The Vlasov equation: Theory and Simulation
Francesco Valentini
Dipartimento di Fisica and CNISM, Università della Calabria, Italy
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 REGIME ω << i Link beween low frequency (fluid) dynamics and high frequency i (kinetic) dynamics ω≥i KINETIC REGIME Frontier problem in plasma physics
Investigation of cross-scale interactions: from fluid to kinetic scales “A challenge in computational plasma physics”
OBSERVATION DATA: Magnetic & MHD and HMDH SIMULATIONS electric field, density, velocity, PDF, …
Kinetic effects in the energy transfer across ion cyclotron frequency
Kinetik Vlasov hybrid simulations Today (Giga-teraflop): Vlasov-hybrid
Future (tera-petaflop): full Vlasov! Hybrid Vlasov-Maxwell equations (dimensionless units)
Vlasov equation for ions
Ohm's law
Faraday equation
Current Density (low frequency approximation)
Quasi-neutrality
Equation of state
Valentini et al, JCP 2007 Numerical algorithm
Splitting scheme (Chen and Knorr, JCP, 1977), for the time advance of the Vlasov equation; third order Van Leer scheme for spatial and velocity derivatives For details, see Valentini et al. JPC, 2007; and Mangeney et al. JCP, 2002
Hybrid Current Advance Method for the evolution of the electromagnetic fields
The code is parallelized in Parallelization strategy physical space using the MPI directives.
For details, see Mangeney et al. JPC, 2002 Hybrid Vlasov-Maxwell simulations of 2D plasma turbulence prova
z B background magnetic field
Simulation of decay turbulence Servidio, Valentini, Califano and Veltri, PRL (2012) y Simulation plane x Hybrid Vlasov-Maxwell simulations of 2D plasma turbulence prova
The contour plot of the out of plane current density shows the presence of magnetic reconnection events (black crosses)
Local temperature anisotropy Hybrid Vlasov-Maxwell simulations of 2D plasma turbulence prova
Statistical analysis of the local temperature anisotropy Hybrid Vlasov-Maxwell simulations of 2D plasma turbulence prova First attempt of 3D-3V Vlasov turbulence prova
The first example of Valov turbulence has been realized with relatively low resolution both in physical and velocity space
B This was a preparatory run performed on SP6 Machine at CINECA
On the new FERMI machine at CINECA, within a PRACE research project, we plan to run Hybrid Vlasov-Maxwell simulations wih a phase space resolution:
OR LARGER?!? ASK CAVAZZONI!!!
!!! First attempt of 3D-3V Vlasov turbulence prova That's all !!! Francesco Valentini Dipartimento di Fisica, Università della Calabria prova