Differences between real and particle-in-cell plasmas: effects of coarse-graining
Mickaël Melzani, Rolf Walder, Doris Folini, Christophe Winisdoerffer
OOctober 9, 2018 6:50 WSPC/INSTRUCTION FILE Melzani
International Journal of Modern Physics: Conference Seriesc (cid:13)
World Scientific Publishing Company
DIFFERENCES BETWEEN REAL AND PARTICLE-IN-CELLPLASMAS: EFFECTS OF COARSE-GRAINING
MICKA¨EL MELZANI [email protected]
ROLF WALDER [email protected]
DORIS FOLINI [email protected]
CHRISTOPHE WINISDOERFFER [email protected]´Ecole Normale Sup´erieure de Lyon, Centre de Recherche Astrophysique de Lyon, UMR CNRS5574, Universit´e de LyonLyon, France
Received Day Month YearRevised Day Month YearThe PIC model relies on two building blocks. The first stems from the capability ofcomputers to handle only up to ∼ particles, while real plasmas contain from 10 to 10 particles per Debye sphere: a coarse-graining step must be used, whereby of theorder of p ∼ real particles are represented by a single computer superparticle. Thesecond is field storage on a grid with its subsequent finite superparticle size. We introducethe notion of coarse-graining dependent quantities, i.e. physical quantities depending onthe number p . They all derive from the plasma parameter Λ, which we show to beproportional to 1 /p .We explore three examples: the rapid collision- and fluctuation-induced thermaliza-tion of plasmas with different temperatures, that scale with the number of superparticlesper grid cell and are a factor p ∼ faster than in real plasmas; the high level of elec-trostatic fluctuations in a thermal plasma, with corrections due to the finite superparticlesizes; and the blurring of the linear spectrum of the filamentation instability, where thefastest growing modes do not dominate the total energy because of a high level of fluc-tuations.We stress that the enhanced collisions and correlations of PIC plasmas must be keptnegligible toward kinetic physics. Keywords : Collisionless plasmas; Particle-In-Cell codes.PACS numbers: 11.25.Hf, 123.1K 1 a r X i v : . [ a s t r o - ph . H E ] N ov ctober 9, 2018 6:50 WSPC/INSTRUCTION FILE Melzani M. Melzani, R. Walder, D. Folini, C. Winisdoerffer
1. Introduction
Particle-in-cell simulations have brought tremendous new insights into collisionlessastrophysical plasmas, for example regarding instabilities, kinetic turbulence, orparticle acceleration in shocks and in magnetic reconnection. However, there remaina number of questions concerning the degree to which PIC models are able tocompletely mirror real plasmas. In this manuscript, and in more detail in Ref. 1, weaddress the incidence of coarse-graining, i.e., of the fact that each computer particlein the PIC plasma actually represents many real plasma particles. The need forcoarse-graining becomes evident when looking at the number of plasma particlesper Debye sphere, given by the plasma parameter Λ. In a real plasma Λ rangesfrom 10 to 10 (e.g., Λ ∼ in solar coronal loops; 10 in the magnetotail,magnetopause, or in typical Crab flares; 10 in AGN jets), while in computerexperiments, where we have to simulate thousands to millions of Debye spheres,Λ reaches hardly a few tens. The corresponding number of particles per computerparticles – that we will denote as superparticles – then reaches p ∼ to 10 .
2. Principles of the PIC Algorithm
A PIC code solves Maxwell equations with current and charge densities computedfrom the superparticles, and the equations of motion with the Lorentz force foreach superparticle. The fields are stored on a grid, while the superparticles evolvecontinuously in position and velocity space.Information between the superparticles inside the cells and the fields stored atgrid nodes is communicated via interpolation, and this interpolation is equivalentto considering the superparticles as clouds of charge with a finite extent , . This inturn implies a vanishing two-point force between superparticles at short distances,and thus reduces drastically the influence of collisions. It helps the PIC plasma toremain collisionless even if its plasma parameter is quite low . Discretization alsobrings numerical stability issues, that have been explored in depth in Refs. 2, 3.
3. Fluid versus Coarse-Graining Dependent Quantities
We assume that we model a real plasma composed of N real particles by usingin the code N/p superparticles. The number of real particles represented by eachsuperparticle is thus p . We also denote by ρ the initial number of superparticlesper grid cell in a simulation, and by X the physical size of a cell.The set of equations solved by the PIC code is not invariant under coarse-graining because there are physical quantities that depend on p . Such quantities willbe termed coarse-graining dependent quantities . By contrast, physical parametersthat do not depend on p will be refereed to as fluid quantities . All the fluid physicsis thus accurately described by the PIC plasma, while some care is to be taken forthe coarse-graining dependent physics.Fluid quantities are invariant under the change of variables ( m, q, n ) → ( m × p, q × p, n/p ), which amounts to pass from the real plasma of particles of massctober 9, 2018 6:50 WSPC/INSTRUCTION FILE Melzani Differences Between Real and Particle-In-Cell Plasmas ρ sp = ρ sp = ρ sp = / T pe ) ρ sp = ρ sp = ρ sp = ρ sp = − T ∞ T − T ∞ T r − T ∞ ( un it s o f m e c ) ρ sp t h e r m a li za ti on ti m e ( un it s o f T p e ) t th for electronst th / (m i / m e ) / for ions ρ sp = Fig. 1.
