Introduction

It has been well known for many years that standard of living is directly proportional to energy consumption. Energy is essential for producing food, heating and lighting homes, operating industrial facilities, providing public and private transportation, enabling communication, etc. In general a good quality of life requires substantial energy consumption at a reasonable price.

Despite this recognition, much of the world is in a difficult energy situation at present and the problems are likely to get worse before they get better. Put simply there is a steadily increasing demand for new energy production, more than can be met in an economically feasible and environmentally friendly manner within the existing portfolio of options. Some of this demand arises from increased usage in the industrialized areas of the world such as in North America, Western Europe, and Japan. There are also major increases in demand from rapidly industrializing countries such as China and India. Virtually all projections of future energy consumption conclude that by the year 2100, world energy demand will at the very least be double present world usage.

A crucial issue driving the supply problem concerns the environment. In particular, there is continually increasing evidence that greenhouse gases are starting to have an observable negative impact on the environment. In the absence of the greenhouse problem the energy supply situation could be significantly alleviated by increasing the use of coal, of which there are substantial reserves.

The purpose of Appendix B is to present a derivation of the formula for the power radiated by an accelerating charge. The derivation requires a complex analysis. An outline of the analysis is as follows. First, a careful definition is given of the “radiation” component of a specified electromagnetic field in the context of Maxwell's equations. Second, a derivation is presented of the formulas that determine the vector and scalar potentials A and φ arising from a general distribution of current and charge densities in the time dependent case. The resulting integral relations are generalizations of the well-known Biot–Savart law and Gauss' theorem from magnetostatics and electrostatics. Third, A and φ, as well as the corresponding electromagnetic fields E and B, are calculated assuming the charge and current densities correspond to a single accelerating charged particle. Finally, these results are combined to evaluate the outward Poynting flux on a surface far from the particle. The integral of the Poynting flux over this surface yields the desired expression for the power radiated by the accelerating charge.

Definition of the radiation field

Assume the existence of a set of localized, time dependent charges and currents. Far from the sources the electric and magnetic fields decay away with distance. Part of these fields corresponds to the “radiation” field and part to the non-radiating “near-field.” What is the precise definition that distinguishes the radiation component of the field from the components of the near-field?

Introduction

A major goal of this book is to provide an understanding of how magnetic fields confine charged particles in a fusion plasma. As such, one would like to develop an intuition about the detailed behavior of particle orbits in self-consistent magnetic fields. In particular, it must be demonstrated that charged particles stay confined within the plasma and do not become lost drifting across the field and hitting the first wall.

As a first step towards this goal this chapter focuses on the motion of charged particles in prescribed magnetic and electric fields. No attempt is made at self-consistency – for example, to include the currents and corresponding induced magnetic fields resulting from the flow of charged particles. The fields are simply specified as known quantities. They are assumed to be smooth, slowly varying functions in order to be compatible with the requirement that plasmas be dominated by long-range collective effects. The question of self-consistent fields is deferred to future chapters after appropriate models have been developed.

In the process of studying single-particle motion it will become apparent that there is a well-separated hierarchy of frequencies that characterize the different types of motion that can occur. The fastest and dominant behavior corresponds to gyro motion in which particles move freely along magnetic field lines and rotate in small circular orbits perpendicular to the magnetic field. This motion provides perpendicular confinement of charged particles and makes a toroidal geometry necessary in order to avoid parallel losses.

The basic issues of macroscopic equilibrium and stability

The first major issue in which self-consistency plays a crucial role is the macroscopic equilibrium and stability of a plasma. One needs to learn how a magnetic field can produce forces to hold a plasma in stable, macroscopic equilibrium thereby allowing fusion reactions to take place in a continuous, steady state mode of operation. This chapter focuses on the problem of equilibrium. The issue of stability is discussed in Chapters 12 and 13.

The analysis of macroscopic equilibrium and stability is based on a single-fluid model known as MHD. The MHD model is a reduction of the two-fluid model derived by focusing attention on the length and time scales characteristic of macroscopic behavior. Specifically, the appropriate length scale L is the plasma radius (L ∼ a) while the appropriate time scale τ is the ion thermal transit time across the plasma (τ ∼ a/vTi). This leads to a characteristic velocity u ∼ L/τ ∼ vTi, which is the fastest macroscopic speed that the plasma can achieve – the ion sound speed.

The derivation of the MHD model from the two-fluid model is the first topic discussed in this chapter. Also presented is a derivation of MHD starting from single-particle guiding center theory. The purpose is to show that the intuition leading to MHD is indeed consistent with single-particle guiding center motion.

