Lagrangian Formulation of the Snowplow Model and Operating Point for Z pinch Devices

TL;DR

This work develops a Lagrangian framework for the snowplow model of Z pinch discharges with two generalized coordinates, $r(t)$ and $Q(t)$, yielding the Lagrangian $\mathcal{L}=\frac{1}{2}M\dot r^2+\frac{1}{2}L\dot Q^2-\frac{1}{2C_0}Q^2$ and corresponding Euler-Lagrange equations that reproduce the modified snowplow dynamics when a nonconservative force $F_k(t)$ is included; dimensionless reformulation connects to the original equations via $\alpha$ and $\beta$ and recovers the standard snowplow model in the limit $F_k=0$. The authors introduce three energy-based performance metrics, $\eta_1$, $\eta_2$, and $\eta_3$, defined from the energy delivered by the capacitor bank, transferred to the plasma, and converted to internal energy, respectively, and apply them by sweeping the capacitor-bank energy $E_0$ in a prototype device to locate an operating point that optimizes source-load coupling. They report maxima in the performance curves near $E_0\approx 0.33$ kJ for $\eta_1$, $0.19$ kJ for $\eta_2$, and $0.14$ kJ for $\eta_3$, with corresponding voltages and plasma temperatures, and they find a fitted relation $k_B T \propto E_0^{0.64}$ at large $E_0$ that informs adiabatic heating limits. Overall, the paper demonstrates that a Lagrangian approach provides a structured view of Z-pinch dynamics and energy coupling, while highlighting the need for experimental validation of the proposed operating-point concept.

Abstract

We write down the lagrangian for the snowplow model of the Z pinch system. Then, we develop the Euler-Lagrange equations to find out the corresponding equations of motion. Next, we set a criterion for quantitatively estimating the performance of Z pinch devices. Finally, we apply this criterion to a specific Z pinch system.

Figures (2)

• Figure 1: The ratios $\eta_1$,$\eta_2$ and $\eta_3$ plotted against $E_0$.
• Figure 2: The $\eta$ ratio plotted against $E_0$. Simulation: solid line, Fitting: dashed line.