Kinetic Theory for Gases and Plasmas Lecture 2

Kinetic Theory for Gases and Plasmas: Lecture 2: Plasma Kinetics Russel Caflisch IPAM Mathematics Department, UCLA IPAM Plasma Tutorials 2012

Review of First Lecture • • Velocity distribution function Molecular chaos Boltzmann equation H-theorem (entropy) Maxwellian equilibrium Fluid dynamic limit DSMC – DSMC becomes computationally intractable near fluid regime, since collision time-scale becomes small IPAM Plasma Tutorials 2012

Temp. (e. V) Where are collisions signifiant in plasmas? Example: Tokamak edge boundary layer 1000 500 0 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) From G. W. Hammett, review talk 2007 APS Div Plasmas Physics Annual Meeting, Orlando, Nov. 12 -16. IPAM Plasma Tutorials 2012 R (cm) Edge pedestal temperature profile near the edge of an H-mode discharge in the DIII-D tokamak. [Porter 2000]. Pedestal is shaded region.

Basics of mathematical (classical) plasma physics IPAM Plasma Tutorials 2012

Gas or Plasma Flow: Kinetic vs. Fluid Kinetic description Fluid (continuum) description • Discrete particles • Density, velocity, temperature • Motion by particle velocity • Interact through collisions • Evolution following fluid eqtns (Euler or Navier-Stokes or MHD) When does continuum description fail? IPAM Plasma Tutorials 2012

Debye Length • Charged particles rather than neutrals Electrons: e. FACM 2010

Debye Length • Quasi-neutrality: nearly equal number of oppositely charged particles Electrons: e. FACM 2010 Ions: H+

Debye Length • Pick out a distinguished particle Electrons: e. FACM 2010 Ions: H+

Debye Length • Debye length = range of influence, e. g. , for single electron λD Electrons: e. FACM 2010 Ions: H+

Debye Length • In neighborhood of an electron there is deficit of other electrons, suplus of positive ions Electrons: e. FACM 2010 Ions: H+

Debye Length • Replace positive charged particles by continuum, for simplicity Electrons: e; test particle smoothed. FACM 2010 ; Ions:

Debye Length: Derivation • Distribution of electrons and ions – – charge q; temperature T; dielectric coeff ε 0; potential φ, energy is -q φ electrons in Gibbs distribution (in space) Uniform ions distribution • Poisson equation (linearized) – Single electron at 0 • Solution With length scale λD = Debye length: FACM 2010

Interactions of Charged Particles in a Plasma • Plasma parameter g = (n λD 3)-1 – Plasma approximation g<<1 – Many particles in a Debye sphere – Otherwise, the system is an N-body problem • Long range interactions – r > λD (λD = Debye length) – Individual particle interactions are not significant – Interaction mediated by electric and magnetic fields • Short range interactions – r < λD – Coulomb interactions FACM 2010

Levels of Description • Magneto-hydrodynamic (MHD) equations – Equilibrium – Continuum • Vlasov-Maxwell equations – Nonequilibrium – No collisions • Landau-Fokker-Planck equations – Nonequilibrium – Collisions IPAM Plasma Tutorials 2012

MHD Equations Fluid equations with Lorenz force Ohm’s law Maxwell’s equations IPAM Plasma Tutorials 2012

Plasma kinetics IPAM Plasma Tutorials 2012

Vlasov Equations • Velocity distribution function – for each species – Convection – Lorentz force – Collisionless m=mass, q=charge IPAM Plasma Tutorials 2012

Landau Fokker Planck Equation • Velocity distribution function – for each species – Convection – Electromotive force – Collisions IPAM Plasma Tutorials 2012 m=mass, q=charge

Coulomb Collisions • Collision of 2 charged particles (i=1, 2) with – Position xi, mass mi, charge qi has solution in which – (r, θ) are polar coordinates for x 1 -x 2 – v 0 is incoming relative velocity, – b is impact parameter IPAM Plasma Tutorials 2012

Derivation of Fokker-Planck Eqtn • Binary Coulomb collision – (with m 1=m 2, q 1=q 2) – relative velocity v 0 , displacement b before collision – deflection angle θ – scattering cross section (Rutherford) θ b v 0 IPAM 31 March 2009

