A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
larger than in current direct-drive laser-capsule experiments elsewhere or in dynamic hohlraum
experiments on Z, this being mainly due to the higher implosion velocities.
Recently Welch et al [495] have produced kinetic particle-in-cell (PIC) simulations of
the Sandia deuterium experiment [492] and have found both thermonuclear and beam–plasma
contributions to the neutron yield. More needs to be done with this type of modelling. Can
ion viscosity or even electron viscosity [496] be included in such a model?
However, as in nearly all high current Z-pinches it appears that the final thermal energy
of the pinch is over double that arising from the kinetic energy. Thus in [494] to obtain
agreement with the experimental neutron yield, noting that the experimentally measured
electron temperature is 2.2 keV and that the calculated equipartition time exceeds twice the
Alfvén transit time τ A (i.e. the total bounce time of the plasma column) an ion temperature
of 10 keV (i.e. greatly exceeding the Bennett value) and an ion density n D of 3 × 10 20 cm −3
(compared with the measured 2.4 × 10 20 cm −3 [492]) is required.
In contrast a new calculation based on the Bennett pressure balance at stagnation gives
T i = 5.5 keV. Using the experimental density for two ion transit times (each of 8 ns) and
σv = 2.3 × 10 −19 cm 3 s −1 the neutron yield is 4 × 10 13 , close to experiment. Here we have
used the formula
Y n = 1 4 πa2 z 0 n 2 D σv × (2τ A) (7.1)
where a is the pinch radius and z 0 its length. The kinetic energy prior to stagnation, based on the
ion radial velocity of 70 cm µs −1 [492] is 5.1 keV per ion with very low electron temperature.
The equipartition time is 25 ns (≫τ A ), allowing the ion temperature to exceed the electron
temperature over the period of neutron emission. In this model it is assumed that the increase
in ion thermal energy is the result of ion-viscous heating associated with small amplitude,
short wavelength m = 0 MHD instabilities [130] rather than just reversible P dV heating. The
latter in any case would be included in current 2D MHD simulations, and is usually insufficient
to account for preferred ion heating. As noted in [494] these simulations generally include
fudge factors to prevent radiative collapse, which can be physically prevented by ion-viscous
heating [130] or some extra phenomenological ion heating source [329–332].
A puzzling result is that of Kroupp et al [497] who, from detailed time-resolved Doppler
shifted spectroscopic measurements of optically thin lines in neon, showed that there is in this
experiment a balance between the ion kinetic energy and the radiated energy, via electron energy
through equipartition. There is no rise in T i at stagnation in contrast to that measured by LePell
in [130]. However it can be shown that at this relatively low current (320 kA at stagnation)
gas-puff Z-pinch there is negligible viscous heating and all short wavelength MHD modes are
essentially resistively damped, i.e. the magnetic Prandtl number is much less than one [253].
There are several ways of arriving at this conclusion. One is to examine where in the I 4 a–N i
stability regimes diagram, namely figure 16 (but altered for the higher Z value according to
equations (3.4), to (3.13), this experiment at stagnation lies. I 4 a is found to be 4.2×10 18 (A 4 m)
at N i = 1.27 × 10 19 (m −1 ). The electrons are weakly magnetized with e τ ei = 4.6 (using the
spatial average of B θ ) while the ions are unmagnetized at i τ i = 2.4 × 10 −3 making the ratio
a i /a = 0.025 irrelevant. The magnetic Reynolds number S is 91, i.e. ∼100, on the border of
resistive stabilization effects [153, 154] while the Reynolds number R is 1.7 × 10 4 taking into
account the contribution from electron viscosity. γτ i = 1.2 × 10 −4 ≪ 1 is closely related to
R −1 , and shows that viscous effects are negligible.
In another approach considering the relative damping of Alfvén waves by resistivity
and viscosity it is found that the competing diffusivities are η/µ 0 = (µ 0 σ) −1 and ν (the
effective kinematic viscosity), respectively. For [497] η/µ 0 is 0.386 m 2 s −1 while ν is
2.0 × 10 −3 m 2 s −1 showing that resistive damping dominates over viscous damping. Thus
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