A Comprehensive Analytical Model of the Dynamic Z-Pinch

We present an analytical 1D axisymmetric model describing the evolution of the dynamic z-pinch. This model is capable of predicting the trajectories of the imploding sheath's magnetic piston and preceding shock front, along with the velocity, pressure, density, and axial magnetic field profiles, for any time-dependent current, spatially-varying initial density profile, and weak initial axial field. The implosion is divided into stages, with each stage described by a pair of coupled ordinary differential equations derived from the ideal MHD equations. Comparison with experimental data from the COBRA pulsed-power installation is quite promising, and implies this model could prove useful in designing and analyzing future pulsed-power experiments.