Landau-Fokker-Planck equation for collisions • Coulomb interactions – collision rate ≈ u-3 for two particles with relative velocity u • Fokker-Planck equation IPAM Plasma Tutorials 2012

Derivation of Fokker-Planck Eqtn • Coulomb collisions are predominantly grazing – Differential collision rate is singular at θ≈0 since – Total collision rate • Aggregate effect of the collisions – measured by the momentum transfer, is – integrand is only marginally singular IPAM 31 March 2009

Derivation of Fokker-Planck Eqtn • Debye cutoff – Screened potential is – Approximate the effect of screening by cutoff in angle • Cross section for momentum transfer is • Corresponding Boltzmann collision operator Qλhas collision rate IPAM 31 March 2009

Derivation of Fokker-Planck Eqtn • Analysis of Alexandre & Villani – “On the Landau approximation in plasma physics” Ann. I. H. Poincaré – AN 21 (2004) 61– 95. – Boltzmann eqtn without Lorentz force – Rescale (x, t) → (c/log Λ) (x’, t’) and drop ’, – to obtain – As Λ→∞, • total angular cross section for momentum transfer goes to c|v-w|-3 • all collisions become grazing collisions • the scaled Boltzmann collision operator converges to the Landau. Fokker –Planck collision operator IPAM 31 March 2009

Derivation of Fokker-Planck Eqtn • Scaling of Alexandre & Villani – They find that the relevant time scale T and space scale X are is – On this time and space scale, they prove that solution of the Boltzmann equation (without Lorentz force) converges to a solution of the LFP equation IPAM 31 March 2009

Derivation of Fokker-Planck Eqtn • Scaling difficulty – Alexandre and Villani are unable to find a scaling such that both the LFP collision operator and the Lorentz force terms are significant – On a scale for which the Lorentz force is O(1), the collision term is insignificant IPAM 31 March 2009

Collisions in Gases vs. Plasmas • Collisions between velocities v and v* • Gas collisions – hard spheres, σ = cross section area of sphere – collision rate is σ | v - v* | – any two velocities can collide → smoothing in v • Plasma (Coulomb) collisions – – – very long range, potential O(1/r) collisions are grazing, localized as in Landau eqtn Collision rate | v - v* |-3 small for well separated velocities differential eqtn in v, as well as x, t waves in phase space Landau damping (interaction between waves and particles) IPAM Plasma Tutorials 2012

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 = (sin θ)-4 – FP “collisions” are each aggregation of many small deflections – described as drift and diffusion in velocity space IPAM 31 March 2009

Simulation methods IPAM Plasma Tutorials 2012

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

Binary Collision Methods for LFP • Bobylev-Nanbu (PRE 2000) – Implicit-like transformation of LFP over a single time step – Expansion of scattering operator in spherical harmonics – Approximation at O(Δt) with tractable binary collision interpretation – Resulting binary collisions • Every particle collides once per time step • Collisions depend on Δt IPAM Plasma Tutorials 2012

Bobylev-Nanbu Analysis • Boltzmann eqtn, as scattering operator in which IPAM Plasma Tutorials 2012

Implicit-like transformation • First order approximation • “Implicit” approximation • Optimal choice of ε • Result – Every particle collides once in every time step IPAM Plasma Tutorials 2012

Transformation to Tractable Binary Form • Implicit-like formulation – with (for Landau-Fokker-Planck) – D is an expansion in Legendre polynomials in |u| u·n – D can be greatly simplified by approximation at O(Δt) IPAM Plasma Tutorials 2012

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(dt 1/2). 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 -- simulation - theory – 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 IPAM 31 March 2009

Simulation for Plasmas: Test Cases • • • Relaxation of an anisotropic Maxwellian Bump-on-tail Sheath Two stream instability Computations using Nanbu’s method and hybrid method IPAM Plasma Tutorials 2012

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

Hybrid Method for Bump-on-Tail FACM 2010

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) FACM 2010

Conclusions and Prospects • Landau Fokker Planck collision operator • Infinite rate of grazing interactions → finite rate of aggregate collisions • Monte Carlo simulation methods for kinetics have trouble in the fluid and near-fluid regime • Math leading to new methods that are robust in fluid limit IPAM Plasma Tutorials 2012