Multiscale Methods for Coulomb Collisions in Plasmas

1. Multiscale Methods for Coulomb Collisions in Plasmas Russel Caflisch IPAM Mathematics Department, UCLA IPAM 31 March 2009

2. Collaborators • UCLA Richard Wang Yanghong Huang • Livermore Labs Andris Dimits Bruce Cohen • U Ferrarra Lorenzo Pareschi Giacomo Dimarco IPAM 31 March 2009

3. 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

4. Coulomb Collisions in Plasma • Collisions between charged particles • Significant in magnetically confined fusion plasmas • edge plasma IPAM 31 March 2009

5. Edge boundary layer very important & uncertain Temp. (eV) 1000 500 0 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 pedestaltemperature 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. 12-16. IPAM 31 March 2009

6. 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 IPAM 31 March 2009

7. 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 IPAM 31 March 2009

8. 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) θ v0 b IPAM 31 March 2009

9. 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 IPAM 31 March 2009

10. 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 space IPAM 31 March 2009

11. 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

12. 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 IPAM 31 March 2009

13. 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). IPAM 31 March 2009

14. 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 IPAM 31 March 2009

15. 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

16. Convergence Study of T&A vs. Nanbu Nanbu • Stochastic error • Variance σ2 • σ≈ O(N-1/2) • Independent of time step dt • Same for T&A and Nanbu T&A IPAM 31 March 2009

17. Convergence Study of T&A vs. Nanbu 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 IPAM 31 March 2009

18. 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) IPAM 31 March 2009

19. Variable thermalization across phase space • Bump-on-tail instability • Persistent because Coulomb cross section decreases as v increases IPAM 31 March 2009

20. 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? IPAM 31 March 2009

21. 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 IPAM 31 March 2009

22. Choice of Probabilities pdand 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) IPAM 31 March 2009

23. 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

24. Hybrid Method for Bump-on-Tail IPAM 31 March 2009

25. Variation of Hybrid Parameters IPAM 31 March 2009

26. Efficiency vs. Accuracy for Hybrid Method IPAM 31 March 2009

27. 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

28. 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

29. Generalization of Hybrid Method Hybrid 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) IPAM 31 March 2009

30. 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 IPAM 31 March 2009