Weakly collisional electrostatic simulations for laboratory astrophysics applications

Abstract

Modern plasma physics research is increasingly realizing the potential for cutting edge laser-produced plasma experiments to be exploited in order to investigate astrophysically relevant phenomena. Of particular interest are collisionless shocks, and the physics of the weakly collisional regime generally. While ubiquitous in nature, collisionless shocks remain relatively poorly understood. Recent and ongoing experiments are proving increasingly successful at isolating and studying these astrophysically relevant plasma processes in a controlled laboratory setting. However, simulations are also essential to increase our understanding in this area, and this mandates the development of a suite of numerical techniques for both theoretical and experimental analysis. This thesis presents the development of an original and novel numerical approach to accurately model high energy plasma dynamics, with arbitrary collisionality. This approach is particularly suited to modelling the weakly collisional regime. The code developed to implement this novel simulation method adopts a hybrid approach, in that it solves the Vlasov-Fokker-Planck equation exactly for the ions by exploiting recently developed pseudo-spectral methods, while applying a fluid model to the electrons. With this code it is possible to perform a detailed study of wave and shock dynamics in the weakly collisional regime. Through insight gained from these simulations, along with experimental data, our current understanding of collisionless shocks can be greatly advanced. Chapter 2 introduces essential basic plasma physics concepts which are prerequisites to understanding the principles of the computational techniques presented in this thesis and plasma simulation in general. Chapter 3 expands on this, giving an overview of standard shock physics, which is valuable insight to possess when seeking to understand, interpret, and design shock physics simulations. Chapter 4 outlines a hydrodynamical computational scheme developed in order to grasp the fundamentals of plasma uid modelling. This was a necessary step since a hybrid code which exactly solves the VFP equation must accurately capture both kinetic microscale and bulk uid motions. Therefore a valid hydrodynamical model was a useful resource to verify results against. Chapter 5 presents the main theory behind the numerical approach taken to the ions in the simulation method developed for this thesis, detailing the equations of the kinetic description of plasmas, as well as collision dynamics. It introduces the concept of expanding the distribution function in terms of some basis functions. Also discussed is the importance of the terms one includes in the generalized Ohm's law to the physics the scheme can capture. Chapter 6 lays out the discretized numerical approach to computing the solution of the relevant equations, explains the overall structure of the code, and describes the different algorithms used for each individual operator. It also explains the reasons for the choice of each technique, their advantages, and limitations. Chapter 7 then presents the main innovations in our novel numerical approach, constituting major improvements to existing methods. These include an operator-splitting approach to solving the equations; a reformulation of the expanded VFP equation by variable substitution; a new approach to the isotropic collisional operator that addresses well-established problems with the low-velocity boundary; and a powerful approximation that can be used with the anisotropic collisional operator which improves the speed of the simulation drastically. Chapter 8 shows that this simulation method can capture various physical phenomena and waves. We explore the relationship between the physics that can be captured and the order of the expansion of the Vlasov equation. Then we perform some basic studies of the weakly collisional regime, showing the role of collisions in the formation and propagation of waves and shocks. We show that weak collisions can keep the growth of anisotropy in a system in check, so that a low-order of distribution function expansion is sufficient to capture and model a collisionless shock.