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 ith loop having flux Fi and current Ii, the self and mutual helicities are proportional to the self and mutual inductances with the constant of proportionality determined by αi = Fi/μ0Ii. In a constant-α 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 α1 > α2 initially, α1 increases and α2 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 αs, and otherwise is somewhat simpler than when helicity is neglected.