Additive Self Helicity as a Kink Mode Threshold

In this paper we propose that additive self helicity, introduced by Longcope and Malanushenko (2008), plays a role in the kink instability for complex equilibria, similar to twist helicity for thin flux tubes (Hood and Priest (1979), Berger and Field (1984)). We support this hypothesis by a calculation of additive self helicity of a twisted flux tube from the simulation of Fan and Gibson (2003). 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 $H_A/Φ^2\approx 1.5$ (where $Φ$ is the flux of the tube and $H_A$ is additive self helicity). We compare additive self helicity to twist for a thin sub-portion of the tube to illustrate that $H_A/Φ^2$ is equal to the twist number, studied by Berger and Field (1984), when the thin flux tube approximation is applicable. We suggest, that the quantity $H_A/Φ^2$ could be treated as a generalization of a twist number, when thin flux tube approximation is not applicable. A threshold on a generalized twist number might prove extremely useful studying complex equilibria, just as 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.