Multiscale Methods for Coulomb
Collisions in Plasmas
Russel Caflisch
IPAM
Mathematics Department, UCLA
IPAM 31 March 2009
Collaborators
• UCLA
Richard Wang
Yanghong Huang
• Livermore Labs
Andris Dimits
Bruce Cohen
• U Ferrarra
Lorenzo Pareschi
Giacomo Dimarco
IPAM 31 March 2009
Outline
• Coulomb collision in plasmas
– Motivation
– binary collisions
– comparison to rarefied gas dynamics
• Fokker-Planck equation
– Derivation
– Simulation methods
– Numerical convergence study of Nanbu’s method
• Hybrid method for Coulomb collisions in plasmas
–
–
–
–
Combine collision method with MHD
Thermalization/dethermalization
Bump-on-tail example
Spatially dependent problems
• Conclusions
IPAM 31 March 2009
Coulomb Collisions in Plasma
• Collisions between charged particles
• Significant in magnetically confined fusion
plasmas
– edge plasma
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Temp. (eV)
Edge boundary layer very important & uncertain
Scrape-off
layer
1000
500
Kinetic
Effects
0
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)
From G. W. Hammett, review talk 2007
APS Div Plasmas Physics
Annual Meeting, Orlando, Nov. 12-16.
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R (cm)
Edge pedestal temperature profile near the
edge of an H-mode discharge in the DIII-D
tokamak. [Porter2000]. Pedestal is shaded
region.
Interactions of Charged Particles
in a Plasma
• Long range interactions
– r > λD
(λD = Debye length)
– Electric and magnetic fields (e.g. using PIC)
• Debye length = range of influence, e.g., for single electron
– charge q; electron, ion densities ne = ni; temperature T; dielectric coeff ε0;
– electrons in Gibbs distribution, ions uniform
– potential φ
 2  (q   0 )( ( x)  ne e q  kBT  ni )
with (linearized) solution
  (4 )1 r 1e r  
D2  ne q 2 ( 0 k BT ) 1
D
• Short range interactions
– r < λD
– Coulomb interactions
– Fokker-Planck equation
IPAM 31 March 2009
Interactions of Charged Particles
in a Plasma
• Short range interactions
– r < λD
– Coulomb interactions
• collision rate ≈ u-3 for two particles with relative velocity u
– Fokker-Planck equation
f

1 2
( )col  
Fd ( v) f ( v) 
: D( v) f ( v)
t
v
2 vv
Fd ( v)  c1
H

f ( v ')
 c1 2 
dv '
v
v | v  v ' |
 2G
2
D( v)  c2
 c2
 f ( v ') | v  v ' | dv '
vv
vv
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Derivation of Fokker-Planck Eqtn
• Binary Coulomb collision
–
–
–
–
particles with unit charge q, reduced mass μ
relative velocity v0 , displacement b before collision
q2
deflection angle θ
tan( / 2)  2
 v0 b
scattering cross section (Rutherford)


q2
 ( )  

2
2
2

v
sin
(

/
2)
0


2
θ
b
v0
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Derivation of Fokker-Planck Eqtn
• Multiple Coulomb collisions
– mean square deflection of charged particle
– F(∆θ)d (∆θ) = # collisions → angle change ∆θ
– traveling distance unit distance
( )
2

 max
 min
( )2 F ( )d ( )
 cv04 ln 
– Coulomb logarithm
ln   ln(bmax / bmin )
– leads to Fokker-Planck eqtn: for small change in velocity
f (v  v)  f (v)  v  v f (v)
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Comparison F-P to Boltzmann
• Boltzmann
– collisions are single physical collisions
– total collision rate for velocity v is
∫|v-v’| σ(|v-v’| ) f(v’) dv’
f (v)
(
)col     f (v1 ') f (v ')  f (v1 ) f (v) v  v1  (v  v1 )d dv1
t
• FP
– actual collision rate is infinite due to long range
interactions: σ = (|v-v’| )-4
– FP “collisions” are each aggregation of many small
deflections
– described as drift and diffusion in velocity space
f

