1 Introduction

In 1958, Sweet proposed a model for solar flares in which four sunspots interacted by transferring flux across a common magnetic field (Sweet, 1958b

). The years following this proposal saw extensive developments fleshing out the theoretical underpinnings of the flux transfer in this hypothesized scenario: magnetic reconnection (Sweet, 1958a ; Parker, 1957 ; Petschek, 1964 ; Sonnerup, 1970 ; Vasyliunas, 1975

). Reconnection is a process whereby topological change in some magnetic field lines facilitates the release of magnetic energy. The vast majority of these initial investigations considered a simplified two-dimensional geometry in which topological change occurred at an isolated X-point or neutral point (see Panel a of Figure 1

). As two field lines were brought into contact with the X-point they appeared to “break” into two pieces each to form four separatrices. New field lines were then forged by joining together pieces from opposite originals. The rate at which this topological change was performed was the rate of flux transfer across the X-point, equivalent to the electric field in the ignorable direction at that point.

It was not until 1980, that Baum and Bratenahl revived Sweet’s original three-dimensional quadrupolar configuration to reveal the subtle inter-relation between its two separatrix surfaces, from the field’s positive and negative magnetic null points, intersecting along a single field line, called the separator (Baum and Bratenahl, 1980 ). While this inter-relation and the terms for the constituents were already being used in the magnetospheric literature (Stern, 1973 ; Yeh, 1976 ), Baum and Bratenahl’s computational investigation marked their introduction to solar physics. They used Sweet’s configuration to show how much more complex was the structure of the separator than the simple X-point which was its two-dimensional analog. To understand reconnection it would therefore be necessary to understand how the simple topological change characterized in two-dimensional models was manifest in three dimensions.

The detailed kinematics of reconnection along a separator were tackled in various papers appearing at the end of that decade (Greene, 1988 ; Gorbachev et al., 1988 ; Lau and Finn, 1990 ). Perhaps the most surprising contrast to two-dimensional models was that reconnection in Sweet’s model did not occur at the field’s null point but rather along the separator field line in the corona. There followed an accelerating flow of investigations using this three-dimensional topological picture to interpret solar flares (Gorbachev and Somov, 1988 , 1989 ; Mandrini et al., 1991 , 1993 ; Démoulin et al., 1993 , 1994 ; Bagalá et al., 1995 ; Longcope, 1996 ). These studies clarified how the morphology of solar flares could be interpreted in terms of topology of a three-dimensional magnetic field.

It is not surprising that three-dimensional magnetic fields are more complex than two-dimensional fields. Indeed, even two-dimensional fields can be called complex if they contain structures over a wide range of length scales. Figure 1 shows two instances of hypothetical two-dimensional field lines. The first (Panel a) is a potential field, while the second (Panel b) is more finely structured, containing current on fine scales and is therefore geometrically more complex. The two fields are, however, topologically equivalent since field lines of one may be deformed into the other without breaking them. The topologies of both fields are characterized by one X-point and four separatrices (dark lines) which separate the other field lines into four distinct classes. As the work begun by Sweet has demonstrated, the analogous topological characterization of a three-dimensional field, even one which is geometrically simple, is far more complex than in two dimensions.

Figure 1:

Two-dimensional magnetic fields which are topologically equivalent. Curves show magnetic field lines, and the dark field lines are the separatrices from a magnetic null point (X-point). Panel a: a null point in a potential magnetic field. Panel b: a non-potential field with small scale structure and current, which is nevertheless topologically equivalent to the potential field in Panel a.

The ever-increasing resolution and cadence of coronal imaging instruments, SMM, Yohkoh, SOHO/EIT and TRACE, have revealed the coronal field to be extremely complex. In a parallel development, the increasing power of computers has opened the way to numerical investigation of three-dimensional magnetic fields at ever-increasing resolution. This combination has led to consideration of solar magnetic fields which are ever more complicated both geometrically and topologically. The increased topological complexity poses a challenge rather different from the increasing geometrical complexity, which reflects only a greater range of scales resolved either observationally or computationally. As these challenges of increased topological complexity are met, by the previously arcane terminology of topological field models, terms such as “spine”, “fan”, “separator”, “bald patch” and “quasi-separatrix layer” are gaining broad use among solar physicists of all descriptions.

This article is intended to review and organize the existing body of literature pertinent to the topological analysis of magnetic fields in the solar corona. In its broadest sense “magnetic topology” encompasses a wide range of purely mathematical work, as well as investigations of magnetic fields in the magnetosphere, astrophysical contexts and laboratory experiments. This review will be limited, however, to applications of direct relevance to solar physics. With that aim in mind, an attempt is made to bypass mathematical rigor with an eye toward results and their applications. In most cases the cited literature can provide caveats and justifications whenever they might be desired.

The scope of the article is intentionally limited to topology, and excludes matters of dynamics and energetics whenever possible. This narrow scope is adopted in the interest of providing a thorough and comprehensive treatment of one subject. The field’s topology does play a critical role in determining its dynamics. This relationship is a complex one and is essential to understanding the significance of topology. Nevertheless, the topology itself is complex enough that it is worth reviewing it alone, before considering its possible influence on energetics or dynamics.

By its very definition, the topology of a field is a robust property which will persist when the field is made more geometrically complex through equilibrium and dynamical currents. The topology may be illustrated in simple fields, such as potential magnetic fields, and still be applicable to fields of far greater sophistication. Thus, while much of the literature invokes simple fields for illustration, the topological analyses reviewed here are applicable to a far wider set of magnetic fields. It is this broad applicability which makes topological field analysis so powerful.

The review is intended to be comprehensive in its coverage of topology of three-dimensional fields, since this is the present state of the art. Two-dimensional or two-and-a-half-dimensional fields are essentially special cases of three dimensions, obtained by invoking an additional symmetry. Topological aspects unique to two dimensions are specifically mentioned where there is particular need.

In an effort to be comprehensive and to be useful to students and non-specialists, Sections 2 and 3 review basic elements of magnetic fields and their topology. This review includes a definition of field lines and various types of magnetic null points, and a summary of circumstances where these theoretical concepts are physically significant. It also reviews the methods of field extrapolation by which model fields are usually constructed. This leads naturally to a discussion of field line mapping and to discontinuities in magnetic fields.

The review introduces an organizing framework which accommodates as much existing literature as possible as one coherent body. Such a comprehensive framework has not, as far as can be determined, been presented before. Existing models are sorted into two broad categories according to their modeling of the photospheric field. In one class, termed here magnetic charge topology, the photospheric field is modeled as an intermittent collection of discrete source regions. Field lines anchored in common sources are deemed topologically equivalent. The other class, called here pointwise mapping models, considers a non-intermittent photospheric field which defines a mapping between photospheric footpoints. The two classes have subtly different definitions of such topological features as separatrices. Prior to drawing this distinction it is difficult to reconcile the uses of these terms across the existing literature.

Sections 4 and 5 review literature on the two classes of models, magnetic charge topology and pointwise mapping model, respectively. An important group of models, which we call submerged poles models, combine elements of both types. We review these separately in Section 6. Section 7 reviews literature concerning coronal magnetic null points, which are features common to all three types of models. Finally, Section 8 reviews, very briefly, those topological elements unique to open field lines, as typically found in global coronal models or heliospheric models.