A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
(ii) Sheared flow could also reduce the growth rate of the RT instability, in analogy with sheared
axial flow on MHD modes. Shumlak and Roderick [285] showed that rather high flows
of order 10 5 ms −1 or greater are required at the outer surface of a 0.5 thick Al foil falling
linearly to zero velocity on the inner edge. Large sheared flows however could trigger the
Kelvin–Helmholtz (KH) instability. The authors suggest that the necessary flows could
be generated with conical liners. Hammer and Ryutov [286] also discuss rotation as well
as axial shear flow and give references to earlier work in this field. Douglas et al [287]
have shown how sheared axial flows can arise through foil shaping like an hour-glass with
a consequent reduction in RT growth.
(iii) Rosenbluth, Rostoker and Krall [288] consider the effect of finite ion Larmor radius on the
gravitational instability. An attempt to extend this to large Larmor radii by Hassam and
Huba [289] was criticized by Lehnert and Scheffel [290], mainly because the ion Larmor
radii were greater than the characteristic scale length and a Vlasov treatment is required.
However, during most linear or wire-array implosions the ions are too collisional for FLR
to be important.
(iv) Viscosity can in principle damp high kRT modes, but the ion temperature in most
implosions at this stage is too low to be effective. Ryutov [291] suggests fine scale
multi-layering normal to g which will lead to sheared velocities and viscous damping on
this scale. But the global mode is hardly suppressed.
(v) Bud’ko et al [292] found that the addition of even a relatively weak axial magnetic field can
reduce the growth rate in dynamic pinches and that experiments by Felber et al [293, 294]
show agreement. However an axial magnetic field requires a substantial energy input
probably through additional coils or twisted wires [295] and the final compressed state
will be weaker. (See sections 5.4 and 8.7.)
(vi) A profiled density of a gas fill can be tailored to suppress the RT instability for a certain
time [296]. It requires that a shock wave propagates into increasing density thus slowing
down the shock which in turn slows down the magnetic piston (not unlike the slug
model), and causing a reversed acceleration. Of course an initial velocity requiring a
brief, large acceleration of the piston is required but the overall growth of kinetic energy
in the perturbation is reduced. The initial shock acceleration leading to a temporary RT
instability is usually termed the Richtmyer–Meshkov instability [297, 298] and is finite
in amplitude. However after the shock has reflected on axis there will be growth of RT
associated with the hot doubly shocked plasma accelerating into the denser plasma at
larger radius. So with this technique the onset of serious RT fragmentation can be delayed
until late on in the discharge. Earlier theoretical work on structured profiles has been
carried out for smooth density and velocity gradients by Bud’ko and Liberman [299].
(vii) The early snowplough and shock dynamics of a uniform gas fill was shown to impede the
growth of RT until the reflected shock and bounce occurred [39, 300]. This was further
explored by Gol’berg and Liberman [301] theoretically.
(viii) Related to these ideas are nested loads, e.g. the double-puff Z-pinch load studied
experimentally by Baksht et al [302]. Multiple shells were studied theoretically by
Cochran et al [303]. Nested wire arrays also give enhanced performance as x-ray radiators
but here we will see in section 5.6 that there are several modes of operation including a
transparent mode. There is however always significant reduction in the RT stability.
4.8. Nonlinear Rayleigh–Taylor instabilities
The nonlinear behaviour of RT instabilities has been studied for several decades especially
in the case of pure hydrodynamics. In two dimensions an initial sinusoidal perturbation at
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