On the relaxation of braided magnetic fields

We discuss the properties of braided magnetic fields, in particular with regard to their relaxation towards equilibrium both in the absence and presence of finite plasma resistivity. We consider a model magnetic field in which a subset of field lines are braided in the mathematical sense (in the form of a pigtail braid), and stretch between two perfectly conducting plates. An ideal relaxation yields an approximate force-free equilibrium without thin current sheets. However, in the presence of a finite resistivity the field spontaneously develops thin (but finite) current layers. Subsequently, the system undergoes a fast relaxation on an Alfvénic timescale via a myriad of thin current layers. The current layer structure becomes increasingly complex, and is increasingly long-lived, for higher magnetic Reynolds numbers. The effect of the resistive relaxation is to unbraid the magnetic field, that is to significantly simplify the mapping of field lines between the end plates. This process involves magnetic flux being reconnected multiple times. The final state of the resistive relaxation approximates a non-linear force-free field, implying the presence of additional constraints beyond the magnetic helicity on the relaxation process. The 'topological dissipation' hypothesis, which proposes to explain the heating of the solar corona via braiding of the coronal field by photospheric motions, is one of the most controversial ideas of solar physics. We therefore consider in some detail the relevance of our results to this debate.