Particle-in-Cell Methods for Simulations of Sheared, Expanding, or Escaping Astrophysical Plasma

TL;DR

This work addresses the challenge of simulating collisionless astrophysical plasmas with local PIC models that must couple to global-scale configurations. It develops three frameworks—the kinetic shearing box with orbital advection (KSB-OA), the kinetic expanding box (KEB), and a leaky-box approach for particle escape—each accompanied by tailored Maxwell solvers and Boris-like pushers, with explicit frame transformations such as $v_s = -s \Omega_0 x \hat{e}_y$, $E' = E + \frac{v_s}{c}\times B$, $u' = u - \gamma' v_s$, and $B' = \ell L^{-1} B$, $E' = \ell L^{-1} E$, $J' = L^{-1} J$. The numerical schemes combine implicit Maxwell solvers and Boris-like momentum updates, including advection upwinding for stability, and practical boundary conditions to maintain physical consistency. Applications demonstrate MRI-driven turbulence in a fully kinetic 3D pair-plasma box, expansion-driven firehose dynamics, and steady-state energy distributions in diffusive escape scenarios, validating the methods and revealing rich kinetic behavior beyond fluid models. Collectively, these methods extend PIC's applicability to more realistic astrophysical environments by enabling consistent coupling of local kinetic evolution to global flows, expansion, and energy sinks.

Abstract

Particle-in-Cell (PIC) methods have achieved widespread recognition as simple and flexible approaches to model collisionless plasma physics in fully kinetic simulations of astrophysical environments. However, in many situations the standard PIC algorithm must be extended to include macroscopic effects in microscale simulations. For plasmas subjected to shearing or expansion, shearing-box and expanding-box methods can be incorporated into PIC to account for these global effects. For plasmas subjected to local acceleration in confined regions of space, a leaky-box method can allow closed-box PIC simulations to account for particle escape from the accelerator region. In this work, we review and improve methods to include shearing, expansion, and escape in PIC simulations. We provide the numerical details of how Maxwell's equations and the particle equations of motion are solved in each case, and introduce generalized Boris-like particle pushers to solve the momentum equation in the presence of extra forces. This work is intended to serve as a comprehensive reference for the implementation of shearing-box, expanding-box, and leaky-box algorithms in PIC.

Figures (3)

• Figure 1: Example PIC simulation of the collisionless pair-plasma MRI, using the KSB-OA method. Left: spatial distribution of the toroidal ($B_y$) magnetic field during the linear MRI stage ($t \approx 3.5P_0$) and during the turbulent, nonlinear quasi-steady state ($t \approx 12P_0$). Right: evolution of the volume-averaged change in magnetic energy for each component of $\hbox{\boldmath{$B$}}$.
• Figure 2: Three-dimensional visualization of the $z$-component of the magnetic-field fluctuations, $\delta B_z/B_g$, shown at (a) the initial time, (b) half an expansion time, and (c) one expansion time. Panel (d) shows the time evolution of the magnetic-field energy (solid blue) and the background guide-field energy (dashed red), normalized to the initial guide-field energy. The dotted black line marks the onset of firehose fluctuations.
• Figure 3: (a) Temporal evolution of the particle energy distribution function in a leaky-box turbulence simulation, where lighter colors indicate later simulation times. The dashed black line represents the time-averaged particle distribution during the steady state. (b) Temporal evolution of the volume-averaged plasma $\beta$. (c) Spatial distribution of the turbulent magnetic-field magnitude at a representative time during the steady state.