Large solar flares are probably the most spectacular eruptive events in cosmical plasmas. Though rather weak in absolute magnitude compared for instance with the enormous energies set free in a supernova explosion, they outshine all other cosmic events for a terrestrial observer. According to the generally accepted picture, a flare constitutes a sudden release of magnetic energy stored in the corona and is therefore primarily an MHD process, though the various nonthermal channels of energy dissipation and deposition, which give rise to the richness of the observations, require a framework broader than MHD theory.

Since the major part of this book is concerned primarily with phenomena in laboratory plasmas, it seems to be convenient for the generally interested reader to find a somewhat broader introduction to this astrophysical topic. The engine driving the magnetic activity in the solar atmosphere is turbulent convection in the solar interior. Section 10.1 therefore gives an overview of our present understanding of the convection zone, in particular magnetoconvection. In section 10.2 we consider the solar atmosphere, its mean stratification, the process of magnetic flux emergence from the convection zone and the magnetic structures in the corona, in particular in active regions. In section 10.3 we then focus in on the MHD modelling of the flare phenomenon.

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).

In this chapter we continue to study electromagnetic fluctuations in homogeneous, magnetized, collisionless plasmas. The new element here is that we consider the zeroth-order distribution function of each plasma component to be Maxwellian with drift velocity v0j parallel or antiparallel to B0 (Equation (3.1.3)). If two components have a relative drift v0 greater than some threshold, the corresponding free energy can lead to instability growth. Section 8.1 outlines the derivation of the dispersion equation for this case; Section 8.2 discusses electromagnetic ion/ion instabilities; Section 8.3 addresses electromagnetic electron/electron instabilities; Section 8.4 considers electromagnetic electron/ion instabilities; and Section 8.5 examines the consequences of electromagnetic effects on ion/ion instabilities that are electrostatic in the limit of zero β. Section 8.6 is a brief summary.

Space plasma heating and acceleration processes typically act on both species and are likely to give rise to beam/core distributions for both electrons and ions. However, in contrast to the case of T⊥ j > T‖j discussed in the previous chapter, the instabilities driven by beam/core free energies do not clearly separate into low frequency ion-driven and high frequency electron-driven modes. Thus, although we treat relative ion drifts and relative electron drifts separately in this chapter, this separation is due more to our desire to clarify the presentation than to any compelling physical arguments. Thus, in Sections 8.2 through 8.5, we consider a two-species, three-component plasma consisting of a relatively tenuous beam (denoted by subscript b), a relatively dense core (c), and a third component of the other species.

Every plasma is inhomogeneous to some extent, and the associated plasma gradients are sources of free energy that can drive plasma instabilities. In this chapter we consider the linear theory of drift instabilities, modes driven unstable by a plasma gradient perpendicular to B0.

In the direction parallel to a magnetic field, pressure gradients give rise to electric fields that lead to currents and bulk plasma motion; that is, such gradients do not correspond to a steady-state description under a macroscopic description of the plasma. However, pressure gradients perpendicular to a magnetic field can correspond to a steady-state situation; that is, ∇P in the momentum equation of a one-fluid description of the plasma can be balanced by the J × B0/c term. Nevertheless, such gradients do not correspond to an equilibrium plasma configuration; the zeroth-order distribution functions are non-Maxwellian and lead to the growth of plasma instabilities which act to dissipate the gradients. In this chapter we consider, as before, collisionless plasmas with a uniform zeroth-order magnetic field B0 = ẑB0. In Section 4.1 we discuss a model distribution function for density gradients perpendicular to a uniform magnetic field, examine the associated linear dispersion equation and discuss the two most popular density drift instabilities. Section 4.2 describes the instability properties that result when a plasma with a density gradient is subject to a uniform acceleration, and Section 4.3 briefly summarizes some properties of temperature drift instabilities.

In this chapter we examine, as before, electromagnetic fluctuations in a homogeneous, magnetized, collisionless plasma. In contrast to the previous chapter, however, we admit anisotropies in the distribution functions. In particular we consider a two-temperature bi-Maxwellian zeroth-order distribution function; this permits the growth of temperature anisotropy instabilities. Section 7.1 outlines the derivation of the dispersion equation; Section 7.2 discusses the properties of modes driven unstable by a proton temperature anisotropy, whereas Section 7.3 discusses the properties of electron temperature anisotropy instabilities. Section 7.4 is a brief summary.

Our emphasis in this chapter is on instabilities driven by T⊥j > T‖ a condition that is observed more often in space plasmas than the converse T‖ > T⊥. The reason for this discrepancy is simple: although space plasmas do not necessarily exhibit a bias toward perpendicular heating processes, perpendicular heating does not much change the mobility of the heated particles, whereas parallel heating enables the particles to move more rapidly along B0. Thus parallel-heated particles may leave the region of energization more quickly, implying that T‖ > T⊥ should be a less frequently observed condition. Of course, parallel-heated particles may appear elsewhere as a magnetic-field aligned beam streaming against a cooler background plasma; the electromagnetic instabilities driven by such configurations have quite different properties from temperature anisotropy instabilities, and are studied in detail in the next chapter.