The Vlasov equation: Theory and Simulation Francesco Valentini Dipartimento di Fisica and CNISM, Università della Calabria, Italy [email protected] EU School on Space Weather fundamental plasma processes, Spineto 4-9 June 2012 OUTLINEOUTLINE FIRSTFIRST PARTPART -The-The physicsphysics ofof thethe interplanetaryinterplanetary medium:medium: thethe solarsolar windwind plasmaplasma -- HowHow toto modelmodel aa plasma?plasma? aa statisticalstatistical approachapproach -- TheThe kinetickinetic dynamicsdynamics ofof aa collision-freecollision-free plasma:plasma: thethe VlasovVlasov equationequation -- TheThe physicsphysics ofof wave-particlewave-particle resonantresonant interactioninteraction (linear(linear andand nonlinearnonlinear regime)regime) -- PlasmaPlasma wavewave echoesechoes SECONDSECOND PARTPART -- TheThe numericalnumerical approachapproach forfor thethe solutionsolution ofof thethe VlasovVlasov equationequation -- FewFew wordswords aboutabout finitefinite differencedifference schemesschemes -- EulerianEulerian algorithmsalgorithms forfor VlasovVlasov simulationssimulations -- DistributionDistribution ofof particleparticle velocitiesvelocities inin spacespace plasmasplasmas -- TheThe tailtail atat shortshort spatialspatial scalesscales ofof turbulenceturbulence inin thethe solarsolar wind:wind: HybridHybrid Vlasov-MaxwellVlasov-Maxwell simulationssimulations -- StateState ofof thethe artart andand workwork inin progressprogress EverywhereEverywhere inin thethe Universe,Universe, fromfrom near-Earthnear-Earth environments,environments, likelike forfor exampleexample planetaryplanetary magnetospheresmagnetospheres andand solarsolar atmosphere,atmosphere, toto furtherfurther systems,systems, likelike radioradio galaxiesgalaxies andand supernovae,supernovae, thethe dynamicsdynamics isis drivendriven byby plasmaplasma processesprocesses A plasma is a collection of discrete charged particles globally neutral, behaving as a collective system, dominated by long range (electromagnetic) interactions Plasma Physics 1. Astrophysics; space plasmas 2. Laboratory plasmas (fusion, ...) 3. Laser plasmas interaction Plasma as an ideal, classical and collisionless gas Plasma parameter Low density and high temperatures Ideal gas with rare collisions The mean free path of a particle in the solar wind is about 1AU Debye length and Debye potential Collective behavior of a plasma Collective behavior of a plasma − + + + + + + + + e A simple view: background of fixed ions + + + + + + + + + + + + + + + + + + + + + + + + − + + + + + + + + e + + + + + + + + + + + + + + + + + + + + + + + + x e− + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + A statisticalA statistical description approach of a plasma Many instead of one particle in phase space (x,v): f x,v,tdxdv Average number of particles in dxdv f = f x,v,t Distribution function Macroscopic variables of a plasmaa nα r , t =∫ f α r , v , t dv The Vlasov Equation In the absence of collisions the Vlasov equations describes the evolution of the distribution function under the effects of self-consistent and external electromagnetic fields when calculated on the characteristics Properties of the Vlasov equation This is consistent with the fact that the Vlasov equation neglects the process (binary collisions) which causes statystical systems to increase their entropy and evolve toward a Maxwell-Boltzmann distribution Vlasov-Maxwell equations (mean-field theory) NO COLLISIONS!!! Vlasov equation coupled to Maxwell equations is a nonlinear problem, whose analytical solution is availaible only in few simplified cases Linear regime of wave-particle interaction Electron distribution function; fixed ions with constant density Vlasov-Poisson (1D-1V) Equilibrium configuration Small amplitude perturbations A simple case of wave damping Electric sinusoidal perturbation (in the lab frame) In the wave frame One can calculate the wave damping coefficient by setting the rate of increase of kinetic energy of resonant particles equal to the rate of decrease of wave energy The solution for the perturbed distribution function A simple case of wave damping Electric sinusoidal perturbation (in the lab frame) In the wave frame One can calculate the wave damping coefficient by setting the rate of increase of kinetic energy of resonant particles equal to the rate of decrease of wave energy A simple case of wave damping Electric sinusoidal perturbation (in the lab frame) In the wave frame One can calculate the wave damping coefficient by setting the rate of increase of kinetic energy of resonant particles equal to the rate of decrease of wave energy The Landau damping rate What happens to the wave energy? However, in absence of collisions, the memory of the Suppose you have a initial