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New formulae for magnetic relative helicity and field line helicity

We consider a magnetic field B occupying the simply connected domain D and having all its field lines tied to the boundary S of D. We assume here that B has a simple topology, i.e., the mapping M from positive to negative polarity areas of S associating to each other the two footpoints of any magnetic line, is continuous. We first present new formulae for the helicity H of B relative to a reference field B r having the same normal component B n on S, and for its field line helicity h relative to a reference vector potential C r of B r . These formulae make immediately apparent the well known invariance of these quantities under all the ideal MHD deformations that preserve the positions of the footpoints on S. They express indeed h and H either in terms of M and B n , or in terms of the values on S of a pair of Euler potentials of B . We next show that, for a specific choice of C r , the field line helicity h of B fully characterizes the magnetic mapping M and then the topology of the lines. Finally, we give a formula that describes the rate of change of h in a situation where the plasma moves on the perfectly conducting boundary S without changing B n and/or non-ideal processes, described by an unspecified term N in Ohm’s law, are at work in some parts of D.