Note on quantitatively correct simulations of the kinetic beam-plasma instability

Abstract

A large number of model particles is shown necessary for quantitatively correct simulations of the kinetic beam-plasma instability with the clouds-in-cells method. The required number of particles scales inversely with the expected growth rate, as in the kinetic regime only a narrow interval of beam velocities is resonant with the wave.PACS numbers: 52.65. Rr, 52.35.Qz Beam-plasma interaction plays an important role in various physical phenomena such as transport of relativistic electrons in the fast ignition scheme of inertial fusion, gamma-bursts, solar type II and III radio bursts, and collisionless shock waves in the space plasma (see review [1] and references therein). Also, beam-plasma collective interaction determines the efficiency of turbulent plasma heating [2,3] and electromagnetic emission [4][5][6][7] in fusion-oriented mirror traps.One of the most popular and effective tools for theoretical studies of the beam-plasma interaction is numerical simulations by the Particle-In-Cell (PIC) method. At present there is a great number of one-, two-, and three-dimensional PIC codes developed. These codes are used to reproduce fine details of the complex chain of intermediary processes that lead to plasma heating or electromagnetic radiation. The beam densities of interest n b are usually much smaller than the plasma density n p . For the fast ignition problem, the electron beams are relatively dense (n b /n p ∼ 0.1) [8][9][10]. Weakly relativistic beams with n b /n p ∼ 10 −4 ÷ 10 −3 are interesting for mirror traps [11][12][13]. Non-relativistic beams of very low density, n b /n p ∼ 10 −8 ÷ 10 −5 are of interest for radiation generation in solar radio bursts [14], though simulated with higher beam densities because of code limitations [14][15][16][17]. The numerical study usually begins from the linear stage of the beam-plasma instability, so the quantitatively correct simulation of this stage at low beam densities is important for the whole process.There are many realizations of the PIC method that differ in the algorithm of charge and current evaluation from positions of model particles. This algorithm is usually referred to as the shape function. One of the simplest shape functions is the triangular one also called CloudsIn-Cells (CIC) or linear interpolation [18]. This shape function is still widely used [14,[19][20][21][22][23][24][25] in spite of availability of more accurate algorithms [26,27].In this paper we show that an uncommonly large number of model particles is necessary for quantitatively correct simulations of the kinetic beam-plasma two-stream instability with the CIC method. The smaller the growth rate the more particles are needed. If the number of particles is insufficient, then the results are only qualitatively correct, that is the beam-plasma system behaves realistically, but the growth rate is underestimated. To prove this statement, we simulate various regimes of the twostream instability with a rather typical three-dimensional CIC code and compare the simulated growth rates with a...

Particle-in-cell simulations of the relaxation of electron beams in inhomogeneous solar wind plasmas

“…This is observed to occur by around tω pe = 9000, which is in excess of the time scale predicted by the aforementioned quasilinear theory formula (τ ql ω pe = (n b /n 0 ) −1 (v b /∆v b ) 2 ). We however, confirmed that this is a realistic relaxation time for this setup in our convergence testing, rather than being artificially hastened as a side-effect of poor computational particle counts as reported by and Lotov et al [2015]. The discrepancy is instead due to the use of the relatively dense and energetic beams which are outside of the formal applicability of the quasi-linear theory, as discussed in Section 2.…”

Self-consistent particle-in-cell simulations of fundamental and harmonic plasma radio emission mechanisms

“…Umeda (2010) reported that increasing pseudoparticle numbers diminished the signal associated with harmonic emission. Under-resolved particle numbers could contribute to excessive noise on the EM dispersion curves (and be mis-interpreted as emission as per the first point) and also result in poor replication of physical time-scales as per Ratcliffe et al (2014) and Lotov et al (2015). Finally, all of the previous works consider systems with high beam-to-background density ratios (all take n b /n 0 > 1%) and are so in the strong beam regime.…”

“…As the instability time scales with the inverse of the beam-to-background density ratio (e.g., the quasilinear relaxation timescale is τ ql = (n 0 /n b ) (v b /Δv b ) 2 ω −1 pe ), direct simulation of astrophysical parameter regimes where beams are typically diffuse (n b /n 0 ≈ 10 −5 −10 −6 ) requires the simulation to run for tens-to-hundreds of thousands of electron plasma periods. Furthermore, this is compounded in that huge particle numbers are required to correctly resolve the expected quasilinear relaxation timescales with the particle-in-cell (PIC) method Lotov et al 2015). Lotov et al (2015) found empirically that for beam-plasma systems the number of pseudoparticles per cell required scales as the inverse of the fraction of (real) particles which are in resonance with the beam.…”

“…Furthermore, this is compounded in that huge particle numbers are required to correctly resolve the expected quasilinear relaxation timescales with the particle-in-cell (PIC) method Lotov et al 2015). Lotov et al (2015) found empirically that for beam-plasma systems the number of pseudoparticles per cell required scales as the inverse of the fraction of (real) particles which are in resonance with the beam. In particular, for diffuse astrophysical beams, this leads to a very high pseudoparticle requirement of tens-to-hundreds of thousands particles per cell.…”

Co-design of Parallel Numerical Methods for Plasma Physics and Astrophysics

“…The simulations were conducted of the relativistic electron beam interacting with plasma [32]. These simulations resulted in the correct value of the two-stream instability increment.…”