Introduction

In previous chapters a plasma has been defined qualitatively as an ionized gas whose behavior is dominated by collective effects and by possessing a very high electrical conductivity. The purpose of this chapter is to derive explicit criteria that allow one to quantify when this qualitative definition is valid. In particular, for a general plasma, not restricted to fusion applications, three basic parameters are derived: a characteristic length scale (the Debye length λD), a characteristic inverse time scale (the plasma frequency ωp), and a characteristic collisionality parameter (the “plasma parameter” ΛD). The sizes of these parameters allow one to distinguish whether certain partially or fully ionized gases are indeed plasmas, for example a thin beam of electrons, the ionosphere, the gas in a fluorescent light bulb, lightening, a welding arc, or especially a fusion “plasma.”

It is shown in this chapter that for an ionized gas to behave like a plasma there are two different types of criterion that must be satisfied. One type involves the macroscopic lengths and frequencies as compared to λD and ωP. The macroscopic length is defined as the typical geometric dimension of the ionized gas (L ∼ a ∼ R0), while the macroscopic frequency is defined as the inverse thermal transit time for a particle to move across the plasma (ωT ≡ vT/L). Plasma behavior requires a small Debye length λD 《 L and a high plasma frequency ωP 》 ωT.

Introduction

Coulomb collisions are the next main topic in the study of single particle motion. The theory of Coulomb collisions is a basic building block in the understanding of transport processes in a fusion plasma. This understanding is important since the transport of energy and particles directly impacts the power balance in a fusion reactor. An overview of the topics covered in Chapter 9 and their relation to fusion energy is presented below.

An analysis of Coulomb collisions shows that there are two qualitatively different types of transport: velocity space transport and physical space transport. In velocity space transport, Coulomb collisions lead to a transfer of momentum and energy between particles in v⊥, v∥ space that tends to drive any initial distribution function towards a local Maxwellian. Generally there are no accompanying direct losses of energy or particles from the plasma (the mirror machine being an exception). In physical space transport, Coulomb collisions lead to a diffusion of energy and particles out of the plasma. These are direct losses affecting the plasma power balance, with energy transport almost always the more serious of the two.

A comparison of the two types of transport shows that velocity space transport almost always occurs on a much faster time scale. Typically, the transport time in velocity space is approximately (rL/a)2 shorter than for physical space. This chapter focuses on velocity space transport. The important issue of physical space transport is addressed in Chapter 12 after self-consistent plasma models have been developed.

Introduction

Based on the results derived in Chapter 3 it is now possible to assemble all the sources and sinks that contribute to the overall power balance in a fusion reactor. Chapter 4 describes the construction and analysis of such a power balance model. The goal is to determine quantitatively the requirements on pressure, density, temperature, and energy confinement of the D–T fuel so as to produce a favorable overall power balance in a reactor: Pout ≫ Pin.

Clearly, power balance plays a crucial role in determining the desirability of magnetic fusion as a source of electricity. An analysis of power balance determines the ease or difficulty of initiating fusion reactions. Specifically, how much external power must be supplied, either initially or continuously, to produce a given amount of steady state fusion power? The input power required must be sufficiently low in comparison to the output power in order that a large net power is produced – this is the basic requirement of a power reactor.

Basic power balance for a magnetic fusion reactor involves the analysis of the 0-D form of the law of conservation of energy from fluid dynamics. The general procedure for deriving the 0-D energy equation is the first topic discussed in Chapter 4. This is followed by a detailed analysis of the 0-D model, leading to quantitative conditions on the pressure, temperature, density, and energy confinement that must be satisfied in order to achieve a favorable power balance.

Introduction

The study of fusion energy begins with a discussion of fusion nuclear reactions. In this chapter this topic is put in context by first comparing the chemical reactions occurring in the burning of fossil fuels with the nuclear reactions that produce the energy in fission and future fusion power plants. The comparison is then taken one level deeper by describing in more detail the basic mechanism of the fission reaction and the reason why this mechanism is not effective for fusion energy. The discussion does, nevertheless, provide the insight necessary to understand the alternative mechanism that must instead be employed to produce large numbers of nuclear fusion reactions. Several fusion reactions, including the deuterium–tritium (D–T) reaction, are described in detail.

Once the analysis of the issues described above has been carried out one is led to the following conclusion. Both the splitting of heavy atoms (fission) and the combining of light elements (fusion) lead to the efficient production of nuclear energy. The opposing energy mechanisms are a direct consequence of the nature of the forces that hold the nuclei of different elements together. The behavior of these nuclear forces is conveniently displayed in a curve of “binding energy” versus atomic number. A simple physical picture is presented that explains the binding energy curve and why it has the shape that it does. This explanation shows why light or heavy elements are good sources of nuclear energy and why intermediate elements are not.

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.