The derivation of Boozer coordinates requires several steps of analysis. First, a general transformation is introduced that converts the familiar laboratory coordinate system into a set of arbitrary flux coordinates. Second, by using the relationships B ⋅ ∇ψ = J ⋅ ∇ψ = 0 and ∇ ⋅ B = ∇ ⋅ J = 0, both B and J can be cast into a cross-product form in flux coordinates, close to the desired form of Boozer coordinates. Third, by means of the relation ∇ × B = μ 0J, it is shown that B can also be written in a gradient form in flux coordinates, close to the desired form of Boozer coordinates. Fourth, it is shown how certain free functions appearing in the representation of B can be eliminated by means of an additional transformation of the angular flux coordinates χ, ζ. The new coordinates correspond to the actual Boozer coordinates. Fifth, the various free functions remaining in the expressions for B are rewritten in terms of physically recognizable quantities. Finally, the magnetic field expressed in Boozer coordinates is used to calculate the guiding center drifts of the particles.

Introduction

The discussion so far has focused on single-particle motion in prescribed, long-range electric and magnetic fields as well as short-range Coulomb collisions. No attempt has been made at self-consistency. That is, no attempt has been made to determine how the current density and charge density generated by single-particle motion feeds back and alters the original applied electric and magnetic field. The development of a self-consistent plasma model is the goal of Chapter 10.

Self-consistency is a critical issue. It is important in: (1) providing the physical understanding of the macroscopic forces that hold a plasma together; (2) determining the transport of energy, particles, and magnetic flux, across the plasma; (3) understanding how electromagnetic waves propagate into a plasma to provide heating and non-inductive current drive; and (4) learning how small perturbations in current density and charge density can sometimes dramatically affect the macroscopic and microscopic stability of a plasma.

In developing self-consistent models one should be aware that various levels of description are possible. The most accurate models involve kinetic theory. These strive to determine the particle distribution functions fe (r, v, t) and fi (r, v, t). Kinetic models are very accurate as well as being inclusive of a wide variety of physical phenomena. They are also more complicated to solve and tend to be somewhat abstract with respect to physical intuition. Consequently, with respect to the introductory nature of the book, kinetic theory is considered to be an advanced topic, awaiting study at a future time.

Introduction

The second main application of the MHD model concerns the problem of macroscopic stability. The starting point is the assumption that a self-consistent MHD equilibrium has been found that provides good plasma confinement. The stability question then asks whether or not a plasma that has been initially perturbed away from equilibrium would return to its original position as time progresses. If it does, or at worst oscillates about its equilibrium position, the plasma is considered to be stable. On the other hand, when a small initial perturbation continues to grow, causing the plasma to move further and further away from its equilibrium position, then it is considered to be unstable.

For a fusion reactor MHD stability, particularly ideal MHD stability, is crucial. The reason is that ideal MHD instabilities often lead to catastrophic loss of plasma. Specifically, the plasma moves with a rapid, coherent bodily motion directly to the first wall. The resulting loss of plasma combined with the potential damage to the first wall has led to a consensus within the fusion community that ideal MHD instabilities must be avoided in a fusion reactor.

How are such instabilities avoided? In general, plasma stability is improved by limiting the amount of pressure or toroidal current. However, high pressure is desirable in order to achieve high pτE in a reactor, and high current, as will be shown, is desirable for increasing τE. MHD stability theory is thus concerned with two basic problems.

Introduction

The laws of physics have shown that a large amount of kinetic energy is released every time a nuclear fusion reaction occurs. Determining the conditions under which this energy can be converted into useful societal applications, such as the production of electricity or hydrogen, requires a substantial amount of analysis and is discussed in Chapters 3–5. The logic of the presentation, starting from the end goal and working backwards, is as follows. The desirability of fusion ultimately depends upon the design of practical, economical reactors that have a favorable power balance: Pout ≫ Pin. Two qualitatively different concepts have been proposed to achieve this goal, magnetic fusion and inertial fusion. This book focuses on magnetic fusion.

