Heating and activity of the solar corona .2. Kink instability in a flux tube

The development of kink instability in a flux tube is investigated numerically, by solving the resistive MHD equations in three dimensions for a setup where a flux tube is stressed by rotating both ends in opposite directions. Two cases are investigated: one where the tube is initially isolated and in pressure equilibrium with surrounding plasma (external kink) and another with an initially uniform magnetic field, where only a smaller part of the boundaries are used to twist the field (internal kink). The twist angle at the onset of the kink instability depends on several parameters, such as rotation velocity, tube diameter, field strength, and magnetic resistivity, but is generally in the range 4 pi-8 pi. Both sets of experiments are followed beyond the point where they become kink unstable into the regime of nonlinear evolution. Of particular interest is the topological evolution. As magnetic dissipation becomes significant, the connectivity between the two boundaries changes from ordered to chaotic, and small-scale current sheets develop. Even though the gross features of the external kink appear to saturate, the total magnetic energy continues to grow, by a steady increase of the free energy in the chaotic region that develops as a result of the kink and by a secular spreading of the magnetic field into the initially field-free region. The internal kink is confined to the cylinder defined by the boundary driving and has only limited influence on the external magnetic field. After the kink, the twist of the magnetic field is reduced, and the internal kink settles into a quasi-steady state where the dissipation on the average balances the Poynting flux input. The average Poynting flux is similar in the external and internal kinks, with a magnitude that corresponds to local winding numbers of the order of unity. Scaling of these results to values characteristic of the solar corona indicate that systematic rotation or shear of the endpoints could be a source of quasi-steady heating in coronal loops.