In the previous chapter we considered the idealized case of a cold plasma in which there were no random thermal motions. In this chapter we introduce a more general framework for analyzing plasmas in which thermal motions must be considered. For a system with a large number of particles it is neither possible nor desirable to determine the motion of each and every particle. Instead, we will use a statistical approach to compute the average motion of a large number of particles. This approach is called kinetic theory. It is not our intention in this chapter to present an exhaustive treatment of plasma kinetic theory. Instead we will introduce the basic concepts of kinetic theory and derive a system of equations known as the moment equations. These equations will then be used to analyze various simple applications that are of interest.

The distribution function

To carry out a statistical description of a plasma, it is convenient to introduce a six-dimensional space, called phase space, that consists of the position coordinates x, y, and z, and the velocity coordinates vx, vy, and vz. At any given time the dynamical state of a particle can be represented by a point in phase space. For a system of many particles, the dynamical state of the entire system can then be represented by a collection of points in phase space, with one point for each particle.

Historically magnetohydrodynamics, abbreviated MHD, preceded the development of modern plasma physics. The original intent of MHD was to treat a plasma as a conducting fluid [see Cowling, 1957]. The governing equations were adapted from fluid mechanics with appropriate modifications to account for electrical forces. To obtain a complete set of equations, it was necessary to specify the current as a function of the applied electric field. This was accomplished by using a linear Ohm's law, such as is often used to describe conducting media. Since, to a first approximation, plasmas are electrically neutral, the net charge density was assumed to be identically zero. Also, since fluid motions tend to be slow compared to the characteristic time scales of a plasma, the displacement current was assumed to be small compared to the conduction current. These assumptions, together with an appropriate equation of state, were sufficient to obtain a closed system of equations.

Although it would be adequate simply to write down the equations of MHD as they were originally postulated, it is useful to try to derive the equations from first principles, using the moment equations developed in Chapter 5. Although this procedure will only be partially successful, it has the advantage of revealing more clearly the underlying assumptions and the range of applicability of MHD. It also has the advantage of providing a theoretical basis for precisely defining certain quantities, such as the fluid velocity and the plasma pressure.

In this chapter we give a brief introduction to various types of non-linear effects that can occur in a plasma. Almost all of the basic equations in plasma physics have non-linear terms. For example, these include the (E + v × B) · ∇vf term in the Vlasov equation (5.2.15), the U · ∇U term in the convective derivative (5.4.26), the ρmU term in the MHD mass continuity equation (6.1.32), and the J × B term in the MHD momentum equation (6.1.33). All of these terms represent potential sources of non-linear effects. There are many more. In our analysis of waves in the previous chapters, we always assumed that the wave amplitude was small, so that the governing equations can be linearized. This assumption provides a remarkably accurate description of many types of small amplitude waves. However, if the wave amplitude becomes large, as always occurs for an instability, the linearization assumption breaks down. Non-linear effects must then be taken into account. There are many such non-linear effects, more than we can possibly discuss in this introductory textbook. In order to limit the scope of the discussion, we will concentrate on two quite different types of non-linear analyses that have a wide range of applications. These are (1) quasi-linear theory, and (2) time-stationary electrostatic potentials. For a more comprehensive discussion of non-linear effects in plasmas, the reader is referred to one of the specialized books on non-linear effects, such as Kadomtsev [1965], Sagdeev and Galeev [1969], and Davidson [1972].

In this chapter we investigate the propagation of small amplitude waves in a hot unmagnetized plasma. Because of the shortcomings of the moment equations, the approach used is to solve the Vlasov equation directly using a linearization procedure similar to that used in the analysis of cold plasmas. Although both electromagnetic and electrostatic solutions exist, the discussion in this chapter is limited to solutions that are purely electrostatic, i.e., the electric field is derivable from the gradient of a potential, E = −∇Φ. Electromagnetic solutions are discussed in the next chapter.

From Faraday's law it is easily verified that electrostatic waves have no magnetic component. This greatly simplifies the Vlasov equation by eliminating the v × B force. For electrostatic waves, it is usually easier to solve for the potential rather than for the electric field. Therefore, in the following analysis, the electric field is replaced by E = −∇Φ, and the potential is calculated from Poisson's equation, ∇2Φ = −ρq/∊0.

The Vlasov approach

In an initial attempt to analyze the problem, we assume that normal modes of the form exp(−iωt) exist and represent them by using Fourier transforms, following the same basic procedure used in Chapter 4. This is the approach used by Vlasov [1945], who first considered this problem. As we will see, the Vlasov approach encounters a mathematical difficulty that can only be resolved by reformulating the problem in terms of Laplace transforms.

