The need for a statistical theory

The linear perturbation theory presented in Chapter 7 makes it clear that a fluid configuration can, under certain circumstances, be unstable to perturbations. Once the perturbations grow to sufficiently large amplitudes, the linear theory is no longer applicable. Hence the linear theory is unable to predict what eventually happens to an unstable fluid system.

To understand the effect of an instability on a general dynamical system, let us employ the notion of a phase space introduced in Chapter 1. Figure 8.1 is a schematic representation of the phase space of a dynamical system, within which let P be a point corresponding to an unstable equilibrium. If the state of the system is represented exactly by P, then the state does not change by virtue of equilibrium. If, however, there is some perturbation around the equilibrium, then the state of the system is represented by some point in the neighborhood of P, and as the perturbation grows, the point in the phase space moves away from P. Thus, depending on whether the initial state was exactly at P or slightly away, the final state after some time can lie in very different regions of the phase space. Because of the limited accuracy in any measurement in a realistic situation, one can only assert that the initial state of a system lies in some finite region of the phase region.

Introduction

In the previous chapter, we developed MHD following a pattern somewhat similar to the pattern followed earlier while developing hydrodynamics. After presenting the basic equations, we first considered the possibility of static equilibrium, and afterwards waves and instabilities were discussed. Although the mathematical analysis in the presence of a magnetic field becomes much more complicated than the corresponding analysis in the pure hydrodynamic case and consequently our discussions in Chapter 14 were often less complete than the earlier corresponding discussions in the pure hydrodynamic case, we have seen that the basic techniques and the methodology were the same.

We now wish to look at a class of MHD problems loosely called topological problems. Let us first consider a situation of ideal MHD, where we have a magnetofluid of zero resistivity. Then, according to Alfvén's theorem, the magnetic field is completely frozen in the plasma. We have pointed out one important consequence of Alfvén's theorem in §14.2. If two fluid elements lie on a magnetic field line, then they would always lie on one field line. We may have two far-away fluid elements in the ideal magnetofluid connected by a magnetic field line. No matter what happens to the magnetofluid or how it evolves in time, this connectivity between the two far-away fluid elements remains preserved if the resistivity is zero. The preservation of such connectivities may introduce some constraints on the dynamics of the system.

Introductory remarks

We now begin the study of plasmas, which are gases in which the constituent particles are charged. In a gas made up of neutral particles, two particles are assumed to interact only when they collide, i.e. are physically very close. Between collisions, the neutral particles move along straight lines. In contrast, the particles in a plasma always interact with each other through long-range electromagnetic interactions and the trajectories of individual particles can be quite complicated. Before investigating how collections of charged particles behave, it is worthwhile developing some ideas about the motions of individual charged particles in electromagnetic fields. This topic is referred as the plasma orbit theory and often turns out to be very useful in handling problems involving plasmas. While developing the theory of neutral gases in Chapters 2–3, it was not necessary to pay much attention to motions of individual neutral particles, as these motions are quite simple. After discussing motions of individual plasma particles in this chapter, we shall, however, follow a course of development roughly similar to that which we followed for neutral fluids. In the next chapter, we shall begin developing theoretical techniques for treating plasmas as collections of charged particles and eventually we shall end up with continuum models in Chapters 14–16.

As soon as we start discussing electromagnetic quantities, we face the vexing question of choosing units.

Thermodynamic properties of a perfect gas

Most of the problems studied in Chapters 4 and 5 did not involve considerations of compressibility, the only exception being §4.4 where the static equilibrium of compressible fluids was considered. We now wish to study the dynamics of compressible fluids. We saw in §4.7 that the irrotational flow of an incompressible fluid around an object gives rise to the Laplace equation, which is an elliptic partial differential equation. The similar problem of high-speed flow of a compressible fluid around an object can give rise to a hyperbolic partial differential equation, provided the flow speed is larger than the sound speed. In other words, the mathematical character of the equations governing high-speed compressible flows can be quite different from that of the equations governing incompressible flows (although we begin from the same hydrodynamic equations!), and consequently the solutions can also be of profoundly different nature. We give here only a brief introduction to gas dynamics, which is the branch of hydrodynamics dealing with compressible flows. Even the subject of supersonic flows past solid objects just mentioned above, which leads to a two-dimensional problem, is not treated in this elementary introduction. See Landau and Lifshitz (1987, Chapter XII) or Liepmann and Roshko (1957, Chapter 8) for a discussion of this subject. We restrict ourselves to a discussion of one-dimensional gas dynamics problems only.

The mathematical analysis of compressible fluids becomes more manageable if we assume the fluid to behave as a perfect gas.