A mathematical model of solar flares

Abstract

The phenomenon of solar flares is modeled assuming that the magnetic field is force-free and that its evolution is quasi-static. This model is simplified so as to be tractable and yields a semi-linear elliptic equation in a halfplane depending on a parameter λ \lambda which describes the time evolution. It is proved that there are (at least) two branches of solutions which have distinct asymptotic behaviors at infinity. The upper branch exists for all λ > 0 \lambda > 0 , but the lower branch exists only on a finite interval [ 0 , λ c ] \left [ {0,{\lambda ^c}} \right ] . As stable solutions must have the same asymptotic behavior as the lower branch of solutions, and as this is impossible after λ c {\lambda ^c} , we contend that no stable solution exists after λ c {\lambda ^c} and that a solar flare is thus triggered.