The end goal of Chapters 3–5 is to present a simple design for a magnetic fusion reactor. In order to develop the design, knowledge of the macroscopic power balance in a magnetic fusion system is required as input. Power balance is discussed in Chapter 4. As might be expected, this analysis involves a variety of physical phenomena representing various sources and sinks of power. Many of these phenomena will be familiar to readers, including thermal conduction, convection, and compression. However, the macroscopic power generated by nuclear fusion reactions, which is clearly the most crucial source term in the system, will probably be less familiar. Similarly, the radiation losses due to the Coulomb interactions between charged particles may also be less familiar.

Plasma Physics and Fusion Energy is a textbook about plasma physics, although it is plasma physics with a mission – magnetic fusion energy. The goal is to provide a broad, yet rigorous, overview of the plasma physics necessary to achieve the half century dream of fusion energy.

The pedagogical approach taken here fits comfortably within an Applied Physics or Nuclear Science and Engineering Department. The choice of material, the order in which it is presented, and the fact that there is a coherent storyline that always keeps the energy end goal in sight is characteristic of such applied departments. Specifically, the book starts with the design of a simple fusion reactor based on nuclear physics principles, power balance, and some basic engineering constraints. A major point, not appreciated even by many in the field, is that virtually no plasma physics is required for the basic design. However, one of the crucial outputs of the design is a set of demands that must be satisfied by the plasma in order for magnetic fusion energy to be viable. Specifically, the design mandates certain values of the pressure, temperature, magnetic field, and the geometry of the plasma. This defines the plasma parameter regime at the outset. It is then the job of plasma physicists to discover ways to meet these objectives, which separate naturally into the problems of macroscopic equilibrium and stability, transport, and heating.

Introduction

The goal of Chapter 13 is to describe the various magnetic configurations currently under investigation as potential fusion reactors. As will become apparent there is a substantial number of concepts to discuss. To succeed, each of these concepts has to successfully overcome the problems not only of MHD equilibrium and stability (p), but also of transport (τE) and heating (T). Even so, it still makes sense to introduce the concepts at this point in the book, immediately following MHD. The reason is that the underlying geometric features that distinguish each concept are primarily determined by MHD behavior. In contrast, transport is a far more difficult issue and significant progress has been made only for the tokamak configuration. With respect to heating, there are several techniques available providing a reasonable number of options. Because of this flexibility, heating can be accommodated in most fusion configurations, and thus is not a dominant driver of the geometry.

To motivate the discussion recall that the main objective of MHD is to discover magnetic geometries that are capable of stably confining sufficiently high plasma pressures to be of relevance to a fusion reactor. The leader for many years in terms of overall performance has been the tokamak which will therefore serve as the reference configuration against which all other concepts must be measured.

Introduction

Power balance considerations have shown that a magnetic fusion reactor should operate at a temperature of about 15 keV and be designed to achieve a value of pτE > 8.3 atm s. Even so, these considerations do not shed any light on the optimum tradeoff between p and τE. Nor do they provide any insight into the geometric scale and magnetic field of a fusion reactor. This is the goal of Chapter 5, which presents the design of a simple magnetic fusion reactor. All geometric and magnetic quantities are calculated as well as the critical plasma physics parameters.

Remarkably, the design requires virtually no knowledge of plasma physics even though for nearly half a century the field has been dominated by the study of this new branch of science. The design is actually driven largely by basic engineering and nuclear physics constraints. These constraints determine the geometric scale of the reactor as well as the size of the magnetic field. Equally important, they make “demands” on the plasma parameters. Plasma physicists must learn how to create plasmas that satisfy these demands (e.g. pressure and confinement time) in order for fusion to become a commercially viable source of energy. Knowledge of the desired plasma parameters is crucial as it defines the end goals of fusion-related plasma physics research, and serves as the guiding motivation for essentially all of the discussion of plasma physics in the remainder of the book.

