Formation of current sheets in magnetic reconnection

An ideal evolution of magnetic fields in three spatial dimensions tends to cause neighboring field lines to increase their separation exponentially with distance ℓ along the lines, δ(ℓ)=δ(0)eσ(ℓ). The non-ideal effects required to break magnetic field line connections scale as e−σ, so the breaking of connections is inevitable for σ sufficiently large—even though the current density need nowhere be large. When the changes in field line connections occur rapidly compared to an Alfvén transit time, the constancy of j||/B along the magnetic field required for a force-free equilibrium is broken in the region where the change occurs, and an Alfvénic relaxation of j||/B occurs. Independent of the original spatial distribution of j||/B, the evolution is into a sheet current, which is stretched by a factor eσ in width and contracted by a factor eσ in thickness with the current density j|| increasing as eσ. The dissipation of these sheet currents and their associated vorticity sheets appears to be the mechanism for transferring energy from a reconnecting magnetic field to a plasma. Harris sheets, which are used in models of magnetic reconnection, are shown to break up in the direction of current flow when they have a finite width and are in a plasma in force equilibrium. The dependence of the longterm nature of magnetic reconnection in systems driven by footpoint motion can be studied in a model that allows qualitative variation in the nature of that motion: slow or fast motion compared to the Alfvén transit time and the neighboring footpoints either exponentially separating in time or not.