Phenomenological theory of the kink instability in a slender plasma column

In this paper we are concerned with the kink instability of a current-carrying plasma column whose radius a is much smaller than its length L. In the limit a ≪L, one can consider the column as a thin filament whose kinking can be adequately described simply by a two dimensional 2D displacement vector, ξ_x=ξ_x(z,t); ξ_y=ξ_y(z,t). Details of the internal structure of the column such as the radial distribution of the current, density, and axial flow can be lumped into some phenomenological parameters. This approach is particularly efficient in the problems with nonideal (sheath) boundary conditions (BC) at the end electrodes, with the finite plasma resistivity, and with a substantial axial flow. With the sheath BC imposed at one of the endplates, we find instability in the domain well below the classical Kruskal-Shafranov limit. The presence of an axial flow causes the onset of rotation of the kink and strong axial "skewness" of the eigenfunction, with the perturbation amplitude increasing in the flow direction. The limitations of the phenomenological approach are analyzed and are related to the steepness with which the plasma resistivity increases at the plasma boundary with vacuum.