M = 1 ideal internal kink modes in a line-tied screw pinch

Abstract

It is well known that the radial displacement of the m=1 internal kink mode in a periodic screw pinch has a steep jump at the resonant surface where k∙B=0 [Rosenbluth, Dagazian, and Rutherford, Phys. Fluids 16, 1894 (1973)]. In a line-tied system, relevant to solar and astrophysical plasmas, the resonant surface is no longer a valid concept. It is then of interest to see how line-tying alters the aforementioned result for a periodic system. If the line-tied kink also produces a steep gradient, corresponding to a thin current layer, it may lead to strong resistive effects even with weak dissipation. Numerical solution of the eigenmode equations shows that the fastest growing kink mode in a line-tied system still possesses a jump in the radial displacement at the location coincident with the resonant surface of the fastest growing mode in the periodic counterpart. However, line-tying thickens the inner layer and slows down the growth rate. As the system length L approaches infinity, both the inner layer thickness and the growth rate approach the periodic values. In the limit of small ϵ∼Bϕ∕Bz, the critical length for instability Lc∼ϵ−3. The relative increase in the inner layer thickness due to line-tying scales as ϵ−1(Lc∕L)2.5.