Ordinary nonmagnetic fluids are known to become turbulent at sufficiently high Reynolds numbers and a similar behavior is expected for electrically conducting magnetized fluids, though direct experimental evidence is scarce. Some confusion may arise, however, owing to the convention, widespread in the fusion research community, of calling the Lundquist number S = LvA/η the magnetic Reynolds number, the latter being correctly defined by Rm = Lv/η, where v is some average fluid velocity. S ≫ 1 simply means that the resistivity is small, while the system may well be nonturbulent, or even static corresponding to Rm ≃ 0. S is an important theoretical parameter characterizing growth rates of possible resistive instabilities. But only when large fluid velocities are generated in the nonlinear phase of an instability or by some external stirring Rm can become large, making the system prone to turbulence. MHD turbulence can thus be expected only in strongly dynamic systems, e.g. disruptive processes in tokamaks or flares in the solar atmosphere.

Though the behavior at Reynolds numbers close to the critical value, where the transition from laminar flow to turbulence occurs, has recently attracted much attention, the strongest interest is in the high-Reynolds-number regime, where turbulence is fully developed, which is characteristic of most turbulent fluids in nature.

Tokamaks constitute the best plasma physics laboratory available today. The largest devices (e.g. JET and DIII-D) confine plasmas of considerable volume (many m3), high densities (ne ∼ 1020 m-3) and high temperatures (Te ∼ 10 keV) under quasi-stationary conditions (for an introduction to the general physics of tokamaks see Wesson, 1987). Tokamak plasmas exhibit a rich variety of MHD phenomena, being investigated by numerous diagnostic tools with high spatial and temporal resolution, which make theoretical interpretation a challenging task.

Particularly conspicuous MHD effects are the different kinds of disruptive events which affect global plasma confinement more or less severely. In this chapter we consider the three most important disruptive processes. Section 8.1 deals with the sawtooth oscillation, a quasi-periodic internal relaxation process, which is observed in most tokamak discharges. Their main effect is to limit the central temperature increase, generating a more uniform average temperature distribution. They also have the beneficial effect of preventing the central accumulation of impurity ions.

Section 8.2 considers major disruptions, which constitute the most violent processes in a tokamak plasma. Disruptions occur when certain limits in the plasma parameters are exceeded, causing loss of a large fraction of the plasma energy, which often leads to the termination of the discharge.

Plasma physics has sometimes been called the science of instabilities. In fact during the last three decades of plasma research, stability theory was probably the most intensively studied field. The reason for this widespread activity is the empirical finding that in general plasmas, especially those generated in laboratory devices, are not quiescent but spontaneously develop rapid dynamics which often tend to terminate the plasma discharge. MHD instabilities are considered as particularly dangerous because they usually involve large-scale motions and short time scales. Though a realistic picture of dynamic plasma processes requires a nonlinear theory, the knowledge of the basic linear instability is usually a very helpful starting point, in particular since linear theory has a solid mathematical foundation.

The organization of the chapter is as follows. Section 4.1 presents the linearized MHD equations. In section 4.2 we consider the simplest case of linear eigenmodes, waves in a homogeneous plasma. The energy principle is introduced in section 4.3. In section 4.4 we then derive in some detail the theory of eigenmodes in a circular cylindrical pinch, which contains many qualitative features of geometrically more complicated configurations. In section 4.5 this theory is applied to the cylindrical tokamak model. The influence of toroidicity, which most severely affects the n = 1 mode, is discussed briefly in section 4.6.

Magnetohydrodynamics (MHD) describes the macroscopic behavior of electrically conducting fluids, notably of plasmas. However, in contrast to what the name seems to indicate, work in MHD has usually little to do with dynamics, or at least has had so in the past. In fact, most MHD studies of plasmas deal with magnetostatic configurations. This is not only a question of convenience — powerful mathematical methods have been developed in magnetostatic equilibrium theory — but is also based on fundamental properties of magnetized plasmas. While in hydrodynamics of nonconducting fluids static configurations are boringly simple and interesting phenomena are in general only caused by sufficiently rapid fluid motions, conducting fluids are often confined by strong magnetic fields for times which are long compared with typical flow decay times, so that the effects of fluid dynamics are weak, giving rise to quasistatic magnetic field configurations. Such configurations may appear in a bewildering variety of shapes generated by the particular boundary conditions, e.g. the external coils in laboratory experiments or the “foot point” flux distributions in the solar photosphere, and their study is both necessary and rewarding.

In addition to finding the appropriate equilibrium solutions one must also determine their stability properties, since in the real world only stable equilibria exist.

Magnetohydrodynamics (MHD) is the macroscopic theory of electrically conducting fluids, providing a powerful and practical theoretical framework for describing both laboratory and astrophysical plasmas. Most textbooks and monographs on the topic, however, concentrate on two particular aspects, magnetostatic equilibria and linear stability theory, while nonlinear effects, i.e. real magnetohydrodynamics, are considered only briefly if at all. I have therefore felt the need for a book with a special focus on the nonlinear aspects of the theory for some time.

In contrast to linear theory which, in particular in the limit of ideal MHD, rests on mathematically solid ground, nonlinear theory means adventures in a, mathematically speaking, hostile world, where few things can be proved rigorously. While in linear stability analysis numerical calculations are mainly quantitative evaluations, they obtain a different character in the study of nonlinear phenomena, which are often even qualitatively unknown. Hence this book frequently refers to results from numerical simulations, as a glance at the various illustrations reveals, but consideration is focused on the physics rather than the numerics.

