Solar flares and coronal mass ejections (CMEs) are closely related phenomena (see Section 5.3.5), and it now seems very likely that they are simply different manifestations of a single, underlying physical process, namely, the release of magnetic energy stored in the magnetic field of the solar atmosphere. In the past there has been considerable controversy about the relation between CMEs and flares. Some authors have argued that flares cause CMEs by creating high enough temperatures to eject both plasma and magnetic field into the interplanetary medium. However, most CMEs are not associated with what is normally considered a flare (Gosling, 1993), and even in those cases that are, the thermal pressure is never enough to force the field open (Low, 2001).

As discussed in the previous chapter, flares occur over a span of energy scales that ranges from very small (microflares at the observable limit) to very large (>1032 ergs). Some time ago, Švestka and Cliver (1992) suggested that the main factor that determines whether a CME will be associated with a flare or not, is the strength of the magnetic field in the erupting region. If the ambient magnetic field strength is weak, then the emitted radiation, although still present, is just too faint to be considered a flare according to the traditional definition (Zirin 1988). According to some models, it is possible to have two CMEs with nearly the same trajectories and speeds but with an order of magnitude or more difference in the peak intensities of their light curves (Reeves and Forbes, 2005).

Preamble

The widespread interest in reconnection results from the fact that it is a fundamental process that occurs in magnetized plasmas whenever the connectivity of the field lines changes in time. Reconnection is most commonly associated with geomagnetic and solar activity because such changes in field line connectivity can be directly observed, but there are many other, less well-known, applications ranging from meteorites and comet tails to accretion disks and galactic jets. Reconnection is also found in laboratory devices that have been built to study the feasibility of controlled thermonuclear reactors, as well as in several experiments that have been specifically designed to study reconnection as a basic plasma process. Those aspects of magnetic reconnection that depend primarily on the topology of the magnetic field tend to be of universal application. However, aspects that depend on the detailed characteristics of the plasma itself, such as its temperature and density, tend to be restricted to the specific application where such characteristics occur. Thus, there is no universal theory that can be applied to all situations.

Basic concepts

The term magnetic reconnection was introduced by Dungey (1953a), who was interested in the problem of particle acceleration in the Earth's magnetosphere. Earlier studies (Giovanelli, 1946; Hoyle, 1949) had considered the acceleration of particles at magnetic neutral points in the presence of an electric field produced by plasma convection, but these studies did not include the magnetic field produced by the current associated with the motion of the particle.

Like most fundamental concepts in physics, magnetic reconnection owes its appeal to its ability to unify a wide range of phenomena within a single universal principle. Virtually all plasmas, whether in the laboratory, the solar system, or the most distant reaches of the universe, generate magnetic fields. The existence of these fields in the presence of plasma flows inevitably leads to the process of magnetic reconnection. As we shall discuss in more detail later on, reconnection is essentially a topological restructuring of a magnetic field caused by a change in the connectivity of its field lines. This change allows the release of stored magnetic energy, which in many situations is the dominant source of free energy in a plasma. Of course, many other processes besides reconnection occur in plasmas, but reconnection is probably the most important one for explaining large–scale, dynamic releases of magnetic energy.

Figures 1.1–1.4 illustrate the rich variety of plasma environments where reconnection occurs or is thought to occur. The evidence of reconnection in laboratory fusion machines such as the tokamak [Fig. 1.1 (a)] and the reversed–field pinch [Fig. 1.1 (c)] is so strong that there is no longer any controversy about whether reconnection occurs, but only controversy about the way in which it occurs (§9.1). However, as one considers environments which are further away from the Earth, the evidence for reconnection becomes more circumstantial. Most researchers who study the terrestrial aurorae [Fig. 1.2(a)] believe that they are directly or indirectly the result of reconnection in the Earth's magnetosphere, but the evidence for similar phenomena in other planetary magnetospheres [Fig. 1.2(b)] is much smaller (§10.6).

When solving partial differential equations, either analytically or numerically, the form, value, and number of the boundary conditions is of crucial importance. Indeed, often much physics is incorporated in the boundary conditions and, in the setting up of a numerical experiment with nonstandard boundary conditions, it is often the implementation of the boundary conditions that causes the most trouble. Petschek's mechanism, in which the boundary conditions at large distances are implicit, has been generalised in two distinct ways by adopting different boundary conditions to give regimes of almost-uniform reconnection (§5.1) and non-uniform reconnection (§5.2). Whereas Petschek's mechanism may be described as being almost-uniform and potential (§4.3), the first of these new families is in general nonpotential and the second is nonuniform. Also, surprisingly late in the day, a theory of linear reconnection was developed, which occurs when the reconnection rate is extremely slow (§5.3).

Almost-Uniform Non-Potential Reconnection

Vasyliunas (1975) clarified the physics of Petschek's mechanism by pointing out that the inflow region has the character of a diffuse fast-mode expansion, in which the pressure and field strength continuously decrease and the flow converges as the magnetic field is carried in. (This characterization of the inflow does not mean that a standing fast-mode wave is present in the inflow, since such a standing wave is not possible in a sub-fast flow.) A fast-mode disturbance has the plasma and magnetic pressure increasing or decreasing together, while a slow-mode disturbance has the plasma pressure changing in the opposite sense to the magnetic pressure.

We introduced briefly in Section 1.3.2 the idea of a current sheet as a narrow region across which the magnetic field changes rapidly. In this chapter we consider in detail the formation of such sheets in a medium where the magnetic field is frozen to the plasma (§1.4), and then in Chapter 4 we describe how they diffuse through the plasma.

