The fourth “Canary Islands Winter School of Astrophysics” was held in Playa de las Américas (Adeje, Tenerife) from the 7th to 18th December 1992, organized by the Instituto de Astrofísica de Canarias and the Universidad Internacional Menéndez-Pelayo. A total of 85 participants from 21 countries world-wide attended the meeting.

This volume contains a series of nine courses which were delivered during the School. The aim of the lectures was to portray a thorough, up-dated view on the field of Infrared Astronomy.

The last School dedicated to Infrared Astronomy before this one (the legendary ISA Course held in Erice, Italy) took place in 1977, fifteen years ago. Since then, dramatic changes in our understanding of the Infrared Universe -pushed forward by the corresponding advances in telescopes, instruments and detector capabilities- have strongly influenced all branches of Astrophysics. One of the primary goals at the present School was to put the new generation of astrophysicists into contact with the extremely important new findings which have surfaced from recent infrared research, and to present and discuss the fundamental physical ideas emerging from those results.

We wish to express our gratitude to the Commission of the European Community (Human Capital and Mobility Programme -Euroconferences), the Dirección General de Investigación Científica y Técnica (Spanish Ministry of Education and Science) and the Government of the Canary Islands for helping us to fund the School. They, together with HOTESA and their “Hotel Gran Tinerfe”, made it possible to allocate grants to over 75% of attendees.

INTRODUCTION

The mathematical difficulties in deriving solutions for growing magnetic fields in spherical geometries have long puzzled dynamo theoreticians. In contrast to the solutions of the kinematic dynamo problem found by Roberts (1970, 1972) and others in the case of periodic velocity fields in infinitely extended electrically conducting fluids, dynamo action often seems to disappear as soon as insulating boundaries are introduced. Motivated by this observation Bullard & Gubbins (1977) have investigated kinematic dynamos in a spherical domain of constant conductivity with insulating exterior for velocity fields with alternating signs as a function of radius. As expected the critical magnetic Reynolds number decreases significantly as the number of sign changes increases and the limit of a periodic velocity field is approached. When the radial velocity component does not change sign, a dynamo solution was not obtained unless an outer shellular region of finite conductivity was introduced.

INTRODUCTION

It is widely though erroneously believed that one can see the Milky Way Galaxy. In fact, one's image of the Milky Way depends more on how one looks at it than on what is available to be seen. For reasons which are related to population biology more than to astrophysics, our eyes are optimised to detect the peak energy output from thermal sources with a surface temperature near 6000K. Thus, unless such an object is typical of the entire contents of the Galaxy, there is no reason why we should be able to see by eye a representative part of whatever may be out there. If we had X-ray or UV sensitive eyes we would ‘see’ only hotter objects, if infrared or microwave eyes only cooler objects.

No single section of the electro-magnetic spectrum provides the ‘best’ view of the Galaxy. Rather, all views are complementary. However, some views are certainly more representative than are others. The most fundamental must be a view of the entire contents of the Galaxy. Such a view would require access to a universal property of matter, which was independent of the state of that matter. This is provided by gravity, since all matter, by definition, has mass. Mass generates the gravitational potential, which in turns defines the size and the shape of the Galaxy. While the most reliable and comprehensive, such a view is also the hardest to derive. Nonetheless, we will repeatedly return to the gravitational picture of the Galaxy in these lectures.

INTRODUCTION

The solution of model Z has been found in many cases with account taken of both viscous and electromagnetic core-mantle coupling (Braginsky 1978; Braginsky & Roberts 1987; Braginsky 1988; Braginsky 1989; Cupal & Hejda 1989). Apart from Braginsky (1989), the time evolution of the solution was used simply as an aid to obtain the steady-state solution. Cupal & Hejda (1992) found numerically a transient solution of model Z having the form of a decaying oscillation. The accuracy of such solutions depends on the numerical method used, on the density of space and time discretization, and for that matter, on the character of the solution itself. An important question is which characteristics of the time behaviour of the solution reflect the real (physical) behaviour of the system and which follow from the limitations of the numerical method.

