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.