Additive self-helicity as a kink mode threshold

In this paper, we propose that additive self-helicity, introduced by Longcope and Malanushenko, plays a role in the kink instability for complex equilibria, similar to twist helicity for thin flux tubes. We support this hypothesis by a calculation of additive self-helicity of a twisted flux tube from the simulation of Fan and Gibson. As more twist gets introduced, the additive self-helicity increases, and the kink instability of the tube coincides with the drop of additive self-helicity, after the latter reaches the value of H_A /Φ² 1.5 (where Φ is the flux of the tube and H_A is the additive self-helicity). We compare the additive self-helicity to twist for a thin subportion of the tube to illustrate that H_A /Φ² is equal to the twist number, studied by Berger and Field, when the thin flux tube approximation is applicable. We suggest that the quantity H_A /Φ² could be treated as a generalization of a twist number, when the thin flux tube approximation is not applicable. A threshold on a generalized twist number might prove extremely useful studying complex equilibria, just as the twist number itself has proven useful studying idealized thin flux tubes. We explicitly describe a numerical method for calculating additive self-helicity, which includes an algorithm for identifying a domain occupied by a flux bundle and a method of calculating potential magnetic field confined to this domain. We also describe a numerical method to calculate twist of a thin flux tube, using a frame parallelly transported along the axis of the tube.