2.5 Topological changes: Reconnection

Field lines may be found for any magnetic field whatsoever, by integrating Equation (1

) from a set of initial points. Only, however, in cases with frozen field lines (i.e. a perfect conductor) can a given field line be unambiguously followed in time (Newcomb, 1958 ). In the frozen-field-line case each field line moves with the plasma itself. Provided the flow field $v(x,t)$ is reasonably continuous, each field line will be continuously deformed by the flow. Continuous deformation precludes any topological changes. This means, for example, a closed field line will always remain closed and an isolated positive null point will always remain an isolated positive null point. Herein lies the utility of field line topology: Each field line’s topology is preserved by arbitrary plasma motion, provided the plasma is a perfect conductor.

As soon as the assumption of perfect conductivity is abandoned it becomes impossible to unambiguously follow field line from one time to the next, and topology loses its practical utility⁸ . Since plasmas are generally very good at eliminating electric fields it is common to retain the assumption of perfect conductivity everywhere with the possible exception of a few localized regions where $E'/= 0$ . The perfectly conducting approximation can be justified by estimating the magnitudes of each term which might balance $E'$ in the generalized Ohm’s law (see Vasyliunas, 1975 , for one discussion of this scaling). Such estimates assume that all fields, including $B$ , vary on length scales comparable to those of the global geometry. In certain cases, however, the self-consistent dynamical solution will spontaneously develop localized structure on much smaller scales, such as a shock or a tangential discontinuity. Revised estimates using this much smaller length scale reveal that significant electric fields are possible within these localized structures. We will not delve into the literature (already substantial and rapidly expanding) concerning the self-consistent generation of $' E$ due to various terms in Ohm’s law. Instead we will merely assume that it is possible for $E'/= 0$ , but only within localized non-ideal regions. In this modified picture a field line may be followed up to the time it encounters, along some part of its length, a non-ideal region. During this encounter, the field line is “cut” into two distinct pieces, each ending at the outside of the non-ideal region, and moving with the flow outside. When these pieces later decouple from the region each will most likely find itself connected to some other partial field line. This is the topological manifestation of reconnection: Field lines are “cut” and then “reconnected” to other segments (Vasyliunas, 1975 ; Hesse and Schindler, 1988 ). The discontinuous change, the cutting and reconnecting, must occur within the non-ideal region since that is the only place not bound by the frozen flux rule.

While it is not possible to follow a field line identified within the non-ideal region, it is possible to trace field lines into this region, which gain their identity from outside. Doing so provides a view of kinematic reconnection, capturing the topological change in action (Greene, 1988 ; Hesse and Schindler, 1988 ; Lau and Finn, 1990 ; Priest et al., 2003 ). This generalization of field line evolution can provide valuable insight into reconnection, but one should always bear in mind that the field-line motion within the non-ideal region is basically a useful fiction. One consequence of the ambiguity inherent in this fiction is the following. Field lines traced from one side of the non-ideal region evolve differently from those traced from the other side.