A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
the interface grows into a bubble and spike. The bubble is rising low density fluid driven by
buoyancy, while the spike is high density fluid falling almost freely under gravity. In turn the
sides of the spike can be unstable to KH instabilities leading to a characteristic roll-up as the
heavier fluid passes through the light fluid. In the case where the light fluid is a magnetic field,
no KH instabilities will occur.
An overview of classical hydrodynamic RT in the nonlinear state has been given by
Sharp [304]. 2D simulations of the nonlinear growth leading to turbulent mixing has been
reviewed by Youngs [305, 306] while experiments using solid fuel rocket motors to accelerate
downwards rectangular tanks of two immiscible fluids have been reported by Read [307] with
accelerations of 750 m s −2 over distances of 1.25 m. The main conclusion of the experiments is
that the denser fluid penetrates the lighter fluid at a rate 0.07Agt 2 in 3D. Narrow tanks were also
employed to give comparison with 2D simulations where the coefficient of 0.07 is replaced by
0.06 in experiment compared with 0.04 to 0.05 in simulations. The bubbles of lighter fluid had
displacements also proportional to Agt 2 , with a constant of 1/2 to 1/3 that of the spike growth.
Following the bubble and spike formation, the next stages are bubble amalgamation, spike
break-up and turbulent mixing, discussed briefly in [304]. Of course bubbles and spikes are
purely 2D: in 3D spherical implosions Town and Bell [308] have shown that, dependent on
the nature of the initial perturbation, either spikes and (connecting) valleys, or bubbles and
(connecting) ridges are formed. Such spherical simulations are important for ICF capsule
modelling, ideally with thermal conduction and ablation also present. A 2D representation
of a spherical shell implosion with these processes present is given by Henshaw et al [309],
where the bubble evolution is particularly important for shell integrity. The nonlinear growth
of bubbles is shown to occur immediately for a perturbation amplitude greater than 0.04 times
the wavelength. 2D nonlinear RT has been explored in ablation-driven shells by McCrory
et al [310], who showed spike amplitude saturation due to ablation while the distortion of the
remaining shell by the bubble development poses a limit to 3D spherical compression.
Some interesting analytic approaches to nonlinear RT in 2D are given by Ott [311],
Basko [312] and by Infeld and Rowlands [313]. These have been used by Haines [280] in
developing a heuristic model of the wire-array pinch. The MRT relevant to Z-pinch implosions
has mainly been studied in the r–z plane as a 2D problem, assuming that the unperturbed plasma
is a uniform shell. This will be discussed in more detail in section 5.5.
5. The wire-array Z-pinch
5.1. Overview
It might seem rather odd that the dynamic Z-pinch with the highest and most reproducible
performance starts off as a cylindrical array of many wires stretched between two electrodes
only 1 or 2 cm apart. Figure 36 is a photograph of nested arrays as typically used in high
performance shots at Sandia. There is however great precision in the mounting and spacing
of these wires. Even though each wire plasma at early times develops axial instabilities,
the wavelength of the mode appears to only be material dependent, but their phases on
different wires are uncorrelated. Thus as proposed in [280] the resulting amplitude of the
seed perturbations at merger of the wire plasmas varies as n −1/2 where n is the number of
wires, a result verified by experiments at Imperial College [314]. The physical nature of this
early instability which leads to many separate inward flowing plasma jets is will be discussed
later in section 5.4. The single most important experimental discovery was by Sanford et al [20]
who showed that above a critical wire number, or, rather below a critical size of gap of the
order 1 mm, the resulting x-ray power increases dramatically as shown in figure 37. This is
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