Title:

An exploration of the effects of radiation reaction on waves propagating through a warm plasma

In this thesis we consider the implications of radiation reaction for the behaviour of electric and electromagnetic waves propagating through a plasma. A plasma contains a very large number of particles, and obtaining a description of the dynamical behaviour of each individual particle is impractical. In Section 2 we detail how one can model such a plasma by treating the plasma as a fluid, and rather than examining the individual particles we instead look at the bulk properties of the fluid. Such a model is based upon the equation that describes the motion of a single particle, hence we introduce this first in Section 1. As such, Section 1 should be viewed as an introduction to the necessary background one needs in order to understand the subsequent sections. We begin by reviewing the Lorentz force equation, from which one can determine the motion of a charged point particle in the absence of radiation reaction. Finding solutions to this equation, and plotting the particle's subsequent trajectory not only allows us to introduce notation that will be used throughout this thesis, but is also a point of comparison that can be referred back to in the subsequent sections. The concept of radiation reaction is first introduced in Section 1.1.2, where we learn that the Lorentz Force Law does not describe the motion of a charged particle completely. It is here that we describe the origin of radiation reaction, as well as introducing the AbrahamLorentzDirac (ALD) equation, the first covariant equation derived that determines the motion of a particle when the effects of radiation reaction are included. Not only is this equation of great historical significance, but it is used in the derivation of the equations of motion of Section 2.2, and hence is pivotal to the work carried out within. It is well known that not all solutions to the ALD equation are physically reasonable. Considering the importance of the ALD equation in this thesis, it is prudent to review this property, and we do so in Section 1.1.2. Many alternative models to the ALD equation have been proposed in an attempt to eliminate these unwanted solutions. We must also attempt to eliminate such unwanted behaviours from our solutions, and so Section 1.1.3 reviews the approach used to generate the LandauLifshitz (LL) equation. We carry out a similar procedure with our equations, and so a review of the LL equation is called for. This completes the necessary background in radiation reaction, however we still need to introduce some fundamentals of a plasma. All of the work within this thesis is carried out within the warm fluid approximation, which we detail in Section 2.2. This allows us to use a perturbative approach when seeking solutions; we assume that the solution we seek is that of the cold fluid, plus a small correction term. As such, it is necessary to first review the properties of a cold plasma. We end the introductory section with an example of where experiments are currently taking place that involve electromagnetic waves travelling through plasma. Section 2 reviews the creation of the model we use in our description of a plasma. In Section 2.2 we build upon a recently developed kinetic model of a collection of charged point particles that incorporates radiation reaction. From this model, we proceed to generate an infinite hierarchy of moment equations that describes our system. We subsequently introduce a new closure mechanism to this model, inspired by closure mechanisms associated with the warm fluid approximation, thus obtaining a finite system of equations. Although this kinetic model has been used previously to generate a system of moment equations, the method used to close them was ad hoc, and the solutions predicted by the fluid model did not match up with that predicted by the kinetic model itself. The closure mechanism we use is a simple extension of that of the warm fluid, and needs no additional assumptions regarding the nature of the system. Additionally we show that the results it predicts are identical to those derived directly from the kinetic theory upon which it is based. Hence, the finite system of moment equations we derive is new work. The remainder of Section 2 is focussed on using this model to determine the bulk properties of such a fluid in equilibrium (solutions which we perturb around in subsequent sections) and also represents entirely new work. In Sections 3 and 4 we use what we have learned in the previous sections to model small amplitude electric and electromagnetic waves propagating through the plasma. We examine the dispersion relations of such waves, as well as (when possible) how such waves modify the bulk properties of the plasma. Finally, in Section 5, we turn our attention to electric waves of arbitrarily strong amplitude. Sections 3  5 represent entirely new work.