1 2
( )col  
Fd ( v) f ( v) 
: D( v) f ( v)
t
v
2 vv
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Collisions in Gases vs. Plasmas
• Collisions between velocities v and v*
– u=| v - v* | relative velocity
– collision rate = u σ
• u has influence in two ways
– relative flux of particles =O(u)
– residence time T over which particles can interact =O(1/u)
• Gas collisions
–
–
–
–
hard spheres
(nearly) instantaneous, so that T is independent of u
total collision effect, e.g., scattering angle =O(u)
weak dependence on u
• Plasma (Coulomb) collisions
–
–
–
–
very long range, potential O(1/r)
residence time effect very strong
total collision effect, e.g., scattering angle =O(u-3)
strong dependence on u
• a source of multiscale behavior!
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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)
dv  Fd dt  σdb
– 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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Takizuka & Abe Method
• T. Takizuka & H. Abe, J. Comp. Phys. 25 (1977).
• T & A binary collision model is equivalent to the collision term in LandauFokker-Planck equation
– The scattering angle θ is chosen randomly from a Gaussian random variable δ
  tan( 2)
– δ has mean 0 and variance
– Parameters
2
 2  (e2 e2 nL log 8 02 m
u 3 )t
• 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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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
sin 2 (  N / 2  (1  e  s ) / 2
-- simulation
- theory
s  N 2 /2
– 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
s  c3 u 3 (ln )t
– Determine A from
coth A  A1  e  s
– Probability that postcollison relative velocity is scattered into dΩ is
A
f (  )d  
e A cos  d 
4 sin hA
– Implemented in ICEPIC by Wang & REC
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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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Convergence Study of T&A vs. Nanbu
• Stochastic error
–
–
–
–
σ2
Variance
σ ≈ O(N-1/2)
Independent of time step dt
Same for T&A and Nanbu
Nanbu
T&A
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Convergence Study of T&A vs. Nanbu
Nanbu
T&A
• Average error
– err(Nanbu) ≈ err(T&A)/2
– err ≈ O(dt1/2)
– consistent with error estimate of O(dt) by Bobylev & Nanbu Phys. Rev. E 2000?
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Accelerated Simulation Methods
for Coulomb collisions
• δf methods: f = M + δf
– simulate (small) correction to approximate result (Kotschenruether 1988)
– δf can be positive or negative
– Particle weights: “quiet” and partially linearized methods (Dimits & Lee
1993)
– Stability problems
• Hybrid method with thermalization/dethermalization
– Hybrid representation (as in RGD)
F (v )  m  g
• m = equilibrium component (Maxwellian)
• g = kinetic (nonequilibrium) component
– Thermalization rate must vary in phase space
• α = α(x,v) = fraction of particles in m
• (um, Tm) ≠ (uF, TF)
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Variable thermalization across
phase space
• Bump-on-tail
instability
– Persistent because
Coulomb cross section
decreases as v increases
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Thermalization/Dethermalization Method
• Hybrid representation (as in RGD)
F (v )  m  g
• Thermalization and dethermalization (T/D)
– Thermalize particle (velocity v) with probability pt
• Move from g to m
– Dethermalize particle (velocity v) with probability pd
• Move from m to g
– Derivation?
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Hybrid collision algorithm
• Hybrid representation (as in RGD)
F (v )  m  g
– g represented by particles n
g    (v  vk (t ))
• Collisions
k 1
– m-m: leaves m unchanged
– g-g: as in DSMC
– m-g: select particle from g, sample particle from m, then perform
collision
• T/D step
– Particle from g is thermalized (moved to m) with probability pt
– Particle sampled from m is dethermalized (moved to g) with
probability pd
• Change (ρm, um, Tm) to conserve mass, momentum, energy
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Choice of Probabilities pd and pt
• T/D step
– Fn = F(n dt) = mn + gn
– One step
m1  (1  pd )m0  pt g 0
g1  pd m0  (1  pt ) g 0
• Detailed balance requirement (?)
F0  M  m  g
 F1  M  m  g
 g  pd m  (1  pt ) g
 g  ( pd / pt )m
 M  (1  pd / pt )m
 (1  pd / pt )  c exp( v /  )
2
– Assuming uM = um = 0
• Simple choice
– pt = 1 for v < v1 (i.e., complete thermalization)
– pd = 1 for v > v2 (i.e., complete dethermalization)
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Application to Bump-on-Tail
Problems
• Bump-on-tail
– central Maxwellian m
– bump on tail of m
• Dynamics
– fast interactions
•
•
•
•
with small |v-v’|
m with m
bump with bump
describe with MHD
– slow interactions
• with large |v-v’|
• bump with m
• describe with particle
collisions
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Hybrid Method for Bump-on-Tail
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Variation of Hybrid Parameters
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Efficiency vs. Accuracy for
Hybrid Method
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Ion Acoustic Waves
– kinetic description
needed for ion
Landau damping and
ion-ion collisions
– wave oscillation and
decay shown at right
– agreement with
“exact” solution
from Nanbu
Nanbu ( ), hybrid ( ), older hybrid method ( )
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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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Generalization of Hybrid Method
Hybrid representation using two temperatures
– Tparallel, Ttransverse
– anisotropic Maxwellian
f (v)  (2 )3/2 (Tt 2Tp )1/2 exp((vx2  vy2 ) / Tt  vz2 / Tp )
– temperature evolution follows Trubnikov solution for relaxation of anisotropy
– application to bump-on-tail below
• temperature evolution (left), velocity distribution (right)
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Conclusions and Prospects
• Fokker-Planck equation for Coulomb collisions
– particle methods
• drift/diffusion method
• binary collision method
– acceleration methods
• δf
• hybrid method
• Hybrid method for Coulomb collisions
– Thermalization/dethermalization probabilities
– Probabilities vary in phase space (x,v)
– Applications
• Bump-on-tail
• Ion acoustic waves
• Ion sheath
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