## 2. Minimum energy of topologically complex magnetic fieldsOf 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
We employ The magnetic energy E stored in the transverse field
is constrained by the complexity (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 If we express © European Southern Observatory (ESO) 1998 Online publication: August 6, 1998 |