Left : Electron temperatures for the cold (1) and hot (2) plasmas, from four simulationswith different numbers of superparticle per cell ρ sp (that include all species). Right : Half-thermalization times for ions and electrons, against number of superparticles per cell. m , charge q , number density n , to the plasma of superparticles. Examples includethe plasma pulsation ω = nq / ( (cid:15) m ), the thermal velocity v th , the Debye length λ D = v th /ω p , and more generally any quantity derived from fluid theory (fluidtheories include MHD, two-fluid models, Vlasov-Maxwell system).The prototype of coarse-graining dependent quantities is the plasma parameterΛ = 4 πn e λ D e . Since Λ is proportional to the number of electrons per Debye sphere,and since the Debye length is p -independent, the plasma parameter of the PICplasma reads Λ PIC p = Λ /p, (1)with Λ = Λ PIC p =1 the real plasma parameter. Another way of seeing this is to recallthat the plasma parameter Λ is the ratio of the particles’ kinetic energy to theirelectrostatic potential energy of interaction and, as such, varies as 1 /p becausekinetic energy is proportional to the superparticles’ mass m sp ∝ p while chargeinteraction energy involves their charge q ∝ p . Other coarse-graining dependentquantities can be built from Λ: the thermalization time or the slowing down timeof fast particles, both due to collisions and correlations, scale as t th ∝ Λ /ω p , orthe level of electrostatic fluctuations ε ∝ / Λ. Other coarse-graining dependentparameters can be found. Any number of particles per fluid volume will have thesame dependency as Eq. 1, for example n e ( c/ω p ) .In the following, we explore situations where coarse-graining effects have visibleand important consequences.
4. Coarse-Graining and Thermalization Times
In a PIC plasma, the behavior of plasma quantities depending on Λ can be guessedby replacing Λ by Λ
PIC p . This is the case for the thermalization time of a plasma bygrazing Coulomb collisions or by electric field fluctuations , which is on the order of t th ∼ T P × Λ (with T P the plasma period). This has two important consequences: (i)ctober 9, 2018 6:50 WSPC/INSTRUCTION FILE Melzani M. Melzani, R. Walder, D. Folini, C. Winisdoerffer e l ec t r i ce n e r gyno r m a li ze d t ok i n e ti ce n e r gy f = ρ sp ˜ λ / (˜ λ D − arctan(˜ λ D )) l og ˜ λ D . . . . . − . / (55 Λ p )numerical experiments10 − − − − Λ p = ρ sp ˜ λ − − − / (71 f)numerical experiments e l ec t r i ce n e r gyno r m a li ze d t ok i n e ti ce n e r gy PIC
Fig. 2. Each point is the measure of the fluctuation level ε from a PIC simulation. Here ˜ λ D = λ D /X is the Debye length normalized by the cell size. we expect t PICth to depend on resolution and coarse-graining, roughly as t PICth /T P ∝ Λ p = ρ sp ( λ D /X ) , and (ii) since Λ PIC p = Λ /p is several orders of magnitude smallerthan the real plasma parameter Λ, we expect the thermalization due to grazingcollisions and fluctuations to be vastly more efficient in PIC codes than in reality.We present here simulations with initially two thermal ion-electron plasmas(mass ratio m i /m e = 25) with two different initial temperatures. The four speciesinteract via collisions and correlations (no sign of plasma kinetic instabilities werefound) and tend to reach the same final temperature T ∞ .Results are shown in Fig. 1, for four simulations with different numbers of su-perparticles per cell. To evaluate the thermalization times we use the exponentialvariation of the temperature curves . We clearly see a slower thermalization as ρ sp increases, with a scaling t PICth ∝ ρ sp roughly correct for both electrons and ions.We also emphasize the difference with a real plasma, where t th /T pe ∼ Λ reaches10 or more, while it is on the order of Λ p ≤ in PIC simulations.
5. Coarse-Graining and Fluctuation Levels
We now study the level of electric field fluctuations in a PIC thermal plasma, andshow that in addition to the substitution Λ → Λ PIC p , effects due to finite superpar-ticle size must be taken into account. In a real plasma in thermal equilibrium, it isgiven by ε = (cid:104) (cid:15) E / (cid:105) nT / ∼ , (2)where the symbol (cid:104)·(cid:105) denotes an average over space, E is the electric field, and n and T the plasma number density and temperature.We perform simulations of thermal plasmas at rest and measure the level ofenergy in the electric field. We use thermal velocities from 0.04 c to 0.10 c , ρ sp fromctober 9, 2018 6:50 WSPC/INSTRUCTION FILE Melzani Differences Between Real and Particle-In-Cell Plasmas
1 2 3 4 Time in total electron plasma period (t / T pe ) s u m o f a ll m o d e s e n e r gy τ s i m u f a s t m od e = . T p e τ s i m u t o t e n = . T p e x = , n t = x = , n t = x = , n t = x = , n t = ρ sp τ simutoten − τ theorymax τ theorymax Fig. 3.
