Conservation of both current and helicity in a quadrupolar model for solar flares

Abstract

A model for a solar flare, involving magnetic reconnection transferring flux and current between current-carrying magnetic loops connecting two pairs of footpoints, is generalized to include conservation of magnetic helicity during reconnection, as well as conservation of current at all four footpoints. For a set of force-free loops, with the $i$th loop having flux $F_i$ and current $I_i$, the self and mutual helicities are proportional to the self and mutual inductances with the constant of proportionality determined by $\alpha_i=F_i/\mu_0I_i$. In a constant-$\alpha$ model, the change in magnetic energy is proportional to the change in helicity, and conservation of helicity implies conservation of magnetic energy, so that a flare cannot occur. In a quadrupolar model, with $\alpha_1>\alpha_2$ initially, $\alpha_1$ increases and $\alpha_2$ decreases when flux and current are transferred from loops~1 and~2 to loops~3 and~4. A model that conserves both current and helicity is constructed; it depends on the initial $\alpha$s, and otherwise is somewhat simpler than when helicity is neglected.