A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
where the lower figure refers to Z →∞and the higher to Z = 1. The fastest growing mode
has a wavelength λ of
[
λ ∼ σκmi
] 1/2
= 2π = (1.52–0.56) × 10
20
T 2 A 1/2
(3.52)
3n 2 e 3 n e Z ln ei
and growth rate of
γ ∼
m e α 0
= 1.2 = (1.03–1.77) × 10 −12 m e n e Z ln ei
m i τ ei m i Te
3/2
(3.53)
This theory satisfactorily explained the occurrence of an instability in a theta pinch during
the early plasma formation [213], agreeing well on the variation of wavelength with density.
Here the electron density at very early times was in the range 5 × 10 20 to 3 × 10 21 m −3 , the
electron temperature 2 to 3 eV, the growth rates 5 × 10 5 to 8 × 10 7 s −1 and wavelengths 7.6
to 0.75 cm. A similar break-up of the current into filaments would be expected in the gaseous
compressional pinch or plasma focus very soon after plasma formation if condition (3.51)
holds. Such an instability however could also occur at stagnation of a highly radiating wirearray
Z-pinch when the conditions ((3.50), (3.51)) can sometimes be satisfied, because of the
high electron density achieved, leading to azimuthal variation with m ∼ 5–9. However care
must be exercised in this interpretation because such structure might be triggered by the slots
in the return current conductor.
The reason for the optimum wavelength is that short wavelengths are damped by thermal
conduction, κ, while for long wavelengths the combined effects of Faraday’s law and Ampère’s
law ensure that an electric field perturbation will depend on k −2 , and this field will oppose the
change in current density.
This theory has been extended to the case of nonlinear heat flow [214], particularly as it
applies to the heat flux in a laser-driven implosion for inertial confinement. Here the heat flow
reaches about 10% of the free-streaming limit. Indeed this is necessarily the case because the
inward heat flow, the source of energy for the implosion, must exceed the outward enthalpy
flux associated with the ablation process. This strong heat flux can be approximated to a
relatively collisionless inward flow of hot electrons (the mean-free path varies as V 4 ) and an
equal and opposite current of cold collisional electrons. This cold return current is driven by
an inward electric field—the thermoelectric field, which only arises because of the velocity
dependence of the collision frequency. Indeed the outward acceleration of ions—the ablation
process—is impeded by this electric field which is of the wrong sign to accelerate the ions,
and it is the frictional force between the cold return current and the ions which causes the
ablation [215]. J · E for the hot electrons is negative and is equal and opposite to the J · E for
the cold return current. Thus, if the conditions (3.49) and (3.51) are satisfied, the Joule heating
associated with the cold return current will cause an electrothermal instability associated now
with heat flux [214]. This instability is the likely cause of fine-scale filamentary structures
observed at first by laser shadowgraphy [216] and recently by MeV proton probing [217]
of plasmas produced by laser–solid interactions. The latter demonstrated that the instability
originates at the ablation surface. In the theory [214] the hot electrons exhibit a Weibel-like
response to the perturbing electric and magnetic fields, especially if they are beam-like, but
if their temperature is comparable to their directed energy they have no major effect on the
electrothermal instability.
Similar filamentary structures are observed on the surface of the cores of wires in wirearray
Z-pinches [218, 219]. An interesting fact is that when experiments try to stimulate a
different wavelength, the natural wavelength for a particular material still persists [220]. These
experiments will be discussed later in section 5.4.
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