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.

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.