When electron inertia is the only non-ideal effect in the evolution of a magnetic field B→, the field lines of B→ reconnect, but the lines of a related field B→ do not. B→≡B→+∇→×((c/ωpe)2μ0j→) with ωpe the plasma frequency and j→ the current density. Although a full four-dimensional relativistic calculation of B→ has been made, studies of B→ have been focused on systems that depend on only two spatial coordinates. Three results are given: (1) A relatively simple demonstration in three-dimensional space that the lines of B→ do not reconnect when electron inertia is the only non-ideal effect. (2) The guiding center motion of charged particles is modified by a term that is proportional to (c/ωpe)2, which is smaller than the drifts proportional to the gyroradius unless the current density is extremely large. (3) In three-dimensional space, the evolution velocity of B→ is characteristically chaotic, which means neighboring streamlines separate exponentially on a timescale τu. B→ undergoes large scale reconnection on a timescale that is only an order of magnitude or two longer than τu unless all diffusive non-ideal effects, such as resistivity, are absolutely zero.
Allen H. Boozer (Sun,) studied this question.