Summer College on Plasma Physics. 30 July - 24 August, The particle-in-cell simulation method: Concept and limitations

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1 Summer College on Plasma Physics 30 July - 24 August, 2007 The particle-in-cell M. E. Dieckmann Institut fuer Theoretische Physik IV, Ruhr-Universitaet, Bochum, Germany

2 The particle-in-cell Concept and limitations Institut für Theoretische Physik IV Ruhr-Universität Bochum D Bochum Institutionen för teknik och naturvetenskap (ITN) Linköpings universitet Campus Norrköping SE Norrköping Book: C K Birdsall & A B Langdon, Plasma Physics Via Computer Simulation, Taylor & Francis Ltd (October 2004) p. 1

3 Contents

4 Particle-in-cell / Vlasov codes can address all collisionless plasma processes Such codes can be used to verify linearized plasma dispersion relations The plasma dynamics can be followed through its non-linear phase s have an unlimited dynamical range for the particle velocities and no boundary conditions along v PIC codes have a limited signal-to-noise ratio and phase space density resolution PIC schemes parallelize well p. 3

5 E = B t (1) B = µ 0 ɛ 0 E t + µ 0J (2) B =0 (3) E = ρ/ɛ 0 (4) Vlasov equation for phase space distribution f(x, v,t) ρ(x,t)=q f t + v xf + q m (E + v B) vf =0 (5) f(x, v,t) dv, J(x, v,t)=q vf(x, v,t) dv (6) p. 4

6 Field equations Replace continuous fields B(x,t), E(x,t) and Maxwell s equations by their discretized counterparts. (1) Introduce finite spatial resolution x j x and finite time resolution t j t, with integer values j. (2) Replace differential operators by difference operators. d Example: dx f(x) (f[(j +1) x] f[j x ])/ x. Replaces differential equations by algebraic equations. Particle equations (PIC codes only) Replace continuous probability distribution by phase space elements N f(x, v,t) S(x x i )δ(v v i ), (7) i=1 where S(x) is a shape function, e.g. a triangle. Replaces phase space probability function by computational particles. p. 5

7 : Maxwellian distributions are often used. Quantitatively different results are sometimes obtained with Vlasov codes (upper phase space distribution) and PIC codes (lower phase space distribution)! If particle weights are equal, the statistical representation of the plasma close to the mean speed of the species is good. The statistical representation is poor, if v v 2v t for a Maxwellian exp ( (v v) 2 / 2 v 2 t ). p. 6

8 The particle-in-cell Field Node Particles Field Node X X X X i i+1 i+2 i+3 The fields are defined on a grid (Field nodes). The particles move on continuous paths. Particles interact with the grid through ρ and J. The grid interacts with the particles through E and B. Interpolation schemes must be specified p. 7

9 A computational electron is located between the cells i, i +1 with the positions X i and X i+1. It has the distance D1 from X i and D2 from X i+1. Electron charge Q is assigned to the grid nodes i, i +1: C i = f 1 (D1,D2) and C i+1 = f 2 (D1,D2). Example: C i = QD2 and C i+1 = QD1 if (X i+1 X i )=1. The particle-in-cell C I C I+1 Electron C Electric field E C I I+1 Force E X D1 D2 X X i i+1 Electron Calculate potential Φ with 2 Φ= ρ/ɛ 0 with ρ i C i and ρ i+1 C i+1. E = Φ. Interpolate E to particle position: E(x) =D2 E(x i )+D1 E(x i+1 ) p. 8

10 1. Initialize plasma phase space distribution: Place particles in space according to density Initialize velocities with random numbers 2. Initialize E and B fields 3. From E,B fields calculate acceleration 4. Multiply acceleration with time step > Velocity increment 5. Multiply velocity with time step > Position increment 6. From new positions and velocities: Calculate new E,B p. 9

11 : Known limitations with known consequence: Field discretization Spatial step x Debye length λ d = v t /ω p. Numerical stability small time step t. For my code: t = N c x / 2 c with 0 <N c < 1. We have x = v t /ω p and for N c =1we get 2 t ω p = v t /c. Low v t requires high sampling rate. Critical speed v t 10 6 m/s. Known limitations with unknown consequence: Particle per cell count rates follow Poisson statistics For N e particles per cell, the relative fluctuations are 1/ N e. The particle-in-cell Fluctuations result in particle-wave collisions. Parametric instabilities speed up. Dispersive properties change. Bernstein modes are damped close to nω c. Signal-to-noise ratio is low. p. 10

12 In continuous infinite plasmas : Fourier integral Infinitely extended and continuous position x and wavenumber k domains. The particle-in-cell f(x) = g(k) = 1 2π f(x) g(k) (8) g(k) exp(ıkx) dk (9) f(x) exp( ıkx) dx (10) 1 Amplitude Position 10 0 Power Wavenumber p. 11

13 In continuous confined plasmas : Fourier series Limited position 0 <x<l, f(x) continuous. Unlimited, discrete k spectrum. f(x) g k = g(k k ) (11) f(x) = g k exp (ık k x) (12) g k = 1 L x=l x=0 k= f(x)exp( ık k x) dx (13) Periodicity of exp (ık i x) The f(x) is periodic, but not g i. Amplitude The particle-in-cell Position 10 0 Power Wavenumber p. 12

