Modeling transport in weakly collisional plasmas using thermodynamic forcing

Abstract

How momentum, energy, and magnetic fields are transported in the presence of macroscopic gradients is a fundamental question in plasma physics. Answering this question is especially challenging for weakly collisional, magnetized plasmas, where macroscopic gradients influence the plasma's microphysical structure. In this paper, we introduce thermodynamic forcing, a new method for systematically modeling how macroscopic gradients in magnetized or unmagnetized plasmas shape the distribution functions of constituent particles. In this method, we propose to apply an anomalous force to those particles inducing the anisotropy that would naturally emerge due to macroscopic gradients in weakly collisional plasmas in which thermal pressure is much larger than magnetic pressure. We implement thermodynamic forcing in particle-in-cell (TF-PIC) simulations using a modified Vay particle pusher and validate it against analytic solutions of the equations of motion. We then carry out a series of simulations of electron-proton plasmas with periodic boundary conditions using TF-PIC. First, we confirm that the properties of two electron-scale kinetic instabilities - one driven by a temperature gradient and the other by bulk-velocity gradient - are consistent with previous results. Then, we demonstrate that in the presence of both macroscopic gradients, heat-flux saturation is mediated by the bulk-velocity-gradient-driven electron firehose instability rather than the temperature-gradient-driven whistler instability. This suggests that saturation mechanisms may differ from our current understanding in the presence of multiple free energy sources. This work enables, for the first time, systematic and self-consistent transport modeling in weakly collisional plasmas, with broad applications in astrophysics, laser-plasma physics, and inertial confinement fusion.

Figures (15)

• Figure 1: Schematic diagram of self-consistent transport modeling in weakly collisional plasmas. The left panel shows the bulk velocities in a simulation of a hydrodynamic galaxy cluster choudhury2022acoustic, while the right panel shows the magnetic field of a kinetically unstable plasma mode, amplified via a kinetic instability, in a thermodynamically forced particle-in-cell simulation and the wavy arrow indicates that electron trajectories are influenced by the fluctuations in magnetic field. In effect, the bulk flows are sources of free energy that drive kinetic plasma instabilities. These instabilities interact with the electrons and ions and affect the particle distribution, which in turn modifies macroscopic momentum and energy fluxes.
• Figure 2: (a) Time evolution of a particle's parallel (with respect to the magnetic field) momentum with and without TF (temperature-gradient) along with analytical prediction. (b) The spatial drift of the particle due to TF (temperature-gradient only) along the magnetic field. (c) The evolution of one component of the particle's perpendicular momentum with time and comparison with and without TF (bulk-velocity gradient only) along with analytical prediction. (d) The trajectory of the particle in the perpendicular plane to the magnetic field with TF (bulk-velocity gradient). The non-relativistic version is in Appendix \ref{['app2']}.
• Figure 3: Comparison between the (a) analytical ($S_{\rm p}t$), and (b) numerical ($f-f_0$) momentum space anisotropy in the parallel direction produced by TF (temperature-gradient), where $f_0$ and $f$ are the distribution functions at the initial time and a later time $t$, respectively, and $S_{\rm p}$ for a $\beta_e=60$ electron-proton plasma is given by (\ref{['eq:Sp1']}) in section \ref{['sec:alignedgrad']}, in two-dimensional momentum space. In the simulations, resonant dark lines are visible that deviate from straight lines around $p_{\parallel}/m_ec =1$, as expected for relativistic resonance; see (\ref{['eq:reso_gen3']}) in Appendix \ref{['sec:resonance']}.
• Figure 4: Perpendicular out-of-plane component of magnetic field (a) at the onset of whistlers and (b) at saturation for $\beta_e=60$. (c) The spectra of net perpendicular field at saturated stage is shown along with the spectra from previous works (red and blue points from Y25 yerger2024collisionless and RC18 PhysRevLett_roberg-clark respectively) for all three simulations. The whistler spectra is peaked at scales $k \sim \rho_e^{-1}$, and at sub-electron scales is consistent with the $k^{-4}_{\parallel}$ power law (dashed black line) observed in prior research. Here, the notation ${\parallel}$ implies relative to the imposed constant magnetic field or $\boldsymbol{\hat{x}}$.
• Figure 5: The time evolution of (a) box averaged net perpendicular field and (b) parallel heat flux for the three simulations with $\beta_e\in[20,40,60]$. (c) The fitted curve to the saturated parallel heat flux ($1.5\beta^{-1}_e$) with initial $\beta_e$ is shown.