3.1 Anchoring and line tying

A particularly simple domain is one where field lines cannot encounter a boundary, either because there are none (the plasma extends infinitely in all directions) or because the field is everywhere tangent to the boundaries ( $n^.B = 0$ ). In such cases field lines can end only at null points, and the vast majority do not end at all. A field line with no end must either form a closed curve, extend to infinity in both directions (when the volume is unbounded) or wander ergodically forever within a fixed volume (Greene, 1992 ). This “endless” situation is a common one in laboratory plasma experiments, which are specifically designed to ensure that $^n .B = 0$ at the experiment wall. Consequently, fusion plasma literature is rife with discussions of periodic, quasi-periodic and ergodic field lines (see, for example, Lichtenberg and Lieberman, 1983 ). Many astrophysical fields can also be considered to be unbounded, however, in these astronomically vast cases, the distinction between a field line which is ergodic and one which closes only after a very long distance is unlikely to be physically meaningful in any of the manners discussed in Section 2.2. For this reason there is less emphasis on topological characterization of astrophysical field lines than in either laboratory or space plasmas. It should be noted that while topology is of little concern, there is a rich literature concerning the structure of galactic fields, as characterized for example by the spectrum of their fluctuation, which has a significant effect on, e.g., the propagation of cosmic rays (Jokipii, 1966 ).

Many space physics applications are modeled as a magnetized plasma with at least one boundary $@V$ where the field’s normal component $Bn = n^.B$ does not vanish. (The surface normal $^n$ will herein be defined to point inward, toward the plasma, rather than outward as other conventions might have it.) The solar corona is the prototypical case, with its lower boundary at the denser layers of the atmosphere. In the simplest coronal models these layers are combined into a single mathematical surface and given the catch-all name “photosphere”. Models of the low corona consider the corona unbounded from above, while some more global models, most notably Source Surface Models (Altschuler and Newkirk Jr, 1969 ; Schatten et al., 1969 , discussed in more depth in Section 8 of this review), include a computational upper boundary to the corona, which the field also crosses. Models of the heliospheric field generally take same source surface as their inner boundary and extend outward to infinity. The present section introduces some basic concepts common to all field models with at least one boundary, but will often refer specifically to the coronal case.

Field lines intersecting the photospheric boundary are said to be anchored and the point of intersection is termed a footpoint. Field lines anchored at both ends to the photospheric boundary are said to be closed⁹ . Closed field lines appear to account for the majority of an active region’s corona. Open field lines, such as in coronal holes, are those with one footpoint in the photosphere and the other end in the source surface or extending to infinity. To interpret the term “open” in cases of real magnetic field lines we recall the physical significance of field line topology discussed in Section 2.2. A field line is open for all practical purposes if it does not return to the photosphere within the mean-free path of high-energy electrons (so no diagnostic could detect its other end) or if it extends beyond the radius of super-Alfvénic solar wind speeds, outside of which dynamical perturbations propagate only outward. A footpoint can have dynamical as well as topological significance. We will draw this distinction by hereafter distinguishing between the concepts of anchoring and line-tying. Anchoring, as just described, refers only to the topology of the field line: It ends at a boundary. This is contrasted to line-tying, where the footpoint is assumed to either remain motionless or to move in a prescribed manner - prescribed independent of the field. According to this usage a field line remains anchored even if its footpoint moves in an unspecified manner across the photosphere. Unfortunately, the literature seems to use these two terms interchangeably, and often fails to distinguish between the topological and dynamical aspects of footpoints. Since the present review is primarily concerned with topology, it becomes necessary to modify the terminology in order to make this important distinction.

A natural reason to consider the photosphere as the lower boundary of the coronal magnetic field is that magnetographs provide spatially resolved measurements at that particular surface. It is beyond the scope of this review to discuss details of the various polarimetric schemes for making magnetograms¹⁰ . Our purpose will be adequately served by assuming that with a magnetogram it is possible to deduce the field’s normal component, $Bn = ^n .B$ , over some portion of the photospheric surface. Many of the widely-used magnetographs measure only the component along the line of sight which matches the vertical only at the center of the solar disk. In these cases the normal field $Bn$ can be derived only after making assumptions about the field’s actual direction. These assumptions become increasingly questionable near the solar limb where the line-of-sight and the vertical become orthogonal to one another. For this reason it is customary to find $Bn$ from line-of-sight measurements only away from the limb.