A Comprehensive Analytical Model of the Dynamic Z-Pinch

TL;DR

The paper advances a fast, 1D analytical framework for the dynamic z-pinch by deriving stage-specific ODEs for the piston and shock radii from ideal MHD, including a spatially varying initial density and a weak axial field. It predicts full sheath profiles, not just front trajectories, by combining RH jump conditions, adiabatic evolution, and mass/flux conservation, then validates the model against COBRA experiments. Key contributions include the Potter-Angus-based staging to avoid initial singularities, explicit expressions for velocity and pressure profiles, and a robust calibration against multiple shots, including varied density profiles and axial fields. This work offers a practical, physics-driven tool for rapid interpretation and planning of pulsed-power z-pinch experiments, complementing more computationally intensive 2D/MHD simulations.

Abstract

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. Comparison with experimental data from the COBRA pulsed-power facility is quite promising and implies this model could prove useful in designing and analyzing future pulsed-power experiments.

Figures (16)

• Figure 1: Z-Pinch Configuration. The z-pinch is shown in cylindrical geometry with time-dependent current $I(t)$, pinching azimuthal field $B_{\theta}(r,t)$, spatially-varying initial density $\rho_{0}(r)$, initial axial field $B_{z0}$, and adiabatic index $\gamma$. The piston $r_{p}(t)$ and shock $r_{s}(t)$ radii separate the system into three distinct radial regions, with the sheath density $\rho_{l}(r,t)$ between them.
• Figure 2: Implosion Stages. Here we see the evolution of the piston and shock radii throughout various stages of the implosion. Each stage is associated with a characteristic model which best describes its underlying physics. Different models either asymptotically agree or are matched with physical boundary conditions at transition times.
• Figure 3: Initial Density and Current Waveform for Shot 5532. Here we see the initial density profile $\rho_{0}(r)$ and current waveform $I(t)$ for shot 5532 as measured by Planar Laser Induced Fluorescence (PLIF) PLIF and a Rogowski Coil around the load, respectively.
• Figure 4: Radial Trajectories for Shot 5532. Here we see a comparison of the calibrated trajectories of the piston and shock radii for shot 5532, together with various diagnostic measurements of their true values, after an MSE fit yielded optimal parameters $(r_{0},\gamma) = (3.50\text{cm},1.37)$.
• Figure 5: Initial Density and Current Waveform for Shot 4959. Here we see the initial density profile $\rho_{0}(r)$ and current waveform $I(t)$ for shot 4959 as measured by Planar Laser Induced Fluorescence (PLIF) PLIF and a Rogowski Coil around the load, respectively.