Introduction

The basic model describing MHD and transport theory in a plasma is the kinetic-Maxwell equations, which consist of a set of coupled electromagnetic and kinetic equations. The electromagnetic behavior is governed by the full Maxwell’s equations (i.e., displacement current and Poisson’s equation are included). In the kinetic model each species is described by a distribution function fα(r, v, t), which satisfies a 6-D plus t integro-differential equation including the effect of collisions. The equations are very general and very, very difficult to solve. They accurately describe behavior ranging from the fast ωcα and ωpα time scales, down to the slower MHD time scale and the even slower transport time scale.

Since the ideal MHD model is based on the kinetic-Maxwell equations, the first step in the theoretical development of MHD is a derivation of the kinetic equation. A simple heuristic derivation is presented below.

The role of MHD in fusion energy

Magnetohydrodynamics (MHD) is a fluid model that describes the macroscopic equilibrium and stability properties of a plasma. Actually, there are several versions of the MHD model. The most basic version is called “ideal MHD” and assumes that the plasma can be represented by a single fluid with infinite electrical conductivity and zero ion gyro radius. Other, more sophisticated versions are often referred to as “extended MHD” or “generalized MHD” and include finite resistivity, two-fluid effects, and kinetic effects (e.g. finite ion gyro radius, trapped particles, energetic particles, etc.). The present volume is focused on the ideal MHD model.

Most researchers agree that MHD equilibrium and stability are necessary requirements for a fusion reactor. If an equilibrium exists but is MHD unstable the result is almost always very undesirable. There can be a violent termination of the plasma known as a major disruption. If no disruption occurs, the result is likely to be a greatly enhanced thermal transport which is highly detrimental to fusion power balance. In order to avoid MHD instabilities it is necessary to limit the regimes of operation so that the plasma pressure and current are below critical values. However, these limiting values must still be high enough to meet the needs of producing fusion power. In fact it is fair to say that the main goal of ideal MHD is the discovery of stable, magnetically confined plasma configurations that have sufficiently high plasma pressure and current to satisfy the requirements of favorable power balance in a fusion reactor.

Introduction

This chapter presents a discussion of the basic properties of the ideal MHD model. These properties include the general conservation laws satisfied by the model as well several types of boundary conditions that are of interest to fusion plasmas. The discussion demonstrates the physical foundations of ideal MHD while providing insight into its reliability in predicting experimental behavior.

The material is organized as follows. First, a short description is given of the three most common types of boundary conditions that couple the plasma behavior to the externally applied magnetic fields: (1) plasma surrounded by a perfectly conducting wall; (2) plasma isolated from a perfectly conducting wall by an insulating vacuum region; and (3) plasma surrounded by a vacuum with embedded external coils. The most complex of these provides a quite accurate description of realistic experimental conditions.

Second, it is shown that despite the significant number of approximations made in the derivation of themodel, idealMHD still conserves mass, momentum, and energy, both locally and globally. This is one basic reason for the reliability of the model.

Finally, a short calculation shows that as a consequence of the perfect conductivity assumption, the plasma and magnetic field lines are constrained to move together; that is, the field lines are “frozen” into the plasma. This leads to important topological constraints on the allowable dynamical motions of the plasma. In fact the property of “frozen-in” field line topology can be taken as the basic definition of “ideal” MHD.

Introduction

Two-dimensional configurations with toroidal axisymmetry have been investigated in Chapter 6. Many fusion concepts fall into this class – tokamaks of all types, the reversed field pinch, the levitated dipole, the spheromak, and the field reversed configuration. One common feature in each of these concepts is the need for a toroidal current to provide toroidal force balance, either using a perfectly conducting shell or a vertical field.

The need for a toroidal current is of particular importance to the tokamak and RFP, the most advanced of the axisymmetric configurations. The reason is that it is not possible to drive a DC toroidal current indefinitely with a transformer, the method now used in pulsed versions of these configurations. This conflicts with the general consensus that a magnetic fusion reactor must operate as a steady state device for engineering reasons to avoid cyclical thermal and mechanical stresses inherent in a pulsed device. In other words, some form of non-inductive current drive is required. This is an active area of research and while a scientifically sound and technologically viable technique may be possible theoretically, success still depends on current and future experimental development. Overall, non-inductive current drive represents a difficult challenge for the tokamak and RFP concepts.

It has been over 25 years since my original textbook on Ideal Magnetohydrodynamics was published. The book, I believe, was well received by the fusion community but, unfortunately, the publisher has long since gone out of business, making it quite difficult to obtain copies. As a result I have often been asked by students and colleagues to write an updated version of the original book and this volume is the result of that effort. The second volume describing extended MHD will be published in the future.

In writing the book I have found some similarities with my original MHD book but many differences as well. One might hope and expect that a considerable amount of new ideas and discoveries will have been developed over the past 25 years. The overall result is that about a third of the book is closely related to my original textbook, about a third has similar subject titles but has been entirely rewritten, and the final third is completely new. The material is largely based on an evolving course on MHD that I have been teaching at MIT for nearly 25 years.

