Coulomb collision model for use in nonthermal plasma simulation

In kinetic simulations of non-Maxwellian plasmas, the calculation of particle scattering due to Coulomb collisions has no simple approximation. In such simulations, the number of collision interactions a particle experiences in a single time step is typically too large for direct calculation. In this work, the cumulative effect of a series of binary collisions is calculated numerically in a stochastic manner, and heuristic trends are produced as functions of the local plasma parameters. The result is a collision model suitable for implementation into a kinetic plasma simulation. The presence of low-probability, high-angle scattering due to close collision encounters is defined and described, and this effect is demonstrated in a test problem simulation of weakly collisional counterstreaming ion beams.

Figure 1

A two-dimensional cross-sectional schematic of the $N$-body simulation for testing the cumulative binary collision approximation. The test particle travels a distance of $v\tau =2\text{mm}$ through a sphere of field particles but only experiences a force from field particles within a distance of $b_{\max }=1\text{mm}$.

Figure 2

Probability distribution functions for varying values of $b_{\max }$ for (a) results of the $N$-body simulation with fixed field particles and (b) results of the cumulative binary collision approximation.

Figure 3

Comparison of scattering angles produced by the cumulative binary collision approximation with scattering angles produced by the three pieces of Eq. (27).

Figure 4

Values of (a) $\tilde{\sigma }$, (b) $\kappa$, (c) $u_{\mathrm{low}}$, and (d) $u_{\mathrm{high}}$ from individual numerical experiments performed at varying values of $N$ and $M$, found by fitting Eqs. (25) and (26) to numerical data produced by Eq. (11). Best-fit curves of Eq. (31) are fit to the data points for $N>{10}^3$ only.

Figure 5

(a)–(e) Values of $\sigma , \tilde{\sigma }, \kappa , u_{\mathrm{low}}$, and $u_{\mathrm{high}}$, respectively, from individual numerical experiments performed at each $(a,N)$ coordinate, found by fitting Eqs. (24) and (26) to numerical data produced by Eq. (8). (f)–(j) Best-fit graphs of Eq. (32) for predicting values of $\sigma , \tilde{\sigma }, \kappa , u_{\mathrm{low}}$, and $u_{\mathrm{high}}$ as a function of $a$ and $N$. Selected contours of constant value are plotted to aid in comparison.

Figure 6

A comparison of the probability distribution functions for the scattering angle between the cumulative binary collision approximation (Sec. 3), the $N$-body simulation (Sec. 4), the Nanbu method, the Takizuka-Abe method, and the present method (Sec. 5).

Figure 7

The percent error of the mean scattering angle as compared to the results of the cumulative binary collision approximation for (a) the Nanbu method and (b) the present method.

Figure 8

Time-averaged density for simulations of counterstreaming beams using (a) the $N$-body method, (b) a particle-in-cell simulation with the present collision model, (c) a particle-in-cell simulation with Nanbu's collision model, and (d) a collisionless particle-in-cell simulation. The plots are axisymmetric about the $z$ axis and plane-symmetric about the $r$ axis. In all simulations, the envelope of the beam sourced at $z=5\text{mm}$ is visible as a dark shade, and the envelope of the beam sourced at $z=-5\text{mm}$ is visible as a light shade. The density resulting from high-angle scatters permeates the remainder of the domain and is displayed using contour lines of constant value. Densities down to ${10}^6{\text{m}}^{-3}$ are resolved by time-averaging the density over $0.5\text{ms}$. Densities below ${10}^6{\text{m}}^{-3}$ are not resolved.

Figure 9

A comparison of scattering angle probabilities with the probabilities of $u_{\mathrm{fusion}}$ (a fusion event), $u_{\mathrm{de}\mathrm{Broglie}}$ (significant interaction of matter waves), and $u_{\mathrm{potential}}$ (potential energy exceeding kinetic energy) occurring.