ipam 31 march 2009 multiscale methods for coulomb collisions in plasmas russel caflisch ipam...

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

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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 ( )

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