Introduction

Although the magnetic configurations of fusion interest are toroidal, one can begin to develop physical intuition by first investigating their one-dimensional cylindrically symmetric analogs: the θ-pinch, the Z-pinch, and the general screw pinch. These can be considered to be the basic building blocks of MHD equilibrium. Focusing on cylindrical systems allows the two basic problems of MHD equilibrium – radial pressure balance and toroidal force balance – to be separated, so that each can be studied individually.

The one-dimensional model focuses entirely on radial pressure balance. The question of toroidal force balance does not enter since by definition the geometry is a linear cylinder. For many configurations, once radial pressure balance is established, toroidicity can be introduced by means of an inverse aspect ratio expansion, from which one can then investigate toroidal force balance.

Chapter 5 provides a description of the basic one-dimensional configurations and how they provide radial pressure balance in a plasma. In particular, it is shown that both toroidal and poloidal fields as well as combinations thereof can easily accomplish this goal.

Included in the analysis are descriptions of two present day fusion concepts: the reversed field pinch, and the ohmic tokamak. These configurations are singled out since both their radial pressure balance and MHD stability are reasonably well described by the one-dimensional cylindrical model. Toroidal effects can be treated perturbatively and make small quantitative, but not qualitative, corrections to the cylindrical equilibrium and stability results.

Introduction

Ideal MHD is the simplest model that describes the macroscopic equilibrium and stability of high-temperature fusion plasmas. A self-consistent derivation of the model has been presented in Chapter 2. The derivation requires that one restrict attention to the MHD length and time scales. The main assumptions for validity of ideal MHD are (1) small ion gyro radius, (2) high collisionality, and (3) negligible resistive diffusion. As pointed out, the high collisionality assumption is never satisfied in fusion-grade plasmas, which makes it perhaps surprising how accurate and reliable the model is in predicting experimental behavior.

The goal of Chapter 9 is to begin to address this unsettling situation. While there are a number of alternate MHD models in various regimes of collisionality, the strategy here is to focus on two specific models that describe MHD behavior in the collisionless regime. These are the models that are most relevant to fusion plasmas. Ultimately, in Chapter 10 a set of general stability “comparison theorems” is derived that enables one to make quantitative comparisons between the predictions of the collision dominated and collisionless models. The present chapter, however, focuses solely upon the introduction of the two alternate models. Also presented is a review of ideal MHD which serves as a reference. The specific models discussed are as follows:

• Ideal MHD: collision dominated fluid

• Kinetic MHD: collisionless kinetic plasma

• Double adiabatic theory: collisionless fluid.

Introduction

Chapter 11 describes the MHD stability of one-dimensional cylindrical configurations, specifically the general screw pinch. The analysis involves both the Energy Principle and in some cases the normal mode eigenvalue equation. The goal is to learn about the properties of a magnetic geometry that lead to favorable or unfavorable MHD stability. Even in a cylindrical geometry a great deal of insight can be obtained regarding MHD stability, although there are important toroidal effects that are described in the next chapter.

The discussion begins with the special case of the θ-pinch. Here, a trivial application of the Energy Principle shows that the θ-pinch has inherently favorable stability properties. Also described is “continuum damping” which has many similarities to the well-known phenomenon of Landau damping of electrostatic plasma oscillations. This analysis is simplified by the introduction of the “incompressible MHD” approximation. It is shown that even though the continuum lies entirely on the real ω axis, an initial perturbation will be exponentially damped.

Introduction

Three models have been introduced to investigate the MHD equilibrium and stability properties of a general multidimensional magnetic fusion configuration: ideal MHD, kinetic MHD, and double adiabatic MHD. Ideal MHD is by far the most widely used model although there is concern since the collision dominated assumption used in the derivation is not satisfied in fusion-grade plasmas. The collisionless kinetic MHD model provides the most reliable description of the physics but is difficult to solve in realistic geometries because of the complex kinetic behavior parallel to the magnetic field. Double adiabatic MHD is a collisionless fluid model that is much easier to solve than kinetic MHD but the closure assumptions cannot be justified by any rigorous mathematical or physical arguments.

Based on this assessment one sees that the situation is not very satisfactory from a theoretical point of view. In practice, ideal MHD, because of its mathematical simplicity, is the model that is most widely used to design, predict, and interpret fusion experiments. Many years of experience have shown, perhaps surprisingly, that the model is far more accurate and reliable than one might have anticipated.

Introduction

In the remainder of the book, it is assumed that an MHD equilibrium has been calculated, either analytically or numerically. The next basic question to ask is whether or not the equilibrium is MHD stable. Qualitatively, the question of stability can be stated as follows. The existence of an MHD equilibrium implies a plasma state in which the sum of all forces acting on the plasma is zero. Assume now that the plasma is perturbed from this state producing a set of corresponding perturbed forces. If the direction of these forces is such as to restore the plasma to its original equilibrium position then the plasma is stable. If, on the other hand, the direction of the forces tends to enhance the initial perturbation then the plasma is unstable.

The question of ideal MHD stability is a crucial one, since plasmas, in general, suffer serious degradation in performance, ranging from enhanced transport to catastrophic termination, as a consequence of such instabilities. Not surprisingly, there is consensus in the international fusion community that a plasma must be MHD stable to be viable in a fusion reactor. Indeed, it is fair to say that MHD stability considerations are a primary driver in the design of virtually all the magnetic geometries that have been proposed as fusion reactors.