14 In PIC plasmas : Discrete Fourier transform The PIC code cannot represent a Fourier series The k-spectrum is truncated. f n = g k = 1 N N 1 k=0 N 1 n=0 f n = f(x n ) g k = g(k k ) (14) g k exp (2πıkn/N), n =0,..., N 1 (15) f n exp ( 2πıkn/N), k =0,..., N 1 (16) The particle-in-cell 1 Amplitude Position 10 0 Power Wavenumber p. 13

15 The particle-in-cell has rapid oscillations: F (k s,x)=sin(k s x). Grid imposes a maximum k Ny = π/ x. A box with length L has the minimum k M =2π/L. (a) k 1 = k Ny /25 and k 2 = k Ny /100. Integer values for k 1 /k M and k 2 /k M. Amplitude Position in x 10 0 Power k / k Ny p. 14

16 (b) k 1 = k Ny /25.25 and k 2 = k Ny / Non-integer k 1 /k M and k 2 /k M. Amplitude Position in x Power k / k Ny Wave oscillation is well-resolved. Finite-box effects fill up the k-spectrum: power leaks to other k (aliasing). p. 15

17 (c) k 1 = k Ny /2 and k 2 =3k Ny /2. Integer values for k 1 /k M and k 2 /k M. Amplitude Position in x Power k / k Ny Peaks are identical in power but have different phases. p. 16

18 Some consequences: The aliasing is introduced by mapping oscillations with an unlimited k spectrum onto a grid with k <k Ny. Case (3) showed that k 1,k 2 with k 1 k 2 = k Ny differ only in phase. Truncated Fourier series forced periodicity in x and k. Signals with k 1 and k 2 differing by 2 k Ny can t be distinguished! Relation to crystals / metals: Brillouin zones First Brillouin zone k x <π. Other zones differ by 2 k Ny = k g = 2π/ x. In s, elementary sine waves are actually not (!) eigenfunctions of the system. p. 17

19 Counterstreaming two-stream instability 1 ω2 p ω 2 = Physical resonance term: ω kv b =0 Aliasing: ω (k + pk g )v b =0, with integer p. Ω 2 b (ω kv b ) 2 + Ω 2 b (ω + kv b ) 2 (17) The term k g v b has the dimension of a frequency. It is the grid crossing frequency ω g of the particle beam. The beam drives many resonances (ω pω g) k = v b.onlyp =0is physical. Consequence: In non-relativistic 1D simulations with k v b the nonphysical instabilities are electrostatic and only heat the beam Harmless. p. 18

20 of relativistic beams in 2D The phase speeds of the waves are ω k = v b + p ω g k. For speeds 1 ɛ<v b /c < 1 and ɛ 1, ap 0introduces superluminal waves that couple directly to the beam. The resonance is sharp. Particle oscillations due to a wave with k v b couple through γ to other velocity components. Quiver motion acts as antenna! In 2D/3D simulations, obliquely propagating electromagnetic waves couple to the fast moving charge density modulation Electromagnetic radiation similar to free electron laser. p. 19

21 Spatially homogeneous system in x-y plane, periodic boundary conditions. Initially, E =0and B =(0, 0,B 0 ) with eb 0 /m e =Ω p /10 (Ω p =plasma frequency). Bulk electrons and protons at rest, number densities n e and n p. Counter-streaming proton beams with velocity vector ±(v b, 0, 0) and v b = 0.8 c. Beam density: n b each. Both beams differ in their temperature. n e = n p +2n b. Rectangular simulation box, sidelength = 6 πv b / Ω p.resolvedbya mesh. The system is advanced in time for tω p = 194. p. 20

22 Results 1: Left plot, upper row: 10-logarithmic wavenumber (power) spectrum of physical wave. v b is parallel to the x-axis. Right plot, upper row: 10-logarithmic wavenumber (power) spectrum of grid instability waves. Left plot, lower row: Low-pass filtered E x + ıe y in a rectangular subinterval with the side length 2 πv b /Ω p. Right plot, lower row: E x + ıe y in a rectangular sub-interval with the side length πv b / 2Ω p. p. 21

23 Results 2: Two straight lines going through k =0, Ω=0: Beam dispersion relation. Two curved lines going through k =0, Ω=0(on this scale): Light modes. Straight line at large Ω parallel to the lower beam dispersion relation: Sideband separated from beam mode by grid crossing frequency. p. 22

24 The particle-in-cell simulation method is currently the tool of choice for the investigation of non-linear processes in kinetic, collision-less plasma. In contrast to fluid / magnetohydrodynamic codes, its resolution is set by the plasma Debye length it can not be scaled to fit best a problem. Its signal-to-noise ratio is limited by noise. Noise can introduce unknown modifications. The statistically poor representation of plasma by s can give different results compared to Vlasov simulations. Example: Electron phase space holes can be more stable in Vlasov simulations: Dieckmann, Eliasson, Stathopoulos, Ynnerman, Phys. Rev. Lett. 92, (2004) The finite grid instability turns out to be a severe simulation constraint for the investigation of relativistic plasma flows. It can be reduced only moderately by finer grids. Dieckmann, Frederiksen, Bret, Shukla, Phys. Plasmas 13, (2006). p. 23