Overview

We begin this chapter by developing the concept of conservation of particles in phase-space and then use this concept as the basis for establishing the three main models of plasma dynamics, namely Vlasov theory, two-fluid theory, and magnetohydrodynamics (MHD). The Vlasov model is the most detailed and characterizes plasma dynamics by following the temporal evolution of electron and ion velocity distribution functions. The two-fluid model is intermediate in complexity and approximates plasma as a system of mutually interacting, finite-pressure electron and ion fluids. The MHD model is the least detailed and approximates plasma as a single, finite-pressure, electrically conducting fluid. The question of which of these models to use when analyzing a given situation is essentially a matter of selecting the best tool for the task and furthermore, just as a mechanic might alternate between using a screwdriver and a pair of pliers for a specific task, it is often advantageous to alternate between these models when analyzing a specific problem. As we develop these three models, we will also take the opportunity to explore some immediate and important fundamental consequences of these models, most notably the strong dependence of a collisionless plasma on its past history (Vlasov model) and the freezing of magnetic flux into the arbitrarily moving frame of a perfectly conducting plasma (MHD).