MOTIVATION

In the absence, so far, of fully hydrodynamic dynamo models representative of the Earth or the planets, the main focus of planetary dynamo theory remains with (axisymmetric) mean-field dynamo models in which the contribution from the nonlinear interaction of the non-axisymmetric components of the problem are parameterized through a prescribed α-effect (see for example Roberts 1993).

MOTIVATION

The interaction between convection and magnetic fields plays a central role in the theory of stellar dynamos. In order to investigate this interaction in detail, we consider a simplified problem: two-dimensional convection in a vertical magnetic field. To represent the astrophysical situation, in which there are no sidewalls, we consider a box with periodic boundary conditions in the horizontal direction, allowing horizontal flows. It is found that convection can be unstable to a horizontal shearing motion.

INTRODUCTION

The study of passive scalar advection provides fundamental understanding of various phenomena such as convection and mixing that are ubiquitous in nature. In particular, the probability distribution of amplitude and its spatial gradients are of vital importance in relation to recent active studies of non-Gaussian probability density functions (PDFs) endemic in turbulence.

Since Castaing et al. (1989) reported exponential-like tails on the PDF of temperature fluctuations in thermal convection at very high Rayleigh numbers, there has been increasing interest in the mechanism of the non-Gaussian tails on PDFs of amplitudes. In a recent paper, Pumir, Shraiman & Siggia (1991) have suggested that the non-Gaussian tails for an advected passive temperature field may be induced by the presence of a mean-temperature profile. A simple physical mechanism for this is proposed in the present paper. The resultant non-Gaussian statistics will be shown by numerical simulations and theoretical analysis for a transport equation without molecular diffusion. In this paper, the result on PDFs is summarized; other details will be presented elsewhere (Kimura & Kraichnan 1993).

This volume contains papers contributed to the NATO Advanced Study Institute ‘Theory of Solar and Planetary Dynamos’ held at the Isaac Newton Institute for Mathematical Sciences in Cambridge from September 20 to October 2 1992. Its companion volume ‘Lectures on Solar and Planetary Dynamos’, containing the texts of the invited lectures presented at the meeting, will appear almost contemporaneously. It is a measure of the recent growth of the subject that one volume has proved insufficient to contain all the material presented at the meeting: indeed, dynamo theory now acts as an interface between such diverse areas of mathematical interest as bifurcation theory, Hamiltonian mechanics, turbulence theory, large-scale computational fluid dynamics and asymptotic methods, as well as providing a forum for the interchange of ideas between astrophysicists, geophysicists and those concerned with the industrial applications of magnetohydrodynamics.

The papers included have all been refereed as though for publication in a scientific journal, and the Editors are most grateful to the referees for helping to get all the papers ready in such a short time. They also wish on behalf of the Scientific Organising Committee to record their appreciation of the dedication of the staff of the Isaac Newton Institute, who coped cheerfully with many bureaucratic complexities, and to give special thanks to the Deputy Director, Peter Goddard, for making the whole meeting possible.

INTRODUCTION

Given the existence of a naturally occurring magnetic field, be it astrophysical or geophysical, it is natural to ask whether the field is generated by dynamo action or if instead it is a fossil field, trapped in the body since its formation. In certain contexts it is possible to give a definitive answer. For example, the Ohmic diffusion time of the Earth's core is of the order of 10 years whereas paleomagnetic records show that the magnetic field of the Earth has existed for 109 years. Consequently, since the field has been maintained for so many Ohmic decay times it must be generated by some sort of dynamo process. For astrophysical bodies on the other hand, for which typically the Ohmic time is comparable to the lifetime of the body itself, it is not so straightforward to assert that a field is dynamo-generated. Of course, there may be other factors suggesting the origin of the field, but simply on the basis of the Ohmic decay time the issue often cannot be decided. What we would like therefore is a test to distinguish between these two possibilities.

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.