A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
Figure 55. End-on laser probing of 32 (a) and 64 (b) Al wires showing collision of plasma streams
from neighbouring wires [352, figure 9]. Copyright © 2001 Cambridge University Press.
et al [354]. This showed that less than 1% of the current could be present in the precursor on
axis for W or Al wire arrays of 16 wires.
In a more academic problem, Haines [355] considers the equilibrium problem of a cylinder
of plasma under pressure balance in which the Joule heating was balanced by radially inward
heat flow to a cold wire acting as a heat sink. The nonlinear second order differential equation
has just one free parameter, which is the ratio of the applied axial electric field to the mean
radial temperature gradient. The inverse of this ratio scales as T 2 n −1/2 , thus demonstrating
again that low magnetic Reynolds’ number, a low Hall parameter and a mean-free path less than
the collisionless skin depth occurs in the corona close to the wire core. This last condition is
a criterion for the onset of nonlinear heat-flow driven electrothermal instabilities [214]. These
dimensionless parameters are discussed further in [315].
A rising total current during the precursor phase and a low R m ensures a skin current and
a very small axial electric field in the flowing precursor, which in effect becomes a super-
Alfvénic flow e.m. accelerator. Resler and Sears [356] considered various regimes for steady
1D resistive MHD flow, and this was extended by Haines and Thompson [357] for a spatially
varying and self consistent magnetic field, finding exact integrals for the flow. In a related
model of steady cylindrical 1D precursor flow Chittenden et al [358] have compared a steadystate
analytic model with 2D r–θ simulations of precursor flow. Figure 56 is a 2D simulation
of plasma ablation from one of 32 wires in an array, showing a concentration of current density
on the outside of the wire core and the precursor jets flowing towards the axis (on the left of
the diagram). The simulations [359] tend to show far more current in the precursor plasma
than is implied by experiments, especially by the lack of m = 1 instabilities, but more data are
required. Garasi et al [360] found that in 2D simulations in the r–θ plane the mass ablation rate
was overestimated by factors of 10–100, and advects 5× current to the axis in the precursor.
Their 3D modelling was closer to experiment. A 1D Cartesian steady-state analytic ablation
model by Sasorov et al [361] compares thermal and radiation transport. A transition from
subsonic to supersonic flow is claimed in the absence of areal change in the flow, in contrast
to [356, 357]. But in addition in a plasma flowing sonically against the heat flow the latter
should be nonlinear as discussed later.
In contrast to steady models, it can be shown that for a rising current of the form in
equation (4.16) and writing γ = I/I ˙ and a magnetic field of the form exp(γ t − kx) in a
plasma of conductivity σ and ablation velocity V a , k is given by
k =−µ 0 σV a /2+[(µ 0 σV a /2) 2 + µ 0 σγ] 1/2 . (5.3)
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