Magnetic braiding and parallel electric fields

The braiding of the solar coronal magnetic field via photospheric motions-with subsequent relaxation and magnetic reconnection-is one of the most widely debated ideas of solar physics. We readdress the theory in light of developments in three-dimensional magnetic reconnection theory. It is known that the integrated parallel electric field along field lines is the key quantity determining the rate of reconnection, in contrast with the two-dimensional case where the electric field itself is the important quantity. We demonstrate that this difference becomes crucial for sufficiently complex magnetic field structures. A numerical method is used to relax a braided magnetic field toward an ideal force-free equilibrium; the field is found to remain smooth throughout the relaxation, with only large-scale current structures. However, a highly filamentary integrated parallel current structure with extremely short length-scales is found in the field, with the associated gradients intensifying during the relaxation process. An analytical model is developed to show that, in a coronal situation, the length scales associated with the integrated parallel current structures will rapidly decrease with increasing complexity, or degree of braiding, of the magnetic field. Analysis shows the decrease in these length scales will, for any finite resistivity, eventually become inconsistent with the stability of the coronal field. Thus the inevitable consequence of the magnetic braiding process is a loss of equilibrium of the magnetic field, probably via magnetic reconnection events.