There are several ways in which current sheets may form. One is by the collapse of an X-type neutral point (§2.1). Such a formation in two dimensions through a series of static potential field states may be described by complex variable theory, in which the sheet is treated as a branch cut in the complex plane (§2.2). Other techniques are required for three-dimensional axisymmetric fields (§2.2.5), force-free fields (§2.3.1) or more general magnetostatic fields (§2.3.2). The concept of magnetic relaxation as developed by Moffatt is described in Section 2.4, and a self-consistent theory for slow time–dependent formation is discussed in Section 2.5. Finally, two other ways of forming current sheets are described, namely by shearing a field with separatrices (§2.6) and by braiding (§2.7).

X-Point Collapse

As we shall discuss in detail in Section 7.1, an X-type neutral point in a magnetic configuration tends to be locally unstable, provided the sources of the field are free to move (Dungey, 1953).

Magnetic reconnection is a fundamental process with a rich variety of aspects and applications in astrophysical, space, and laboratory plasmas. It is one that has fascinated us both for much of our research careers, so that we have from time to time felt drawn to return to it after working on other topics and to ponder it anew or view it from a different angle.

Indeed, it was reconnection that brought us together in the first place, since one of us (TGF) went to work as a postdoc with the other in 1980 on the subject of reconnection in solar flares. Our initial meeting in Edinburgh at the start of this collaboration was rather amusing, since a friend had misleadingly described Eric Priest as a tall old man with ginger hair, and so the inaccuracy of this description did not exactly help us to find each other in the crowded airport!

At present the whole field of reconnection is a huge, vibrant one that is developing along many different lines, as can be seen by the fact that a recent science citation search produced a listing of 1,069 published articles written on this subject in only the past three years. We are therefore well aware of the impossibility of comprehensively covering the whole field and apologise in advance to those who may be disappointed that we have not found space to discuss their work on reconnection.

This book is devoted almost entirely to magnetohydrodynamic theories of reconnection and does not review the extensive literature on collisionless processes. Such processes are critical for the production of energetic particles by reconnection, but they constitute an enormous topic on which a whole text could easily be written. However, since the production of fast charged particles is one of the main consequences of reconnection in the cosmos, we feel it is important to devote some space to a brief discussion of this subject.

As we shall see, many particle acceleration theories rely, either implicitly or explicitly, on MHD concepts to supply information about the large-scale distribution of magnetic and electric fields. In the simplest theories (§13.1), the energetic particles are directly accelerated by an electric field whose behaviour is assumed to be known. In this case MHD models are often used to justify the particular choices made for the fields. However, even in more complex theories, MHD concepts such as turbulence (§13.2) and shock waves (§13.3) are often invoked. MHD theory is complemented by kinetic theory, which provides a great deal of additional information about the local plasma behaviour.

Energetic particles are common throughout the universe and reveal at once that it is not as a whole in thermodynamic equilibrium. They show up, for example, as cosmic rays incident on our atmosphere; they produce synchrotron emission from distant radio galaxies; and they are detected in the solar wind upstream of shocks produced by the Earth's magnetosphere and by coronal mass ejections.

The theory of reconnection in two dimensions is now fairly well understood and is highly developed, and, as we have seen, the type of reconnection that is produced depends very much on the reconnection rate, the configuration, the boundary conditions, and the parameter values. Many questions do remain, however, such as: what are the properties of turbulent or impulsive bursty reconnection; why does the diffusion region in Petschek reconnection lengthen when the resistivity is uniform; what is the effect of outflow boundary conditions on fast reconnection; how does reconnection occur in a collisionless plasma; and how do the different terms in the energy equation such as radiation and conduction affect reconnection?

The theory of three-dimensional reconnection is much less developed. We have only just started a voyage of discovery that will last many years, but some important directions have already been indicated. Many features are quite different in three dimensions. For example, we discuss here the definition of reconnection (§8.1), the structure of null points (§8.2), the nature of the bifurcations (§8.3), the global magnetic topology (§8.4), and the nature of the reconnection itself (§§8.6, 8.7).

In this chapter we introduce several new concepts. At null points, magnetic reconnection can take place by spine reconnection, fan reconnection, or separator reconnection (§8.6). Regions where magnetic field lines touch a boundary and are concave towards the interior of the volume are referred to as bald patches (§8.4.1). When no null points or bald patches are present, the mapping of field lines from one boundary to another is continuous, so they all have the same topology and there are no separatrices.

Introduction

In a conducting medium a typical current sheet tends to diffuse outward at a slow rate with a time-scale of τd = l2/η, where 2l is the width of the current sheet and η = (μσ)-1 is the magnetic diffusivity. During the process of magnetic diffusion, magnetic energy is converted ohmically into heat at the same slow rate. However, in practice, the magnitude of τd is often far too large to explain the time-scale of dynamical cosmic processes. Nevertheless, Furth et al. (1963) showed how the diffusion can drive three distinct resistive instabilities at a rate which is often fast enough to be physically significant. These instabilities occur when the sheet is wide enough that τd ≫ τA, where τA = l/υA is the time it takes to traverse the sheet at the Alfvén speed υA = B0(μρ0)-½. The instabilities occur on time-scales τd(τA/τd)λ, where, and they have the effect of creating in the sheet many small-scale magnetic loops. In other words, resistive instabilities produce current filaments in current sheets (or, indeed, in any sheared structure) subsequently, the filaments and associated magnetic loops diffuse away, releasing magnetic energy in the process.

The gravitational and rippling modes (§6.3) occur when the density or resistivity varies in the direction across the sheet. They create a small-scale structure in the sheet (Fig. 6.1) and so are relatively harmless as far as the large-scale global stability of the configuration is concerned, although they may produce a turbulent diffusivity.