https://hal-cea.archives-ouvertes.fr/cea-01539178Aly, Jean-JacquesJean-JacquesAlyAIM (UMR_7158 / UMR_E_9005 / UM_112) - Astrophysique Interprétation Modélisation - CEA - Commissariat à l'énergie atomique et aux énergies alternatives - INSU - CNRS - Institut national des sciences de l'Univers - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche ScientifiqueNew formulae for magnetic relative helicity and field line helicityHAL CCSD2017MHDMagnetic helicitySolar and stellar magnetic fields[SDU] Sciences of the Universe [physics]RAINOT, Elisabeth2017-06-14 15:27:412021-10-19 18:59:522017-06-15 11:40:25enPreprints, Working Papers, ...application/pdf1We consider a magnetic field $\bf B$ occupying the simply connected domain $D$ and having all its field lines tied to the boundary $S$ of $D$. We assume here that $\bf B$ has a simple topology, i.e., the mapping $\bf M$ 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 $\bf B$ relative to a reference field $\bf B$$_r$ having the same normal component $B$$_n$ on $S$, and for its field line helicity $h$ relative to a reference vector potential $\bf C$$_r$ of $\bf B$$_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 $\bf M$ and B$_n$ , or in terms of the values on S of a pair of Euler potentials of $\bf B$. We next show that, for a specific choice of $\bf C$$_r$ , the field line helicity $h$ of $\bf B$ fully characterizes the magnetic mapping $\bf M$ 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 $\bf N$ in Ohm's law, are at work in some parts of $D$.