A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
wavelength. Initial fine-scale random perturbations of temperature in the core were introduced
in order to seed the instability. Earlier 2D and 3D simulations by Frese et al [372] as well as
Garasi et al [360] also contributed usefully to understanding the physics. The emphasis here
was to question the notion that merger of plasmas from separate wires occurs and leads to the
formation of a shell, rather than to the inward flow of separate plasma jets.
Axial wave numbers would be characteristic of a thermal instability in a conductor in which
the resistivity η increases with temperature T and the current is in the axial direction. The
instability arises because in a region of increased temperature but current density continuous,
the heating rate ηJ 2 will be increased, so increase the temperature further. However dη/dT
is > 0 for W at solid density only below 1 eV and as the density is dropped the point where
dη/dT = 0 moves to lower temperature. For Al, dη/dT is >0 at solid density below 6 eV
and is less pronounced as the density drops. Short wavelengths would be damped by thermal
conduction, but there does not appear to be a mechanism to give a natural wavelength, and
certainly for W, the dη/dT > 0 effect disappears in typical corona close to the wire core.
Further discussion on this can be found in [738] including the composition with MHD modes.
The fourth candidate is the heat-flow driven electrothermal instability, proposed for
conditions of nonlinear heat flow such as near the ablation surface of an ICF capsule [214],
where the heat flow is 3% of the free-streaming limit. It is discussed in detail in section 3.12.
A second necessary condition is given by equations (3.50) and (3.51). Taking equation (3.52)
and inserting for W with Z = 6 and T e = 12 eV and a wavelength λ of 0.25 mm this instability
will occur at an electron density of n e of 6.8×10 25 m −3 or a density of 3.5 kg m −3 . Referring to
figure 46, it can be seen that this is the density at 50–100 µm from the wire core. Further in, at
high density this and shorter wavelengths will be locally unstable, but the shorter wavelengths
will be damped by thermal conduction as the plasma ablates and the density drops. Perhaps this
occurs in the experiment of Jones et al [219]. The condition (3.51) is well satisfied. The growth
rate (equation (3.53)) is 6.3×10 8 s −1 or an e-folding time of 1.6 ns compared with the ablation
transit time of 13 ns. For Al with Z = 2; T e = 12 eV and a wavelength of 0.5 mm, this will be
the fastest growing mode at a density of 0.43 kg m −3 . Shorter wavelengths will occur at higher
density or lower temperatures. In this theoretical model there is no radiation transport, but very
close to the high density wire cores the thermal conduction is more important. Furthermore
the process of ablation of ions requires, in this range of density, the frictional drag by the cold
return current associated with the nonlinear heat flow [215]. The perturbations in return current
flow will lead to perturbations in the mass ablation rate, leading eventually to gaps. It can also
be argued that radiation transport also leads to a thermoelectric field due to photon momentum
deposition, and hence a return cold current which can be thermally unstable. To summarize, the
electrothermal heat-flow instability will occur close to the wire cores and a natural wavelength
emerges as the fastest growing mode where the plasma exits the core interaction region.
Unfortunately, it cannot be modelled by present fluid simulations, and requires at least a hybrid
code. Probing the wires with MeV protons [217] should resolve the issue as the electrothermal
instability occurs at high density and low temperature, i.e. close to the cores.
In a recent experiment using the Cornell Beam Research Accelerator (COBRA) by Knap
et al [373] the axial wavelength and amplitude of the instability together with the core radius
were measured as a function of time using laser shadowgraphy. It was found that both grew
from ∼0.1 mm to ∼0.5 mm over 30 ns from breakdown. At this time there is saturation, and
the wavelength is essentially the fundamental wavelength that had been measured in earlier
experiments. The saturation is believed to be associated with the inward flow of precursor
plasma and change in magnetic topology, similar to MHD simulations [371]. B-dot probes
show the reduction in the initial private magnetic flux around each wire and the dominance of
the global field (see figure 50(b)) by this time.
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