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
utility^{8}.
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 . The
perfectly conducting approximation can be justified by estimating the magnitudes of each term which might
balance in the generalized Ohm’s law (see Vasyliunas, 1975, for one discussion of this scaling). Such
estimates assume that all fields, including , 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 due to various terms in Ohm’s law.
Instead we will merely assume that it is possible for , 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.

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