In spite of the numerous references to the literature the book is essentially self-contained. Even the individual chapters can be studied quite independently as introductions to or current overviews of their particular topics.

There is hardly a term in plasma physics exhibiting more scents, facets and also ambiguities than does magnetic reconnection or, simply, reconnection. It is even sometimes used with a touch of magic. The basic picture underlying the idea of reconnection is that of two field lines (thin flux tubes, properly speaking) being carried along with the fluid owing to the property of flux conservation until they come close together at some point, where by the effect of finite resistivity they are cut and reconnected in a different way. Though this is a localized process, it may fundamentally change the global field line connection as indicated in Fig. 6.1, permitting fluid motions which would be inhibited in the absence of such local decoupling of fluid and magnetic field. Almost all nonlinear processes in magnetized conducting fluids involve reconnection, which may be called the essence of nonlinear MHD.

Because of the omnipresence of finite resistivity in real systems resistive diffusion takes place everywhere in the plasma, though usually at a slow rate. Reconnection theory is concerned with the problem of fast reconnection in order to explain how in certain dynamic processes very small values of the resistivity allow the rapid release of a large amount of free magnetic energy, as observed for instance in tokamak disruptions or solar flares.

The study of linear stability of plasmas had for a long period been carried by the conception that only stable configurations can exist in nature, since instability would lead to destruction of the equilibrium and loss of plasma confinement, which would be the faster the larger the growth rate. Statements like: “all plasmas (meaning real inhomogeneous plasma configurations) are unstable”, sometimes pronounced by plasma theoreticians in the heyday of instability theory, seemed to imply that magnetic fusion research is basically a futile endeavor. The development in experimental plasma physics during the past two decades proved this conception thoroughly wrong. Tokamak discharges may exist, well confined, in spite of the presence of instabilities, which often lead only to a slight change of the plasma profiles and a certain increase of plasma and energy transport (and which may even have beneficial effects such as the removal of impurities by the sawtooth process). Thus in order to judge the effect of an instability it is evidently necessary to calculate or at least estimate its nonlinear behavior, in particular the saturation level. It will turn out that linear mode properties, in particular growth rates, often have little to say about the nonlinear behavior.

As a general rule an instability is found to be the more “dangerous”, i.e. its effect on the plasma configuration is the more detrimental, the longer the wavelength (global modes).

Introduction

The region of the spectrum in the vicinity of 10 μm wavelength is called the thermal infrared. It is important because many materials have strong vibrational absorption bands there (Chapter 3). In most remote-sensing measurements these bands can be detected only through their effects on the radiation that is thermally emitted by the planetary surface being studied. Many substances have overtone or combination bands at shorter wavelengths, and although the latter bands are observed in reflected light, their depths and shapes may be affected by the thermal radiation that is emitted by the material. Hence, even though the primary subject of this book is reflectance, it is important that the effects of thermal emission be discussed. It will be seen that most of the preceding discussions of reflectance also apply to emissivity at the same wavelength because of the complementary relation between the two quantities.

Figure 13.1 shows the spectrum of sunlight reflected from a surface with a diffusive reflectance of 10%, compared with the spectrum of thermal emission from a black body in radiative equilibrium with the sunlight, at various distances from the sun. Clearly, thermal emission can be ignored at short wavelengths, and reflected sunlight at long, but at intermediate wavelengths in the mid-infrared the radiance received by a detector viewing the surface includes both sources.

Scientific rationale

The subject of this book is remote sensing, that is, seeing “with clearer eyes.” In particular, it is concerned with how light is emitted and scattered by media composed of discrete particles and what can be learned about such a medium from its scattering properties.

If you stop reading now and look around, you will notice that most of the surfaces you see consist of particulate materials. Sometimes the particles are loose, as in soils or clouds. Sometimes they are embedded in a transparent matrix, as in paint, which consists of white particles in a colored binder. Or they may be fused together, as in rocks, or in tiles, which consist of sintered ceramic powder. Even vegetation is a kind of particulate medium in which the “particles” are leaves and stems. These examples show that if we wish to interpret quantitatively the electromagnetic radiation that reaches us, rather than simply form an image from it, it is necessary to consider the scattering and propagation of light within nonuniform media.

One of the first persons to use remote sensing to learn about the surface of a planet was Galileo Galilei.

Introduction

Chapter 8 treated the bidirectional reflectance of an optically thick, plane-parallel particulate medium in which the particles were randomly oriented and could be regarded as embedded in a vacuum. In this chapter we will discuss the effects on the reflectance when each of these restrictions is removed.

Diffuse reflectance from a medium with a specularly reflecting surface

The upper surfaces of many particulate materials may be sufficiently smooth on a scale comparable to the wavelength that light is scattered both quasi-specularly from the surface and diffusely from below the surface. The specular component is known as regular reflection. Because a surface effectively becomes more optically smooth at large angles of incidence (Chapter 6), the regular component may become especially important at large phase angles.

The most familiar example of the combination of diffuse reflection and regular reflection is water containing suspended solids, as in rivers, lakes, and oceans. If a body of water is examined in the geometry for specular reflection from the surface, a bright glare is seen, which is the reflected image of the sun. However, if the same body is examined in an off-specular configuration, it looks dark and may be colored blue, brown, or green, depending on the nature of the suspended solids.