J. M. Buzzi

Microwave generation by intense relativistic electron beams propagating in a circular waveguide

The stability of circular waveguide modes perturbed by a relativistic electron beam is investigated. The electron beam propagates along the waveguide and is guided by an applied uniform axial magnetic field. The electron beam is represented by a distribution of rigid rotators all having the same energy and Larmor radius rL with their guiding centers centered on a circle or radius Rb around the guide axis. Numerical results are presented giving a general view of the competition between synchronous and cyclotron instabilities for TM and TE modes as a function of rL and Rb.

Ion-Acoustic Waves in Maxwellian Plasmas A Boundary-Value Problem

The history of the study of Landau damping of ion-acoustic waves is an interesting and at times colorful one. The initial concept of so-called Landau damping was introduced by L. D. Landau (1946) who demonstrated theoretically that if the speed of an acoustic wave in a plasma is slightly larger than the average thermal speed of one of the charge species (either ions or electrons) in the plasma, a wave-particle interaction should occur which would lead to a net transfer of energy from the wave to the slower moving particles, thereby causing the wave to be damped.

Waves in a Conductivity-Tensor-Defined Medium A Cold-Plasma Example

The aim of this chapter is to introduce and briefly describe the propagation of electromagnetic waves in a medium where anisotropic conductivity tensor σ can be defined. This approach provides a valid description of the wave phenomena possible in many plasmas of interest. Since such a description can be found in many places [see, for example, Denisse and Delcroix (1961), Stix (1962), Allis, Buchsbaum, and Bers (1963), Quemada (1968), Chen (1974), and Nicholson (1983)], we present here only the cold-plasma theory as an example of the utility of this approach. Special emphasis will be given to the concept of dispersion relations and their importance to wave propagation in plasmas. Also, as will be true throughout the rest of the book, we will routinely use the mathematical techniques of Fourier and Laplace transforms.

Kinetic Theory of Forced Oscillations in a One-Dimensional Warm Plasma

In the present chapter we present the classical theory of forced oscillations of an electron gas. This theory, first given by Landau (1946), corresponds to wave excitation by a transparent grid immersed in a uniform plasma and biased at the plasma potential. As in Chapter 7, we will again use the concepts of free-streaming and collective perturbations in the plasma. We will show that, as first pointed out by Hirschfield and Jacobs (1968), the collisionless damping of macroscopic quantities such as electron density, electric field, and potential, is not always due to an energy exchange between waves and particles. In order to demonstrate this point we will compute the kinetic energy deposited in the plasma for the particular case of dipolar excitation.

Finite-Size-Geometry Effects

In all the models we have studied so far we have assumed that the plasmas are infinite and homogeneous. It is found both theoretically and experimentally, however, that the properties of plasma waves predicted by such models can be strongly modified when studied in plasmas whose dimensions are comparable with the wavelength of the waves. These modifications are called finitesize-geometry effects. In this chapter, we study the propagation of two waves we are already familiar with—electron plasma waves and ion-acoustic waves— along a column of plasma infinite in length but finite in radius, which is supported by a strong magnetic field, and which may be contained inside a waveguide, for example.

Computing Techniques for Electrostatic Perturbations

In Chapter 8 we showed that the plasma response to forced oscillations is given by a Fourier transform which, in the general case, can be evaluated only by numerical methods. In this chapter we discuss three numerical techniques for calculating electrostatic perturbations in a plasma.

Ion-Acoustic Waves with Ion-Neutral and Electron-Neutral Collisions

In Chapter 3 we studied the behavior of electrostatic waves propagating parallel to the applied magnetic field in warm plasmas. We explicitly eliminated any kind of wave damping or instability from our model so that both ω and k were real quantities. We want to continue the study of the preceding chapter, except that we now want to include in our model collisions between charged particles and neutrals, and calculate what effects these collisions have on wave motion in the plasma. Although we will consider only one type of wave explicitly, ion-acoustic waves, this case will be considered in sufficient detail that the reader should experience little difficulty in adapting the procedure to other types of waves in other kinds of plasmas.

Electrostatic Waves in a Warm Plasma A Fluid-Theory Example

In Chapter 2 we studied general wave propagation in an infinite, cold, two-component plasma which was characterized by charged particles having no energy except the energy alternately gained and lost as a result of their participation in the wave motion. In this chapter, we restrict our study to the behavior of electrostatic waves propagating parallel to the applied magnetic field in so-called warm plasmas, which are characterized by charged particles having thermal energy. Whereas, for the cold plasma, we found only one purely electrostatic mode, which was simply a (nonpropagating) normal mode of oscillation of the plasma, we find that in a warm plasma the thermal pressure of the particles makes it possible for more than one purely electrostatic mode to exist and causes these modes to be propagating modes.

The Cookbook Everything You Always Wanted to Know About Fourier, Laplace, and Hilbert Transforms, but Were Afraid to Ask ...

Electromagnetic waves in plasmas, like electromagnetic waves in space and in all other media, must obey Maxwell’s equations. Even for propagation in free space the resulting wave equation is a linear second-order partial-differential equation. Thus, the equations describing wave propagation in a complex medium such as a plasma can be complicated and difficult to solve directly. A technique that is widely used for solving such linear differential equations for the case of infinite homogeneous plasmas involves the so-called Fourier and Laplace transforms. Basically, this technique maps the real-space and time variables (r,t) of the wave—which are differentially related in the wave equation—to a complex space (ω,k), where the variables ω and k are algebraically related. In this way, one arrives relatively easily at an expression for E(ω,k), for example, where E is the electric field of the wave as expressed in (ω,k) space. Application of Fourier-Laplace transforms to other equations that must be satisfied by the wave-plasma system allows other quantities, such as f 1(ω,k) and n 1(ω,k), the first-order perturbations of the particle distribution and particle density, respectively, to be similarly computed in (ω,k) space.

Landau Damping an Initial-Value Problem

As we have seen in the preceding five chapters, many wave phenomena in plasmas can be studied using only the fluid-theory model in which the plasma is considered as two or more interpenetrating fluids (one for the electrons and one for each ion species). The main disadvantage of this model, as has been pointed out several times, is that only the velocity average of each plasma specie can be taken into account, so that any velocity-dependent effects, such as the Landau damping to be discussed in this chapter, are not predicted by the theory. From a mathematical point of view, the main advantage of the model is that it is a much simpler model than the kinetic-theory model to be used in the present and remaining chapters of the book. Because of this relative simplicity, unless velocity-dependent phenomena are of specific interest, the fluid-theory model is always used. For non-velocity-dependent effects, the fluid-theory model and the kinetic-theory model give identical results. As we employ the kinetic theory in the remaining chapters of the book, we shall see that not only are velocity-dependent effects predicted, but that we will recover many of the fluid-theory wave phenomena already found.