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 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 set of coupled ordinary differential equations derived from the ideal MHD equations. Comparisons with experimental data from the COBRA pulsed-power facility are quite promising and imply this model could prove useful in designing and analyzing future pulsed-power experiments.

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• High-energy-density plasmas
• Magnetohydrodynamics
• Magnetoinertial fusion
• Shock waves & discontinuities in plasma
• Z-pinches
• First-principles calculations in plasma physics
• Magnetohydrodynamic techniques