The Rise of Kink-unstable Magnetic Flux Tubes and the Origin of δ-Configuration Sunspots

We perform three-dimensional simulations of the rise of twisted magnetic flux tubes in an adiabatically stratified model solar convection zone. The initial flux tube in our simulations is a uniformly twisted, buoyant, horizontal tube located near the bottom of the stratified layer. The twist of the initial flux tube is described by a parameter α, which is defined as the angular rate of field-line rotation about the tube axis per unit length of the tube. We study the nonlinear evolution of the helical kink instability of the flux tube as it rises through the stratified layer. We find from our simulations that in order for the tube to develop significant kinking during its rise, the initial twist of the tube needs to be close to or greater than the critical limit (α_c) for the onset of the kink instability. If the initial twist is significantly below the critical limit (α below about 50% of α_c), we find essentially no kink development and the evolution is similar to the results from previous two-dimensional simulations of the rise of twisted, horizontal flux tubes. On the other hand, if the initial twist is sufficiently greater than the critical limit such that the e-folding period of the fastest growing kink mode is small compared to the rise time of the tube, we find sharp bending and distortion of the tube as a result of the nonlinear evolution of the kink instability. In this case, we find that due to the effect of gravitational stratification, the kinked flux tube arches upward and evolves into a buckled loop with a local change of tube orientation at the loop apex that exceeds 90° from the original direction of the tube. The emergence of this buckled loop can give rise to a compact magnetic bipole with polarity order inverted from the Hale polarity law, similar to the configuration often seen in δ spots. Furthermore, our simulations show that the writhing of the tube axis as a result of the kink instability stretches the flux tube and increases its buoyancy. Hence, the development of the kink instability can speed up the overall rise of the flux tube.