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 the 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 to an ideal force-free equilibrium; that equilibrium is found to be smooth, with only large-scale current structures. However, the equilibrium is shown to have a highly filamentary integrated parallel current structure with extremely short length-scales. 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 a force-free field. Thus the inevitable consequence of the magnetic braiding process is shown to be a loss of equilibrium of the coronal field, probably via magnetic reconnection events.