Numerical and Analytical Studies of the Nonlinear Sausage Instability.

Sausage instabilities of an incompressible, uniform, perfectly conducting Z pinch are studied in the nonlinear regime. In the long wavelength limit analogous to the shallow water theory of hydrodynamics, a simplified set of universal fluid equations is derived, with no radial dependence, and with all parameters scaled out. Analytic and numerical solution of these one dimensional equations show that an initially sinusoidal perturbation grows into a spindle or cylindrical spike and bubble shape, with sharp radial maxima. In the short wavelength limit, the problem is shown to be mathematically equivalent to the planar semi-infinite Rayleigh-Taylor instability, which also grows into spike-and-bubble shape. Since the spindle shape is common to both limits we conclude that it probably obtains in all cases. The results are in agreement with dense plasma focus experiments.