A new method for analyzing line-tied kink modes in cylindrical geometry

A new method for studying linear stability of the m=1 (kink) mode in a cylinder with line-tied boundary conditions is presented. The method is applicable to both resistive and ideal MHD. It is based on expansion in one-dimensional eigenfunctions depending on the radius, satisfying boundary conditions on the cylindrical axis and radial wall. The boundary conditions at the end plates are satisfied by a sum of such radial eigenfunctions. The spectrum of possibly complex axial eigenvalues k is studied and is shown to consist of a continuum part and a discrete part in ideal MHD. Only the discrete part is used to give an approximation to the complete two-dimensional eigenfunction. The method is applied to a special equilibrium magnetic field with constant field line pitch. The role of the individual radial eigenfunctions is explained. It is shown that our method reproduces previously found values of the critical pitch (at marginal stability) for a plasma column in vacuum. The method also suggests important differences between ideal and resistive MHD. (c) 2006 American Institute of Physics.