2. Minimum energy of topologically complex magnetic fields
Of all field configurations with a given normal component at the solar surface, the potential field has the minimum energy. Any equilibrium field with a current has higher energy. A magnetic configuration of a general type tends to relax to the minimum energy state; in a trivial case to a potential field. The minimum magnetic energy in a non-trivial case depends on topological invariants of the configuration, which do not change in the course of magnetic relaxation (Moffatt 1990). An example of such relaxation is the transition of a twisted flux tube into a writhed (coiled) tube, as happens to telephone cords (for a detailed study of this transition see Ricca (1995)). It is the current that induces topological complexity, such as twist, writhe and linking of field lines-usually described by the invariant called "magnetic helicity". The magnetic helicity does not characterize all of the topological complexity of the field: there are an infinite number of high-order invariants (c.f. Ruzmaikin, Akhmetiev 1994). A simple measure of complex magnetic line entanglement of general nature, called "crossing number", was introduced by Freedman and He (1991). Berger (1993) derived a lower bound for the energy of braided magnetic fields as a function of the crossing number and extended the concept of the crossing number to continuous fields.
A braid is defined to be a collection of curves stretching between two parallel planes (Fig. 1). A one-string braid is topologically trivial. Two-string braids can simulate the twist of magnetic lines around each other. With three or more strings the braid can simulate topologically complicated configurations. We identify the strings with thin magnetic flux tubes and their positions at the lower and upper plane with the positive and negative footpoints of the flux tubes. This represents magnetic loops in the solar atmosphere assuming that the positive footpoints are well separated from the negative ones. What is neglected in such representation is the curvature effects of loops. The complexity of a braid is measured by the number of times the strings cross each other as seen in projection. Because the braids are three-dimensional, this crossing number depends on the viewing point. However, the crossing number averaged over the viewing angle (say, a polar angle in the low plane in Fig. 1) is angle independent (Berger 1993). The averaged crossing number is not in itself a topological invariant. It has, however, a positive minimum (called here K) which is, as well as the minimum energy E, a topological invariant (Freedman, He 1991).
We employ K as a measure of topological complexity inside a magnetic loop of length L and radius r. To simulate solar conditions we assume that the axial field B is much stronger than the perpendicular field. Let be a typical coherence length of the perpendicular field, in the radial direction. Then the loop can contain up to flux tubes (strings) braided about each other. The invariant K tells us the minimum number of times that flux tubes wrap around each other between the two ends.
(Berger 1993), where is the magnetic flux of the braid. To qualitatively understand this relationship, let measure the typical transverse field strength, and be the transverse length of a (curved) tube. Then . Now it takes a transverse distance of about for one tube to wrap around another tube. In projection, however, the tube will be seen to cross other tubes. One tube of length then contributes crossings; and for n tubes
If we express µ in terms of K then consistent with Eq. (1).
© European Southern Observatory (ESO) 1998
Online publication: August 6, 1998