Eulerian simulations of collisional effects on electrostatic plasma waves

Abstract

The problem of collisions in a plasma is a wide subject with a huge historical literature. In fact, the description of realistic plasmas is a tough problem to attach, both from the theoretical and the numerical point of view, and which requires in general to approximate the original collisional Landau integral by simplified differential operators in reduced dimensionality. In this paper, a Eulerian timesplitting algorithm for the study of the propagation of electrostatic waves in collisional plasmas is presented. Collisions are modeled through one-dimensional operators of the Fokker-Planck type, both in linear and nonlinear form. The accuracy of the numerical code is discussed by comparing the numerical results to the analytical predictions obtained in some limit cases when trying to evaluate the effects of collisions in the phenomenon of wave plasma echo and collisional dissipation of Bernstein-Greene-Kruskal waves. Particular attention is devoted to the study of the nonlinear Dougherty collisional operator, recently used to describe the collisional dissipation of electron plasma waves in a pure electron plasma column [M. W. Anderson and T. M. ONeil, Phys. Plasmas 14, 112110 (2007)]. A receipt to prevent the filamentation problem in Eulerian algorithms is provided by exploiting the property of velocity diffusion operators to smooth out small velocity scales.

Collisional effects on the numerical recurrence in Vlasov-Poisson simulations

“…Moreover, the analysis has been extended to the nonlinear regime through a 1D-1V Eulerian collisional Vlasov-Poisson code, already tested and used in previous works (see Refs. [20,21]). …”

“…This approach has already been adopted in several works [14][15][16][17][18][19][20][21][22]. In the same spirit, here we focus on the one-dimensional LB operator.…”

“…Time evolution of the distribution function is approximated through the splitting scheme first introduced by Filbet et al [38] [see Refs. [20,21] for details about the numerical algorithm], which is a generalization of the well-known splitting scheme discussed by [1]. We impose periodic boundary conditions in physical space and f is set equal to zero for |v| > v max , where v max = 6v th,e .…”

Nonlinear regime of electrostatic waves propagation in presence of electron-electron collisions

“…As discussed earlier, we model electron-electron collisions through the Dougherty operator 7,8,16 and neglect electron-proton and proton-proton collisions, as their characteristic time is significantly longer than that for electron-electron interactions. 3,12,13 We consider the following dimensionless DoughertyPoisson (DP) equations, in 1D-3V phase space configuration:…”

“…In fact, the computational time t c for 1D-3V (1D in physical space and 3D in velocity space) Eulerian simulations which include the full Landau operator scales as t c $ N 7 (where N is the number of gridpoints, assumed, for simplicity, to be the same for each phase-space coordinate); for the DG operator, the scaling is t c $ N 4 ; this significant reduction of t c allows to run numerical experiments of the self-consistent electrostatic dynamics of a collisional plasma in 1D-3V geometry. In previous works, [10][11][12][13] numerical simulations of electrostatic waves in collisional plasmas have been performed in a phase space of reduced dimensionality (1D-1V).…”