Re-appraisal and extension of the Gratton-Vargas two-dimensional analytical snowplow model of plasma focus - Part II: Looking at the singularity

TL;DR

This paper extends the Gratton-Vargas analytical model of plasma focus, demonstrating it can qualitatively reproduce experimental current waveforms and plasma behavior near the singularity without explicitly modeling plasma phenomena.

Contribution

It shows that the Gratton-Vargas model's solutions remain valid after the current singularity, capturing key plasma dynamics without hydrodynamic assumptions.

Findings

Solutions exist after the current singularity, modeling plasma expansion.

The model reproduces current derivative shapes observed in experiments.

Axial propagation follows a power-law similar to point explosion theory.

Abstract

The Gratton-Vargas snowplow model, recently revisited and expanded (S K H Auluck, Physics of Plasmas, 20, 112501 (2013)), has given rise to significant new insights into some aspects of the Dense Plasma Focus (DPF), in spite of being a purely kinematic description having no reference to plasma phenomena. It is able to provide a good fit to the experimental current waveforms in at least 4 large facilities. It has been used for construction of a local curvilinear frame of reference, in which conservation laws for mass, momentum and energy can be reduced to effectively-one-dimensional hyperbolic conservation law equations. Its utility in global parameter optimization of device parameters has been demonstrated. These features suggest that the Gratton-Vargas model deserves a closer look at its supposed limitations near the singular phase of the DPF. This paper presents a discussion of its development near the device axis, based on the original work of Gratton and Vargas, with some differences. It is shown that the Gratton-Vargas partial differential equation has solutions for times after the current singularity, which exhibit an expanding bounded volume, (which can serve as model of an expanding plasma column) and decreasing dynamic inductance of the discharge, in spite of having no built-in hydrodynamics. This enables the model to qualitatively reproduce the characteristic shape of the current derivative in DPF experiments without reference to any plasma phenomena such as instabilities, anomalous resistance or reflection of hydrodynamic shock wave from the axis. The axial propagation of the solution exhibits a power-law dependence on the dimensionless time starting from the time of singularity, which is similar to the power-law relations predicted by theory of point explosions in ideal gases and which has also been observed experimentally.