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Multiscale Methods for Coulomb Collisions in Plasmas

Multiscale Methods for Coulomb Collisions in Plasmas. Russel Caflisch IPAM Mathematics Department, UCLA. Collaborators. UCLA Richard Wang Yanghong Huang Livermore Labs Andris Dimits Bruce Cohen U Ferrarra Lorenzo Pareschi Giacomo Dimarco. Outline. Coulomb collision in plasmas

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Multiscale Methods for Coulomb Collisions in Plasmas

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  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

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