$m=1$ 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 $\mathbf{k}\cdot \mathbf{B}=0$. 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 $\epsilon \sim B_{\varphi }/B_z$, the critical length for instability $L_c\sim {\epsilon }^{-3}$. The relative increase in the inner layer thickness due to line-tying scales as ${\epsilon }^{-1}(L_c/L{)}^{2.5}$.

To appear in Physics of Plasmas

Cited by in corpus (4)

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• Effects of Line-tying on Magnetohydrodynamic Instabilities and Current Sheet Formation
• Variational Integration for Ideal Magnetohydrodynamics and Formation of Current Singularities
• Boundary layers and noise in magnetized plasmas line-tied at conducting surfaces