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

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Variation of Hybrid Parameters

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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