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The motivation for the work described in this paper is to understand kink-unstable magnetic flux tubes and their role in the formation of δ-spot active regions. It has been proposed that, during their rise to the photosphere, a certain fraction of convection zone flux tubes become twisted to the point where they are unstable to the current driven kink instability. These kink-unstable flux tubes then evolve toward a new, kinked equilibrium as they continue to rise to the photosphere, appearing as δ-spot groups upon emergence. Because of their kinked nature, these flux tubes could be highly susceptible to flaring, explaining the very active nature of δ-spot groups. We study the kinking flux tube problem with a three-dimensional numerical model 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 instability, and follow the evolution into the nonlinear regime: (1) We perform numerical simulations of constant-twist, kink-unstable flux tubes in an initially cylindrical equilibrium configuration in three dimensions, in a high-β pressure-confined environment. We consider many different initial configurations, including the Gold-Hoyle flux tube. (2) These numerical calculations confirm the growth-rate predictions of our earlier work, when viscous dissipation is included. They also confirm our velocity profile predictions. (3) The flux tubes evolve toward new helically symmetric equilibrium configurations. (4) The timescale for saturation to the kinked equilibrium configuration is τsat ~ 10/ω0, where ω0 is the linear growth rate calculated as in the earlier paper. (5) The cylindrically symmetric part of the kinked equilibrium is well described by the m = 0 Chandrasekhar-Kendall functions (i.e., the Lundquist field). The m = 1 helically symmetric part, however, is not well described by the m = 1 Chandrasekhar-Kendall 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 instability can be predicted from the initial excess perpendicular magnetic energy. (8) The amplitudes of the kinked tubes are large enough to give a δ-spot region tilt angle of up to 30° away from that of an unkinked tube.

Nonlinear Evolution of Kink‐unstable Magnetic Flux Tubes and Solar δ‐Spot Active Regions