In this chapter we discuss the propagation of small amplitude waves in a hot magnetized plasma. Just as for a cold plasma, the presence of a static zero-order magnetic field in a hot plasma leads to a wide variety of new phenomena. Because the zero-order motions of the particles in a magnetized plasma consist of circular orbits around the magnetic field, some type of resonance can be expected when the wave frequency is equal to the cyclotron frequency. In a cold plasma, this resonance is the same for all particles of a given charge-to-mass ratio, and gives rise to the well-defined cyclotron resonances described in Chapter 4. In a hot plasma, the frequency “felt” by a particle is Doppler shifted by the thermal motion of the particle along the static magnetic field. For a given parallel velocity, resonance occurs when the frequency in the guiding center frame of reference of the particle is at the cyclotron frequency, i.e., ω′ = ω − k∥ν∥ = ωc. Because of the thermal spread in the particle velocities, the resonance is no longer sharp, as it was in a cold plasma, but is now broadened by the thermal motion. The resonant interaction also produces damping, called cyclotron damping, in a manner somewhat analogous to Landau damping. If the cyclotron radius of the particle is a significant fraction of the wavelength, the phase shift introduced by the periodic cyclotron motion of the particles back and forth along the perpendicular component of the wave vector produces a phase modulation at the cyclotron frequency.

A complete mathematical model of a plasma requires three basic elements: first, the motion of all particles must be determined for some assumed electric and magnetic field configuration; second, the current and charge densities must be computed from the particle trajectories; and third, the electric and magnetic fields must be self-consistently determined from the currents and charges, taking into account both internal and external sources. To be self-consistent, the electric and magnetic fields obtained from the last step must correspond to the fields used in the first step. It is this self-consistency requirement that makes the analysis of a plasma difficult.

To develop an understanding of the processes occurring in a plasma, a useful first step is to forget about the self-consistency requirement and concentrate on the motion of a single particle in a specified field configuration. This approach can be useful in a variety of situations. If the external fields are very strong and the plasma is sufficiently tenuous, the internally generated fields are sometimes small and can be safely ignored. This situation arises, for example, in radiation belts at high energies and in various electronic devices such as vacuum tubes and traveling wave amplifiers. In other situations the self-consistent electric and magnetic fields may be known from direct measurement. In this case, it is often useful to follow the motion of individual tracer particles in the known electric and magnetic fields in order to gain insight into the physical processes involved, such as particle transport and energization.

A plasma is an ionized gas consisting of positively and negatively charged particles with approximately equal charge densities. Plasmas can be produced by heating an ordinary gas to such a high temperature that the random kinetic energy of the molecules exceeds the ionization energy. Collisions then strip some of the electrons from the atoms, forming a mixture of electrons and ions. Because the ionization process starts at a fairly well-defined temperature, usually a few thousand K, a plasma is often referred to as the “fourth” state of matter. Plasmas can also be produced by exposing an ordinary gas to energetic photons, such as ultraviolet light or X-rays. The steady-state ionization density depends on a balance between ionization and recombination. In order to maintain a high degree of ionization, either the ionization source must be very strong, or the plasma must be very tenuous so that the recombination rate is low.

The definition of a plasma requires that any deviation from charge neutrality must be very small. For simplicity, unless stated otherwise, we will assume that the ions are singly charged. The charge neutrality condition is then equivalent to requiring that the electron and ion number densities be approximately the same. In the absence of a loss mechanism, the overall charge neutrality assumption is usually satisfied because all ionization processes produce equal amounts of positive and negative charge. However, deviations from local charge neutrality can occur.

This textbook is intended for a full year introductory course in plasma physics at the senior undergraduate or first-year graduate level. It is based on lecture notes from courses taught by the authors for more than three decades in the Department of Physics and Astronomy at the University of Iowa and the Department of Applied Physics at Columbia University. During these years, plasma physics has grown increasingly interdisciplinary, and there is a growing realization that diverse applications in laboratory, space, and astrophysical plasmas can be viewed from a common perspective. Since the students who take a course in plasma physics often have a wide range of interests, typically involving some combination of laboratory, space, and astrophysical plasmas, a special effort has been made to discuss applications from these areas of research. The emphasis of the book is on physical principles, less so on mathematical sophistication. An effort has been made to show all relevant steps in the derivations, and to match the level of presentation to the knowledge of students at the advanced undergraduate and early graduate level. The main requirements for students taking this course are that they have taken an advanced undergraduate course in electricity and magnetism and that they are knowledgeable about using the basic principles of vector calculus, i.e., gradient, divergence and curl, and the various identities involving these vector operators. Although extensive use is made of complex variables, no special background is required in this subject beyond what is covered in an advanced calculus course.