Electron kinetic simulations using Maxwellians and generalized Laguerre polynomials.

The symbols and special characters used in the original abstract could not be transcribed due to technical problems. Please use the PDF version to read the abstract. Electron heat flow in steep temperature gradients is known to produce distinctly non- Maxwellian electron velocity distributions: In the cold regions, there is a considerable surplus of hot electrons which have streamed from the hot regions before being thermalized. In electron kinetic codes, the velocity distribution function is represented on a grid in velocity space. Expanding the angular dependence in Legendre polynomials greatly reduces the computational requirements, but the energy grid is still a burden. Here, we propose to represent the energy dependence as a sum of one, two or three Maxwellians, each multiplied by a finite sum of generalized Laguerre polynomials, to represent, for example, the streaming of energetic electrons into cold plasma. In the Vlasov-Fokker-Planck electron kinetic equation, if we assume that the spatial variation is in 1 dimension, along χ, and that the velocity variation has azimutal symmetry about the χ axis, and we expand the angular µ dependence in Legendre polynomials, we can derive the kinetic equation (eq. 2 of F. Alouani_Bibi et al., Comp. Phys. Comm. 164, 60 (2004)), assuming there is no magnetic field, except possibly along χ. In the present “FPI” electron kinetic code, these Legendre coefficients fl (χ,, v, t) are advanced in time, using finite difference grids in χ and v. The aim of our project is to replace the grid in v by one, two or three Maxwellians, each multiplied by a sum of generalized Laguerre polynomials on each point of the spatial grid, and then to compute the time evolution of the electron distribution function for different temperature ratios using generalized Laguerre polynomials: [Mathematical equation] where me is the electron mass, v is velocity, n denotes the population, c for cold, m for medium and h for hot. Tn(x,t)’s are the temperatures of each population, anlj(χ, t)’s are the coefficients of the generalized Laguerre polynomials (The Tn(χ, t) and the anlj(χ, t)’s are the fitting parameters). As electron-electron (e-e) collisions are the most complex part of this problem, we focus on this aspect in the present research. We study the relaxation of a hot Maxwellian initially embedded in a colder and denser one towards a single Maxwellian. First, we fitted the distribution functions from the “FPI” finite difference electron code for this problem to a single/two/three Maxwellian(s) each multiplied by a finite sum of generalized Laguerre polynomials, at several times, to show that this expansion is viable. Then, we provided a moment evolving and fitting method for advancing the parameters (the temperatures Tn(χ, t)’s and the coefficients of the generalized Laguerre polynomials anlj(χ, t)’s) in time, and a comparison to “FPI” code results at several times was made. Good agreement was obtained, if proper choices of the number of Maxwellians and of Laguerre terms were made It was found that a single temperature expansion is viable only for low initial temperature ratios (2:1 or less), but that for initial temperature ratios above 10:1, a three temperature expansion is best, and was seen to give good results (i.e. good agreement with “FPI”) for temperature ratios as high as 1000:1.