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

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