  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.
