The Self-Similar Shrinkage of Force-Free Magnetic Flux Ropes in a Passive Medium of Finite Conductivity

Abstract

The twisted magnetic flux tube (magnetic rope) is a typical and the most important element of solar activity. Normally, the magnetic flux ropes in solar atmosphere are surrounded by a quasi-potential magnetic field, which provides the pressure balance in the cross-section. We have found a new exact MHD-solution for dissipative evolution of a thin magnetic flux rope in passive resistive plasma, with the growing gas density in the rope. The magnetic field inside the flux rope is assumed to be force-free: it is a set of concentric cylindrical shells (envelopes) filled with a twisted magnetic field [${B}_{z} = {B}_{0}{J}_{0}(\alpha r),{B}_{\varphi } = {B}_{0}{J}_{1}(\alpha r)$, Lundquist, Phys. Rev. 83, 307–311 (1951)]. There is no dissipation in a potential ambient field outside the rope, but inside it, where the current density can be sufficiently high, the magnetic energy is continuously converted into heat. The Joule dissipation lowers the magnetic pressure inside the flux rope, thereby balancing the pressure of the ambient field; this results in radial and longitudinal contraction of the magnetic rope with the rate defined by the plasma conductivity and the characteristic spatial scale of the magnetic field inside the flux rope. Formally, the structure shrinks to zero within a finite time interval (the dissipative magnetic collapse). The compression time can be relatively small, within a few hours, for a flux rope with a radius of about 300 km, if the magnetic helicity initially trapped in the flux rope (the helicity is proportional to the number of magnetic shells in the rope) is sufficiently large. This magnetic system is open along its axis of symmetry and along the separatrix surfaces, where J ₁(αr) = 0. On the rope’s axis and on these surfaces, the magnetic field is strictly longitudinal, so the plasma will be ejected from the compressing tube along the axis and separatrix surfaces outwards in both directions (jets!) at a rate substantially exceeding that of the diffusion. The obtained solution can be applied to the mechanisms of coronal heating and flare energy release.