# New formulae for magnetic relative helicity and field line helicity

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Abstract : We 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 $D$.
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• HAL Id : cea-01539178, version 1

### Citation

Jean-Jacques Aly. New formulae for magnetic relative helicity and field line helicity. 2017. ⟨cea-01539178⟩

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