Nonlinear evolution of kink-unstable magnetic flux tubes and solar delta-spot active regions

The motivation for the work described in this paper is to understand kink-u nstable magnetic flux tubes and their role in the formation of delta-spot a ctive regions. It has been proposed that, during their rise to the photosph ere, a certain fraction of convection zone flux tubes become twisted to the point where they are unstable to the current driven kink instability. Thes e kink-unstable flux tubes then evolve toward a new kinked equilibrium as t hey continue to rise to the photosphere, appearing as delta-spot groups upo n emergence. Because of their kinked nature, these flux tubes could be high ly susceptible to flaring, explaining the very active nature of delta-spot groups. We study the kinking flux tube problem with a three-dimensional numerical m odel containing only the most basic features of a kink-unstable flux tube. We build on our earlier work describing the linear phase of the kink instab ility, and follow the evolution into the nonlinear regime: (1) We perform n umerical simulations of constant-twist, kink-unstable flux tubes in an init ially cylindrical equilibrium configuration in three dimensions, in a high- beta pressure-confined environment. We consider many different initial conf igurations, including the Gold-Hoyle flux tube. (2) These numerical calcula tions confirm the growth-rate predictions of our earlier work, when viscous dissipation is included. They also confirm our velocity profile prediction s. (3) The flux tubes evolve toward new helically symmetric equilibrium con figurations. (4) The timescale for saturation to the kinked equilibrium con figuration is tau(sat) similar to 10/omega(0), where omega(0) is the linear growth rate calculated as in the earlier paper. (5) The cylindrically symm etric part of the kinked equilibrium is well described by the m = 0 Chandra sekhar-Kendall functions (i.e., the Lundquist field). The m = I helically s ymmetric part, however, is not well described by the m = 1 Chandrasekhar-Ke ndall functions. (6) The equilibrium kink amplitudes are not large, at less than one-third of the tube radius. (7) The peak kinetic energy of the inst ability can be predicted from the initial excess perpendicular magnetic ene rgy. (8) The amplitudes of the kinked tubes are large enough to give a delt a-spot region tilt angle of up to 30 degrees away from that of an unkinked tube.