Instability of Thin Resistive Liners in the Linear Approximation

Magnetic implosion of liners is usually accompanied by the development of the Rayleigh-Taylor instability, which makes it hard to achieve high energy densities. However, the liners, which are thin compared with their skin layer (let us call them resistive), can also experience a different kind of instability, when parallel currents in different liner parts are attracted and draw these parts together dividing the liner into layers and filaments (filamentation or the tearing instability). We consider the problem of perturbation development in an infinitely thin resistive liner accounting for the distribution of the magnetic field, spread of currents, and flow of matter assuming that the perturbations are linear. Considering potential applications for different liner configurations (Zand 8-pinch liners and flat flyers) and the fact that the most destructive instabilities develop from the shortest wavelengths, we restrict our analysis to a planar case, which ignores the curvature of the liner and magnetic field lines. We find that the instability develops from any perturbation wavelengths, similar to the case of a perfectly conducting liner, with the instability growth rate for all the wavenumbers k being of the order of √kg (g is the liner acceleration), but always less than √kg, and with the maximum growth rates for any directions of the wave vector k being greater than the growth rates of the perfectly conducting liner. One can often treat wire-array liners as resistive and, hence, use the results obtained for the description of instability development after the phase of merging the plasmas produced by individual wires.