Left : Growth of individual Fourier modes for b x . Right : Relative difference between thegrowth rate measured on the total energy and the analytical prediction for various simulations.The timestep is T pe /n t and the grid size is cT pe / (2 π ) (with T pe the electron plasma period). ε is not found proportional to 1 / Λ PIC p .To explain this, we compute the electric field produced by particles in a ther-mal plasma . The expression for the electric field is a generalization of the Debyeelectric field for moving particles, and is integrated in k -space from 0 to a maximalwavenumber k max = X − given by the superparticle size X . We then obtain (cid:104) (cid:15) E x / (cid:105) nT / π λ D /X − arctan λ D /X ρ sp ( λ D /X ) , (3)in very good agreement with the simulations (Fig. 2, right panel).We note that for a very high resolution, ˜ λ D (cid:29)
1, the field energy decreases as1 / ( ρ sp ˜ λ ), which is non-trivial and different from what is expected in a real plasmawhere it decreases as 1 / Λ = 1 / ( nλ ).
6. Coarse-Graining and Linear Growth Rates of Instabilities
This section illustrates the incidence of the high level of noise in the linear phaseof instabilities. We perform several simulations of cold and unmagnetized counter-streaming beams of pair plasmas, with parameters such that the dominant unstablemode is the filamentation instability. We measure the growth rate during the lin-ear phase with two methods: by a direct measure on the total energy curve, e.g., (cid:82) d V b x ∝ exp(2 t/τ ), giving an effective growth rate that we denote by τ simutot en , andby following the time evolution of the Fourier modes of the magnetic field.Figure 3 (left) is an example of the temporal evolution of the modes of b x . Thesum of all modes grows at the same effective growth rate as the total energy in b x ,ctober 9, 2018 6:50 WSPC/INSTRUCTION FILE Melzani M. Melzani, R. Walder, D. Folini, C. Winisdoerffer τ simutot en = 0 . T pe . However, the fastest growing modes grow with τ simufast mode = 0 . T pe ,which is close to the analytical cold-fluid result τ theorymax = 0 . T pe . We see fromFig. 3 (left) that the large difference between the effective growth rate τ simutot en andthe growth rate of the fastest modes τ simufast mode is due to a significant contribution ofall the modes during the whole linear phase. The fastest mode thus never dominatesthe total energy in the linear phase.These results hold for all the test simulations that we conducted : the growthrates of the fastest Fourier modes remain at a constant value, while the effectivegrowth rates measured on the total energy present various levels of discrepancieswith theory, between 14% and 67%. Figure 3 (right) shows that there is a systematicdecrease in this discrepancy when the superparticle number per cell ρ sp is increased(all other parameters being kept constant). Since the fluctuation level in the PICplasma decreases with increasing ρ sp , this indicates that the high fluctuation levelexcites all the modes and prevents the fastest ones from dominating the energy.
7. Conclusion
PIC simulations employ a high level of coarse-graining: each superparticle repre-sents up to p ∼ real plasma particles. This would not be an issue for simu-lating collisionless and correlation-less plasmas because the Vlasov-Maxwell systemis invariant under coarse-graining. However, the physics of collisions and correla-tions is coarse-graining dependent, and is strongly enhanced by the smallness of thePIC plasma parameter Λ PIC p = Λ real /p ∼ a few. As a result, noise and fluctuationlevels are larger than in the real plasma by a factor p ∼ , and collision- andfluctuation-induced thermalization is faster by a factor p ∼ . For example, inreal collisionless shocks, the mean free path for collisions l mean free path is far largerthan the shock thickness ∆ shock and the thermalization processes are collisionlesskinetic instabilities. Since the mean free path l PICmean free path ∝ Λ p in a PIC plasmais smaller by a factor of p ∼ than in the real plasma, one has to check that l PICmean free path (cid:29) ∆ shock still holds. More generally, to truly describe a collisionlessand correlation-less plasma with a PIC algorithm, one has to be careful that the un-physically enhanced collisional physics remains slower than the collisionless physics.This can be achieved with large enough numbers of superparticles per cell, or/andby the use of smoother particle shapes that reduce fluctuation levels.We also showed that the behavior of PIC plasma quantities can be guessed bythe substitution Λ → Λ PIC p , but that this recipe is not exact and that the finite sizeof the superparticles is to be taken into account to obtain the precise dependency. References
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