The effect of line-tying on tearing modes

Cylindrical magnetohydrodynamic (MHD) constant-psi or nonconstant-psi tearing modes that are linearly unstable with periodic axial boundary conditions are studied in a line-tied cylinder. Examples of these two respective classes of modes, with m = 1 and m = 2 (m being the azimuthal mode number), are studied. With a suitable MHD equilibrium, the former modes are marginally stable in ideal MHD for periodic axial boundary conditions, and occur as fast tearing modes (resistive kinks) in the presence of resistivity eta. The latter modes are stable in ideal MHD for periodic axial boundary conditions, and with resistivity occur as constant-psi tearing modes, unstable in a range of parameters. In both cases, the results for the line-tied modes show the expected tearing scaling with mu for very long plasmas, but the scaling becomes gamma proportional to eta for smaller cylinder lengths L. These results are consistent with the following interpretation: For L --> infinity, the modes have a tearing width characteristic of tearing, leading to characteristic tearing mode growth. As L decreases, the modes develop a geometric width, which increases as L decreases; the gamma proportional to eta scaling occurs when L is small enough that the geometric width exceeds the tearing width. (C) 2008 American Institute of Physics.