IPAM 31 March 2009
Multiscale Methods for Coulomb Collisions in Plasmas
Russel CaflischIPAM
Mathematics Department, UCLA
IPAM 31 March 2009
Collaborators
• UCLARichard Wang Yanghong Huang
• Livermore Labs Andris DimitsBruce Cohen
• U Ferrarra Lorenzo PareschiGiacomo 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
IPAM 31 March 2009
Scrape-offlayer
Kinetic Effects
Edge pedestal temperature profile near the edge of an H-mode discharge in the DIII-D tokamak. [Porter2000]. Pedestal is shaded region.
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 boundary layer very important & uncertain
R (cm)
Tem
p. (
eV)
0
500
1000
From G. W. Hammett, review talk 2007 APS Div Plasmas Physics Annual Meeting, Orlando, Nov. 12-16.
IPAM 31 March 2009
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
with (linearized) solution
• Short range interactions– r < λD
– Coulomb interactions– Fokker-Planck equation
20( )( ( ) )Bq k T
e iq x n e n
2 2 10( )D e Bn q k T
1 1(4 ) Drr e
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
21( ) ( ) ( ) : ( ) ( )
2col d
ff f
t
F v v D v v
v v v
1 1
( ')( ) 2 '
| ' |d
H fc c d
vF v v
v v v v2 2
2 2( ) ( ') | ' | 'G
c c f d
D v v v v vv v v v
IPAM 31 March 2009
Derivation of Fokker-Planck Eqtn
• Binary Coulomb collision– particles with unit charge q, reduced mass μ
– relative velocity v0 , displacement b before collision
– deflection angle θ– scattering cross section (Rutherford)
2
20
tan( / 2)q
v b
22
2 20
( )2 sin ( / 2)
q
v
b
θ
v0
IPAM 31 March 2009
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
– Coulomb logarithm
– leads to Fokker-Planck eqtn: for small change in velocity
max
min
0
2 2
4
( ) ( ) ( ) ( )
ln
F d
cv
max minln ln( / )b b
( ) ( ) ( )vf v v f v v f v
IPAM 31 March 2009
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’
• 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 space21
( ) ( ) ( ) : ( ) ( )2col d
ff f
t
F v v D v v
v v v
1 1 1 1 1
( )( ) ( ') ( ') ( ) ( ) ( )col
f vf v f v f v f v v v v v d dv
t
IPAM 31 March 2009
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!
IPAM 31 March 2009
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
dd dt d v F σ b
IPAM 31 March 2009
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).
tan( 2)
2 2 2 2 2 30( log 8 )Le e n m u t
IPAM 31 March 2009
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
2
2
sin ( / 2 (1 ) / 2
/ 2
sN e
s N
-- simulation - theory
33 (ln )s c u t
1coth sA A e
cos( )4 sin
AAf d e d
hA
IPAM 31 March 2009
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).
IPAM 31 March 2009
Convergence Study of T&A vs. Nanbu
• Stochastic error– Variance σ2
– σ ≈ O(N-1/2)
– Independent of time step dt
– Same for T&A and Nanbu
T&A
Nanbu
IPAM 31 March 2009
Convergence Study of T&A vs. Nanbu
• 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?
T&A
Nanbu
IPAM 31 March 2009
Accelerated Simulation Methodsfor 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)
• 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)
( )F v m g
IPAM 31 March 2009
Variable thermalization across phase space
• Bump-on-tail instability– Persistent because
Coulomb cross section decreases as v increases
IPAM 31 March 2009
Thermalization/Dethermalization Method
• Hybrid representation (as in RGD)
• 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?
( )F v m g
IPAM 31 March 2009
Hybrid collision algorithm
• Hybrid representation (as in RGD)
– g represented by particles
• Collisions– 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
( )F v m g
1
( ( ))n
kk
g v v t
IPAM 31 March 2009
Choice of Probabilities pd and pt
• T/D step– Fn = F(n dt) = mn + gn
– One step
• Detailed balance requirement (?)
– Assuming uM = um = 0• Simple choice
– pt = 1 for v < v1 (i.e., complete thermalization)– pd = 1 for v > v2 (i.e., complete dethermalization)
1 0 0
1 0 0
(1 )
(1 )d t
d t
m p m p g
g p m p g
0 1
2
(1 )
( / )
(1 / )
(1 / ) exp( / )
d t
d t
d t
d t
F M m g F M m g
g p m p g
g p p m
M p p m
p p c v
IPAM 31 March 2009
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
IPAM 31 March 2009
Hybrid Method for Bump-on-Tail
IPAM 31 March 2009
Variation of Hybrid Parameters
IPAM 31 March 2009
Efficiency vs. Accuracy for Hybrid Method
IPAM 31 March 2009
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 ( )
IPAM 31 March 2009
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)
IPAM 31 March 2009
Generalization of Hybrid MethodHybrid representation using two temperatures
– Tparallel, Ttransverse
– anisotropic Maxwellian
– temperature evolution follows Trubnikov solution for relaxation of anisotropy– application to bump-on-tail below
• temperature evolution (left), velocity distribution (right)
3/2 2 1/2 2 2 2( ) (2 ) ( ) exp( ( ) / / )t p x y t z pf v T T v v T v T
IPAM 31 March 2009
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