rate decreases as Np increases (smaller numerical noise due to ion density) Nonlinear saturation at Te = 40 eV as Np increases (due to ion trapping)

1 2 3 4 5

• 20
• 15
• 10
• 5

5

Cf) typically in one cell, 108-1010 real particles

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0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 100 101 102

Np = 100 Np = 200 Np = 500 Np = 1000 Np = 10000 Np = 30000

• 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 𝑂𝑞 → ∞.

Increasing macroparticle count reduces numerical noise

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100 101 102 103

Np = 100 Np = 200 Np = 500 Np = 1000 Np = 10000 Np = 30000

Initial phase (t < 200 ns) Transition to nonlinear saturation (200 ns - 900 ns)

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• 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?

Energy growth: ions total energy, electron total energy, and maximum electric field (potential energy)

0.5 1 1.5 2 2.5 3 3.5 4 68.8 69 69.2 69.4 69.6 69.8 70 70.2 70.4 70.6

Np = 100 Np = 1000 Np = 10000 Np = 50000

0.5 1 1.5 2 2.5 3 3.5 4 103 104 105 106

Np = 100 Np = 1000 Np = 10000

0.5 1 1.5 2 2.5 3 3.5 4 100 101 102 103

Np = 100 Np = 1000 Np = 10000 Np = 50000

Ion total energy Electron total energy Electric field

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A measure for benchmarking codes: numerical convergence

0.5 1 1.5 2 2.5 3 3.5 4 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6

Np = 100 Np = 1000 Np = 10000 Np = 50000

• 200
• 100

100 200

• 1150
• 1100
• 1050
• 1000
• 950
• 900
• 850
• 800

Np = 100 Np = 500

• Macroscopic quantities can be used for benchmarking purposes of numerical codes and

understanding numerical convergence?

• A phase space of spatially-averaged uex(t) and uey(t) is constructed from the nonlinear saturation

region.

Axial velocity Azimuthal velocity Time Hara, K. (unpublished).

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Statistical error vs Np: convergence? (1)

• PDF of uex(t) vs. uey(t) is constructed; and

investigate the variance and covariance.

• Standard deviation: 𝜏

𝑘 = ± 𝑊𝑏𝑠 𝑤𝑘

• A general trend of 𝑂𝑞

−1/2 convergence is shown.

• Questions:
• Where can we claim that numerical

convergence is achieved?

• It seems like the results are not fully

converged (e.g., hp convergence & round-

• ff error, in CFD)
• What is the measure to use for

convergence? (e.g., variance, covariance)

102 103 104 105 103 104 105

ux,e variance uy,e variance

𝑊𝑏𝑠 ∝ 𝑂𝑞

−1/2

𝑊𝑏𝑠 ∝ 𝑂𝑞

−1

𝜏 = ±100 m/s 𝜏 = ±32 m/s 𝜏 = ±320 m/s Hara, K. (unpublished).

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Statistical error vs Np: convergence? (2)

102 103 104 105 10-4 10-3 10-2 10-1 100 101 102

|𝐷𝑝𝑤| ∝ 𝑂𝑞

−1

|𝐷𝑝𝑤| ∝ 𝑂𝑞

−2

102 103 104 105 10-6 10-5 10-4 10-3 10-2 10-1 100

R =

𝑑𝑝𝑤 𝑌,𝑍 𝜏𝑌𝜏𝑍

Hara, K. (unpublished). Approaching zero “correlation” between uez and uey is good? Correlation Covariance |𝑆| ∝ 𝑂𝑞

−1

|𝑆| ∝ 𝑂𝑞

−2

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• Multidimensional effects (wave structures in sheath / ion front bowing out)
• Note f = 0 V (ref. potential) is assumed at northeast corner of the domain.

2D simulations for cosine ion density profile (1/2)

Potential (f) Ion azimuthal mean velocity (Uyi) Movie

ExB

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• Coupling between “azimuthal” plasma wave and “radial” wall sheath.
• Radial electron flux is no longer locally zero, leading to radial Joule heating/cooling.
• A 3D PIC code by F. Taccogna shows similar results.

2D simulations for cosine ion density profile (2/2)

Electron radial flux (neUxe)

Movie

ExB Plasma density

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• Electrostatic current-driven instabilities using

noiseless grid-based direct-kinetic (DK) simulation for the Vlasov Poisson system. [Tuesday poster in GEC]

• Generation of high-energy ion population

(backward propagating) is observed at 𝑁0 =

𝑣𝑓 𝑤𝑢ℎ,𝑓 ≥ 1.3

• High-energy ions are 3-5 orders of

magnitude smaller than bulk = at least 𝑂𝑞 ≥ 105 is needed if PIC is used.

• We can evaluate sputtering rates etc from

the set of ion VDFs obtained.

• A grid-based kinetic simulation of ECDI

can be helpful for benchmarking?

Current-driven ion acoustic instability

• K. Hara and K. M. Hanquist, PSST 27, 065004 (2018)
• K. Hara and C. Treece, (unpublished)

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• Comparing <ne> with Boltzmann ne
• Simple Experimental Evidence?

Has vq-tail been observed in LIF?

• Realistic Collision Models

DSMC for Neutral/Ionize Charge-Ex ES-PIC and 1D2V Coulomb Collisions Tested on ES-Shock vs. Fluid Models

• Multiscale Methods → Full 3D?

Adapt Quasi-1D Hybrid Codes for Bursts of Electron Kinetics to Estimate m⊥?

• V&V for Dynamical Systems

Future (and Current) Work

0D Equilibration via Coulomb Collisions Nanbu vs. Rosenbluth Vr Vx