A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
Figure 33. A dimensionless plot of pinch radius against time (a) for the snowplough model, (b)
for the shell model and (c) for the slug model with the shock front trajectory given in (d).
numerical solution is tricky at y = 0, but a judicious mix of analytic and numerical methods
leads to the result that y ≈ 1.48 at x = 0. A plot of x versus y is given in figure 33(a).
Various modifications to this model can be implemented, e.g. the coupling of the pinch
current to a circuit which includes the increasing inductance of the pinch with time. In
section 4.4 a new analytic approach allows for the effect of increasing inductance of the
imploding current shell. However the fact that the dimensionless time τ 0 depends only on
the fourth root of the initial density or M l makes the model fairly robust and insensitive to
modifications.
The snowplough model does not conserve energy, and so is strictly only applicable to a
Z-pinch which can radiate the excess energy away during the implosion. It therefore might
be applicable to argon or xenon discharges where the kinetic energy per ion is comparable to
or less than the ionization potential. But otherwise the energy delivered to the ions through
reflection in the radial Hall electric field associated with the skin current will result in shockformation
travelling ahead of the skin current ‘piston’. This is the model considered next. For
this the mean-free path of the particles should be small, as otherwise a collisionless reflected
model [242] would be applicable.
4.2. The shock model
An analytic shock model of a fast pinch was developed by Allen [257] who used a planar
model, valid at early times. Jukes [258] modified this for cylindrical geometry, making the
same assumptions of infinite conductivity, an instantly rising current to a finite value and an
infinitesimal shock thickness. Jukes used the classic work of Guderley [259] to find a similarity
solution, in which the current I decays monotonically while with the rising inductance L the
voltage LI increased almost linearly.
It soon became clear when comparing experiments with theory that to get better agreement
it was necessary, especially for these early experiments in which the implosion energy per ion
was comparable to the ionization potential, that dissociation and ionization energy should be
included. Reynolds and Quinn [260] called this a modified shock model.
4.3. The slug model
Analytic models are however very useful for scaling purposes and for experimental design.
A slug model devised by Potter [261] is a zero-dimensional (0D) model which follows the
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