1. Kinetic Theory for Gases and Plasmas: Lecture 2: Plasma Kinetics
Russel Caflisch
IPAM
Mathematics Department, UCLA
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2. Review of First Lecture
Velocity distribution function
Molecular chaos
Boltzmann equation
H-theorem (entropy)
Maxwellian equilibrium
Fluid dynamic limit
DSMC
DSMC becomes computationally intractable near fluid regime, since collision time-scale becomes small
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3. Where are collisions signifiant in plasmas
Where are collisions signifiant in plasmas? Example: Tokamak edge boundary layer
Temp. (eV)
1000
500
R (cm)
Schematic views of divertor tokamak and edge-plasma region (magnetic separatrix is the red line and the black boundaries indicate the shape of magnetic flux surfaces)
Edge pedestal temperature profile near the edge of an H-mode discharge in the DIII-D tokamak. [Porter2000]. Pedestal is shaded region.
From G. W. Hammett, review talk 2007
APS Div Plasmas Physics
Annual Meeting, Orlando, Nov
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4. Basics of mathematical (classical) plasma physics
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5. Gas or Plasma Flow: Kinetic vs. Fluid
Kinetic description
Discrete particles
Motion by particle velocity
Interact through collisions
Fluid (continuum) description
Density, velocity, temperature
Evolution following fluid eqtns (Euler or Navier-Stokes or MHD)
When does continuum description fail?
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6. Debye Length Charged particles rather than neutrals Electrons: e-
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7. Debye Length
Quasi-neutrality: nearly equal number of oppositely charged particles
Electrons: e Ions: H+
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8. Debye Length Pick out a distinguished particle Electrons: e- Ions: H+
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9. Debye Length
Debye length = range of influence, e.g., for single electron λD
Electrons: e Ions: H+
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10. Debye Length
In neighborhood of an electron there is deficit of other electrons, suplus of positive ions
Electrons: e Ions: H+
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11. Debye Length
Replace positive charged particles by continuum, for simplicity
Electrons: e ; test particle ; Ions: smoothed
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12. Debye Length: Derivation
Distribution of electrons and ions
charge q; temperature T; dielectric coeff ε0;
potential φ, energy is -q φ
electrons in Gibbs distribution (in space)
Uniform ions distribution
Poisson equation (linearized)
Single electron at 0
Solution
With length scale λD = Debye length:
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13. Interactions of Charged Particles in a Plasma
Plasma parameter g = (n λD3)-1
Plasma approximation g<<1
Many particles in a Debye sphere
Otherwise, the system is an N-body problem
Long range interactions
r > λD (λD = Debye length)
Individual particle interactions are not significant
Interaction mediated by electric and magnetic fields
Short range interactions
r < λD
Coulomb interactions
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14. Levels of Description Magneto-hydrodynamic (MHD) equations
Equilibrium
Continuum
Vlasov-Maxwell equations
Nonequilibrium
No collisions
Landau-Fokker-Planck equations
Collisions
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15. MHD Equations Fluid equations with Lorenz force Ohm’s law
Maxwell’s equations
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16. Plasma kinetics
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17. Velocity distribution function
Vlasov Equations
Velocity distribution function
for each species
Convection
Lorentz force
Collisionless
m=mass, q=charge
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18. Landau Fokker Planck Equation
Velocity distribution function
for each species
Convection
Electromotive force
Collisions
m=mass, q=charge
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19. Coulomb Collisions Collision of 2 charged particles (i=1,2) with
Position xi, mass mi, charge qi
has solution
in which
(r,θ) are polar coordinates for x1-x2
v0 is incoming relative velocity,
b is impact parameter
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20. Derivation of Fokker-Planck Eqtn
Binary Coulomb collision
(with m1=m2, q1=q2)
relative velocity v0 , displacement b before collision
deflection angle θ
scattering cross section (Rutherford) θ v0 b
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21. Landau-Fokker-Planck equation for collisions
Coulomb interactions
collision rate ≈ u-3 for two particles with relative velocity u
Fokker-Planck equation
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22. Derivation of Fokker-Planck Eqtn
Coulomb collisions are predominantly grazing
Differential collision rate is singular at θ≈0 since
Total collision rate
Aggregate effect of the collisions
measured by the momentum transfer, is
integrand is only marginally singular
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23. Derivation of Fokker-Planck Eqtn
Debye cutoff
Screened potential is
Approximate the effect of screening by cutoff in angle
Cross section for momentum transfer is
Corresponding Boltzmann collision operator Qλhas collision rate
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24. Derivation of Fokker-Planck Eqtn
Analysis of Alexandre & Villani
“On the Landau approximation in plasma physics” Ann. I. H. Poincaré – AN 21 (2004) 61–95.
