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 Once the perturbations grow to sufficiently large amplitudes, the linear theory is no longer applicable.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.

The fundamental equations

We discussed in the previous chapter how the one-fluid or the MHD model of the plasma can be developed starting from microscopic considerations. It was not possible to give as thorough or as systematic a presentation of the subject as we did in Chapter 3, where the hydrodynamic model for neutral fluids was developed from the microscopic theory. We have not rigorously established the conditions under which the one-fluid model of a plasma holds. We saw in Chapter 3 that frequent collisions make a neutral gas behave like a continuous fluid. Collisions certainly help in establishing fluidlike behaviour. It was, however, mentioned in §11.7 that a strong magnetic field in a plasma can also keep charged particles confined within local regions for sufficient time, thereby giving rise to fluidlike behaviour even in the absence of collisions.

Between the microscopic model based on distribution functions and the macroscopic one-fluid model, there exists the intermediate twofluid model of the plasma discussed in Chapters 11–13. This was referred to in Table 1.1 as the 2½ level. When we consider phenomena in which electrons and ions respond differently (such as the propagation of electromagnetic waves through a plasma), the two-fluid model has to be applied rather than the MHD model. The MHD model is applicable only when charge separation is negligible. The condition for it is that the length scales should be larger than the Debye length and the time scales larger than the inverse of plasma frequency.

Introduction

In this chapter, we shall mainly study what happens when a plasma in thermodynamic equilibrium is slightly disturbed. If the effect of collisions can be neglected in studying the evolution of the disturbance, then we call it a collisionless process. In stellar dynamics, one often studies systems with relaxation times larger than the age of the Universe so that collisions have never been important in the system. Here, however, we shall consider plasmas which are close to thermodynamic equilibrium presumably as a result of collisions. But collisions happen to be unimportant in the particular processes we are going to look at.

If a plasma is close to thermodynamic equilibrium, then hydrodynamic models may be applicable as we pointed out in §11.5. We shall mostly use the two-fluid model which was developed in the previous chapter. Only §12.4 will provide an example of how calculations can be done with the help of the Vlasov equation. We shall see that the Vlasov equation can give rise to a conclusion different from what we get by using the two-fluid model due to rather subtle reasons.

We shall mainly consider waves of high frequencies (so that collisions are unlikely to occur during a period of the wave), as such waves are the most important examples of collisionless processes. For low-frequency processes (with frequency ≪ the plasma frequency ωp defined in §12.2), we shall see in later chapters that charge separation can be neglected and the plasma may be regarded as a single fluid.

Fluids and plasmas in the astrophysical context

When a beginning student takes a brief look at an elementary textbook on fluid mechanics and at an elementary textbook on plasma physics, he or she probably forms the impression that these two subjects are very different from each other. Let us begin with some comments why we have decided to treat these two subjects together in this volume and why astrophysics students should learn about them.

We know that all substances are ultimately made up of atoms and molecules. Ordinary fluids like air or water are made up of molecules which are electrically neutral. By heating a gas to very high temperatures or by passing an electric discharge through it, we can break up a large number of molecules into positively charged ions and negatively charged electrons. Such a collection of ions and electrons is called a plasma, provided it satisfies certain conditions which we shall discuss later. Hence a plasma is nothing but a special kind of fluid in which the constituent particles are electrically charged.

When we watch a river flow, we normally do not think of interacting water molecules. Rather we perceive the river water as a continuous substance flowing smoothly as a result of the macroscopic forces acting on it. Engineers and meteorologists almost always deal with fluid flows which can be adequately studied by modelling the fluid as a continuum governed by a set of macroscopic equations.

THE RECORDS OF CO2 ABUNDANCE AND GLOBAL TEMPERATURES IN MODERN TIMES

Ice cores contain trapped bubbles of air which, provided they can be properly dated, represent a record of the composition of air over time. Because of the weight of overlying layers of ice, compressing the pores in the ice, it is very difficult to extend the record back as far as that for temperature derived from the isotopic composition of the water itself. In fact, the manner in which the air bubbles were originally trapped in ice results in their movement upward or downward relative to the ice itself, making age determination a challenge.

Figure 22.1 displays CO2 values from an ice core collected in Greenland. The dating of the air was achieved by taking advantage of a byproduct of nuclear weapons testing: The isotope 14C reached a peak in Earth's atmosphere, from the detonation of nuclear bombs, in 1963. Using this peak in heavy carbon, geochemists M. Wahlen of Scripps Institution of Oceanography and colleagues determined that the trapped air was displaced by the equivalent of 200 years relative to the ice surrounding it.

With this important correction, the figure shows that, during the Little Ice Age, CO2 values were fairly constant. Beginning in the mid-1800s, carbon dioxide began to increase. Direct measurements from a station in Hawaii, selected to be high above any local industries and hence sampling worldwide CO2 borne by the trade winds, show that the increase accelerates after World War II.