perturbation cannot be lost, but it remains perturbation of the form locked in the distribution function, as a perturbation in the form The energy of this waves can be dissipated through Landau damping Ballistic contribution If a second wave is launched into the plasma after the first wave has been Landau damped A third wave called plasma echo will apperar after a certain delay time Plasma wave echoes Plasma wave echoes Nonlinear regime of wave-particle interaction Nonlinear effects: Particle trapping L. Landau (1946) linearized the VP equations in the perturbation amplitudes O’Neil (1965) showed that nonlinear effects are no longer negligible for times comparable to For an electron moving at velocity close to the wave phase speed: Nonlinear effects get important when resonant particles are trapped in the wave potential well Phase space particle trajectories Trapping process Linear approximation no longer valid in the resonant region Different numerical approaches Eulerian algorithms for Vlasov simulations The Vlasov equation is discretized on a grid in phase space and the field equations on a spatial grid. Eulerian methods Numerical integration of the Vlasov equation in phase space Electrostatic case. fixed ions with constant density. 1D-1V phase space (x,v) for the sake of simplicity, but the generalization to multi-dimensional geometry is straightforward Advection equation in ∂ f ∂ f ∂ f phase space e v e a e =0 ∂ t ∂ x ∂ v eE f = f x,v,t ;a=− VLASOV-POISSON SYSTEM e e m ∞ ∂ E =4πe [ ni−∫ f e x,v,t dv ] ∂ x −∞ Phase space and time discretization v k1 Δv xi=i−1Δx v k=kΔv k i=1,N k=−N ,N x Δx v v L V Δx= x k−1 Δv= max N x N v i−1 i i1 x Time discretization: t n=nΔt n=0,nstep Δx Δv Δt≤min , CFL stability condition {v max amax } Splitting method for the Vlasov equation Distribution function in physical space ∂ f ∂ f x v x =0 Advection in physical space (describes translation ∂ t ∂ x in physical space) ∂ f ∂ f v a v =0 Advection in velocity space (describes ∂ t ∂ v translation in velocity space) Distribution function in The time evolution of the distribution velocity space function is divided into two steps: 1) EVOLUTION in physical space 2) EVOLUTION in velocity space The time splitting algorithm ∂ f x ∂ f x v =0⇒ f xtΔt = Λx Δt f x t ∂ t ∂ x Translation ∂ f ∂ f operators v a v =0 ⇒ f tΔt = Λ Δt f t ∂ t ∂ v v v v Finite differences for the translation operators Centred difference The time splitting algorithm v nΔt Δt Calculate E Calculate E Calculate E x The time splitting algorithm v nΔt Δt Calculate E Calculate E Calculate E x Chen and Knorr, JCP (1977) Mangeney et al. JCP (2002) Vlasov simulations of Landau damping Initial electric perturbation Electric field spectral component E x,t=0 = E0 sin kx 2π E 0=0.02;k= =0 .5 L x Periodic boundary conditions in the physical domain Exponential damping in time 2 2 2 2 ω R=ω p 13k λD Comparison with linear theory of Landau damping Plasma wave echoes (Vlasov simulation) −1 k echo=k 2−k 1=0.4λ D k =0.8λ−1 −1 2 D k1=0.4λD Nonlinear saturation of Landau damping O’Neil found that, due to nonlinear effects, the Landau damping rate of the wave starts oscillating around zero on the trapping time scale with decreasing amplitude. SATURATION OF DAMPING Landau linear prediction E x,t=0= E0 sin kx Vlasov simulations 2π E 0=0.2;k= =0.5 L x Nonlinear saturation of Landau damping E x,t=0= E0 sin kx Phase space distribution function Vlasov simulations 2π E 0=0.2;k= =0.5 L x Signature of trapping Nonlinear saturation of Landau damping E x,t=0= E0 sin kx Phase space distribution function Vlasov simulations 2π E 0=0.2;k= =0.5 L x Signature of trapping Be careful with numerical Vlasov simulations Galeotti and Califano, PRL (2005) = 3 N3 (t) ∫f (x, v,t)dxdv S(t) = −∫f ln(f)dxdv Trapping region Filamentation problem ≈ Lv Δv When the typical size of the velocity structures gets smaller than the numerical resolution… The numerical algorithm is not able to resolve them anymore…The topology of phase space is changed Small-scale structures are generated in the phase space distribution function due to trapping NUMERICAL VISCOSITY Energy conservation Valentini et al. PRE (2005) Valentini et al. JCP (2005) Velocity distributions in space plasmas The velocity distributions in space plasmas are source of free energy for many instabilities and kinetic processes Solar wind protons Solar wind electrons Generation of suprathermal tails Generation of accelerated beams Turbulence and kinetic effects in space plasmas Hydro range Kinetic range Energy cascades towards small scales the understanding of the physical processes that govern the system dynamics at frequency of the order and Turbulent cascade beyond the ion-cyclotron Dissipation (?) frequency is of key relevance in space physics FLUID