Dimensional Analysis Approach to Experiments in Z pinch Devices

TL;DR

The paper addresses the challenge of characterizing Z pinch discharges with many control parameters by applying dimensional analysis, reducing the description to three dimensionless groups $(\alpha^2,\beta,\gamma)$ that define a surface $\gamma=f(\alpha^2,\beta)$ in $3D$ space. It combines the Buckingham $\Pi$ theorem with the snowplow model to derive a closed-form relation $\gamma=\left(\dfrac{\beta}{\alpha^2}\right)F(\alpha^2,\beta)$, linking energy partition to the observed radius dynamics, and shows how plasma temperature $k_B T$ can be inferred from accessible measurements via $U=\gamma E_0$ and $k_B T=\gamma\left(\dfrac{2}{3}\right)\left(\dfrac{E_0}{N_0}\right)$. While direct temperature measurements in small Z pinch experiments remain challenging, the proposed method enables estimation by integrating radius-time data with a validated model, providing a practical pathway to cross-compare experiments and to infer macroscopic plasma properties. The work highlights two routes: map the empirical $3D$ surface as data accumulate, or rely on the snowplow model to approximate the surface when data are sparse, with the latter offering a first-principles-based means to estimate $k_B T$.

Abstract

The physical behavior of discharges in Z pinch devices can be completely deciphered in terms of only three dimensionless parameters. These parameters can be arranged in a way that draw a surface in 3D space. This surface compiles all the accessible information on the macroscopic physical behavior of each possible Z pinch discharge. We analyze the practical problems the drawing of this surface encounters and in view of the situation, we devote the remainder of the article to outline a feasible method for estimating the plasma temperature in Z pinch discharges.