Introduction

The analysis presented in the previous chapters has established the plasma properties necessary for a magnetic fusion reactor. In particular, a combination of engineering and nuclear physics constraints has shown that a fusion plasma must achieve a temperature T ∼ 15 keV, a pressure p ∼ 7 atm, a plasma beta β ∼ 8%, and an energy confinement time τE ∼ 1 s. Furthermore, the plasma must be confined in the shape of a torus with minor radius a ∼ 2 m and major radius R0 ∼ 5 m. The challenge to the fusion plasma physics community is to discover ways to simultaneously achieve these parameters.

Because the behavior of a fusion plasma can be quite complicated and subtle, as well as being far from our everyday intuitive experiences, it is perhaps not surprising that this has led to the development of a new subfield of physics known as “plasma physics.” Only after knowledge of this new state of matter has been mastered will it be possible to produce robust fusion plasmas suitable for a fusion reactor.

The need to master plasma physics is the motivation for the second part of the book. Presented in these chapters is a description of the plasma physics necessary to produce a fusion plasma. The goal is to provide a reasonably rigorous introduction to the field of plasma physics as viewed from the perspective of a nuclear engineer.

It is over a century since Marconi's famous radio transmission across the Atlantic Ocean, an experiment closely followed by Kennelly and Heaviside's suggestions that an ionized layer in the Earth's upper atmosphere had made it possible. From the first, the ionosphere has been put to use, supporting an increasing range of applications from point-to-point communication and broadcasting, to direction-finding, navigation, and over-the-horizon radar. After 75 years of active research, the ionosphere can hardly be considered one of the mysteries of the Universe, but in fact some scientific problems and technical difficulties do remain. Many of them concern the high-latitude regions, which are particularly subject to disturbances arising initially on the sun.

Since radio propagation depends so strongly on the behavior of the ionosphere, we have tried to bring the two topics together into a single monograph about the polar regions. The early chapters (1–4) provide introductions to the ionosphere in general, to the influence of the magnetosphere, to the principles of radio propagation, and to the major techniques of ionospheric observation. Chapters 5–7 describe the various phenomena of the ionosphere that are peculiar to the high latitudes. The final chapters (8–9) present the results of high-latitude propagation experiments, many of which have been published only in reports that were not widely disseminated at the time or have indeed remained unpublished. Short summaries are included at the end of each chapter to aid readers in getting a quick overview of the material in the chapter.

Introduction

By “aurora” people usually mean the emission of light from the upper atmosphere, but in fact there are numerous related phenomena, each being a direct or indirect consequence of energetic particles entering the atmosphere from the magnetosphere. They include

1. (a) luminous aurora;

2. (b) radar aurora, by which is meant the reflection of radio signals from ionization in the auroral region;

3. (c) auroral radio absorption, the absorption of radio waves in the auroral ionization;

4. (d) auroral X-rays, which are generated by the incoming particles and may be detected on high-altitude balloons;

5. (e) magnetic disturbances, due to enhanced electric currents flowing in the auroral ionization, which may be detected by magnetometers;

6. (f) electromagnetic emissions in the very-low- and ultra-low-frequency bands, which are generated in the magnetosphere by wave–particle interactions (Section 2.5.6), and which then propagate to the ground where they may be detected with a radio receiver or a sensitive magnetometer.

Arising as they do from a common cause, the auroral phenomena display several common properties.

1. They all exhibit a general relationship with solar activity, though often there is no specific association with any obvious solar event. From the 1930s the term M region was used to signify a hypothetical and unseen solar region causing aurora and magnetic storms, and this served as a unifying hypothesis for some 40 years. It is now well appreciated, of course, that the unseen agent is the solar wind.