Boltzmann eqtn without Lorentz force
Rescale (x,t) → (c/log Λ) (x’,t’) and drop ’,
to obtain
As Λ→∞,
total angular cross section for momentum transfer goes to c|v-w|-3
all collisions become grazing collisions
the scaled Boltzmann collision operator converges to the Landau-Fokker –Planck collision operator
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25. Derivation of Fokker-Planck Eqtn
Scaling of Alexandre & Villani
They find that the relevant time scale T and space scale X are is
On this time and space scale, they prove that
solution of the Boltzmann equation (without Lorentz force)
converges to a solution of the LFP equation
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26. Derivation of Fokker-Planck Eqtn
Scaling difficulty
Alexandre and Villani are unable to find a scaling such that both the LFP collision operator and the Lorentz force terms are significant
On a scale for which the Lorentz force is O(1), the collision term is insignificant
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27. Collisions in Gases vs. Plasmas
Collisions between velocities v and v*
Gas collisions
hard spheres, σ = cross section area of sphere
collision rate is σ | v - v* |
any two velocities can collide → smoothing in v
Plasma (Coulomb) collisions
very long range, potential O(1/r)
collisions are grazing, localized as in Landau eqtn
Collision rate | v - v* |-3 small for well separated velocities
differential eqtn in v, as well as x,t
waves in phase space
Landau damping (interaction between waves and particles)
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28. Comparison F-P to Boltzmann
collisions are single physical collisions
total collision rate for velocity v is
∫|v-v’| σ(|v-v’| ) f(v’) dv’ FP
actual collision rate is infinite due to long range interactions: σ = (sin θ)-4
FP “collisions” are each aggregation of many small deflections
described as drift and diffusion in velocity space
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29. Simulation methods
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30. Monte Carlo Particle Methods for Coulomb Interactions
Particle-field representation
Mannheimer, Lampe & Joyce, JCP 138 (1997)
Particles feel drag from Fd = -fd (v)v and diffusion of strength σ = σ(D)
numerical solution of SDE, with Milstein correction
Lemons et al., J Comp Phys 2008
Particle-particle representation
Takizuka & Abe, JCP 25 (1977), Nanbu. Phys. Rev. E. 55 (1997) Bobylev & Nanbu Phys. Rev. E. 61 (2000)
Binary particle “collisions”, from collision integral interpretation of FP equation
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31. Binary Collision Methods for LFP
Bobylev-Nanbu (PRE 2000)
Implicit-like transformation of LFP over a single time step
Expansion of scattering operator in spherical harmonics
Approximation at O(Δt) with tractable binary collision interpretation
Resulting binary collisions
Every particle collides once per time step
Collisions depend on Δt
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32. Bobylev-Nanbu Analysis
Boltzmann eqtn, as scattering operator
in which
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33. Implicit-like transformation
First order approximation
“Implicit” approximation
Optimal choice of ε
Result
Every particle collides once in every time step
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34. Transformation to Tractable Binary Form
Implicit-like formulation
with (for Landau-Fokker-Planck)
D is an expansion in Legendre polynomials in |u| u·n
D can be greatly simplified by approximation at O(Δt)
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35. Takizuka & Abe Method T. Takizuka & H. Abe, J. Comp. Phys. 25 (1977).
T & A binary collision model is equivalent to the collision term in Landau-Fokker-Planck equation
The scattering angle θ is chosen randomly from a Gaussian random variable δ
δ has mean 0 and variance
Parameters
Log Λ = Coulomb logarithm
u = relative velocity
Simulation
Every particle collides once in each time interval
Scattering angle depends on dt
cf. DSMC for RGD: each particle has physical number of collisions
Implemented in ICEPIC by Birdsall, Cohen and Proccaccia
Numerical convergence analysis by Wang, REC, etal. (2007) O(dt1/2).
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36. Nanbu’s Method
Combine many small-angle collisions into one aggregate collision
K. Nanbu. Phys. Rev. E. 55 (1997)
Scattering in time step dt
χN = cumulative scattering angle after N collisions
N-independent scattering parameter s
Aggregation is only for collisions between two given particle velocities
Steps to compute cumulative scattering angle:
At the beginning of the time step, calculate s
Determine A from
Probability that postcollison relative velocity is scattered into dΩ is
Implemented in ICEPIC by Wang & REC
-- simulation
- theory
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37. Simulation for Plasmas: Test Cases
Relaxation of an anisotropic Maxwellian
Bump-on-tail
Sheath
Two stream instability
Computations using Nanbu’s method and hybrid method
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38. Numerical Test Case: Relaxation of Anisotropic Distribution
Specification
Initial distribution is Maxwellian with anisotropic temperature
Single collision type: electron-electron (e-e) or electron-ion (e-i).
Spatially homogeneous.
The figure at right shows the time relaxation of parallel and transverse temperatures.
All reported results are for e-e; similar results for e-i.
Approximate analytic solution of Trubnikov (1965).
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39. Hybrid Method for Bump-on-Tail
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40. Hybrid Method Using Fluid Solver
Improved method for spatial inhomogeneities
Combines fluid solver with hybrid method
previous results used Boltzmann type fluid solver
Euler equations with source and sink terms from therm/detherm
application to electron sheath (below)
potential (left), electric field (right)
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41. Conclusions and Prospects
Landau Fokker Planck collision operator
Infinite rate of grazing interactions
→ finite rate of aggregate collisions
Monte Carlo simulation methods for kinetics have trouble in the fluid and near-fluid regime
Math leading to new methods that are robust in fluid limit
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