Current sheet formation along three-dimensional magnetic separators

It was shown by Syrovatskii [Sov. Phys. JETP 33, 933 (1971)] that a perfectly conducting two-dimensional magnetic field stressed externally develops a current sheet at an X point. A generalization of this analysis shows that a generic three-dimensional magnetic field develops a closed ribbon of current along separators already existing in the field. A separator is a field line that begins and ends at different magnetic null points. A current ribbon will occur if there are two distinct separators beginning and ending at the same two nulls, thus forming a loop. The magnetic field containing the current ribbon is in force-free equilibrium. An expression is found for the net current induced in the ribbon, and the magnetic energy stored by it, in terms of the amount of external stress. An electric field parallel to the current will facilitate reconnection and will dissipate the current and release the stored energy.