2. […]

Circulation of the high-latitude F region

Introduction

The high-latitude ionosphere is greatly influenced by the outer magnetosphere and the solar wind, the essential connection being via the geomagnetic field. Through this connection the high-latitude F region is exposed to the interplanetary medium and thence to disturbances originating in the Sun. The circulation of the magnetosphere (Section 2.4.1) establishes a corresponding circulation pattern in the high-latitude F region. Although production by solar EUV is still important, these added features lead to a more complex ionosphere, which exhibits some striking differences both from the middle- and from the low-latitude zones. In describing the F region at high latitude, therefore, we shall be particularly concerned with two underlying factors:

1. (a) the dynamic nature of the high-latitude ionosphere, the pattern of circulation of the F region being mainly controlled by the solar wind and its variations, and

2. (b) the influence of energetic particles from the magnetosphere and the solar wind, to which the region is generally more accessible than is the ionosphere at lower latitudes.

The auroral zones, which occur within the high-latitude region, are particularly complex, and the trough of depleted ionization on its equatorward side has its own pattern of behavior. The present chapter deals with the behavior of the highlatitude F region, its patterns of circulation, and their consequences. The auroral phenomena are discussed in Chapter 6.

Introduction

The purpose of this chapter is to review the basic techniques (and the newer modifications and adaptations of these techniques) for studying the terrestrial ionosphere, with particular emphasis on the capabilities and limitations of the techniques when they are used to probe the high-latitude ionosphere. We are fortunate to have several books and reports written since 1989 that have addressed the general topic of ionospheric investigations using radio techniques (Kelley, 1989; Liu, 1989; Davies, 1990; Hunsucker, 1991; Hargreaves, 1992; Hunsucker, 1993 and 1999; pp. 502–505), so in this chapter we will emphasize the limitations and capabilities of these techniques and update the information on deployment of ionospheric instrumentation at high latitudes. Figure 3.34 of Chapter 3 shows the frequency–height regimes which various selected radio techniques can probe.

Ground-based systems

Ionosondes

In its simplest form, an ionosonde consists of a transmitter and receiver with coupled tuning circuits, which is swept in frequency (usually in the frequency range of approximately 0.5–25 MHz). It can be either a pulsed or a CW-FM (chirp) system, and the transmitter and receiver can either be co-located (monostatic) or separated (bistatic). After the RF signals have been reflected by the ionosphere they are received and processed by the receiver to produce ionograms. The basic information in the received signal is the transit time for passage between ionospheric layers and the Earth, frequency, amplitude, phase, polarization, Doppler shift, and spectrum shape (see Section 3.2.4).

Introduction

The ionosphere and radio-wave propagation

The ionosphere is the ionized component of the atmosphere, comprising free electrons and positive ions, generally in equal numbers, in a medium that is electrically neutral. Though the charged particles are only a minority amongst the neutral ones, they nevertheless exert a great influence on the electrical properties of the medium, and it is their presence that brings about the possibility of radio communication over large distances by making use of one or more ionospheric reflections.

The early history of the ionosphere is very much bound up with the development of communications. The first suggestions that there are electrified layers within the upper atmosphere go back to the nineteenth century, but the modern developments really started with Marconi's well-known experiments in trans- Atlantic communication (from Cornwall to Newfoundland) in 1901. These led to the suggestions by Kennelly and by Heaviside (made independently) that, because of the Earth's curvature, the waves could not have traveled directly across the Atlantic but must have been reflected from an ionized layer. The name ionosphere came into use about 1932, having been coined by Watson-Watt several years previously. Subsequent research has revealed a great deal of information about the ionosphere: its vertical structure, its temporal and spatial variations, and the physical processes by which it is formed and which influence its behavior.

Each of the following titles addresses a good range of the geophysical topics that have concerned us, including, in particular, chapters or articles on the upper atmosphere, the ionosphere and magnetosphere, and the aurora and substorms. They are therefore especially useful as works of general reference. Obviously, each will reflect the state of knowledge at the time it was written. While the more recent should be the most up to date, the older ones should not be neglected for they are closer to the development of the basic ideas and knowledge upon which the field stands today. Mitra's famous book of 1952 is well worth re-reading. The auroral classics by Harang (1951) and Stormer (1955) are cited in Chapter 6.