A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
for electron temperatures up to 100 keV. Refinement of the Lee and More [392] transport
model for strongly coupled plasmas has been carried out by Dejarlais [340] while for partially
degenerate, magnetized plasmas a complete set of transport coefficients has been derived by
Brown and Haines [393, 394].
Pereira and Davis [51] have reviewed the experimental data of x-rays from dense Z-pinches
and compared the results with theoretical modelling. The experimental I 4 scaling of x-ray
yield for neon K shell and krypton L shell radiation is explained. Also there is a marked radial
distribution of radiation emission, the XUV emission coming broadly from the pinched region
but the harder radiation, i.e. the soft x-rays, being emitted from many hot spots along the axis.
This is particularly true for higher Z materials such as tungsten. Even harder radiation (inner
shell transitions) from 10 to 100 keV electrons is also detected at stagnation. The origin of
this electron acceleration process is still not firmly established, and in [51] is usually assumed
to be caused by a resistive or inductive axial electric field. However it should be noted that
runaway electrons have singular orbits and can only occur within one electron Larmor radius
of the axis. Such orbits are discussed in section 2.2; furthermore as shown in [113] off-axis
energetic relatively collisionless electrons drift towards the axis with a E r /B ϑ drift—a sort
of high energy filter—where they are free to accelerate in the applied electric field. In linear
transport theory the velocity dependence of the collision frequency leads to the Ettingshausen
heat flows (see section 2.6) in the direction of J × B, of which the runaway electron drift is an
extreme example. Thus on the axis when the axial electric field has convected or diffused in
(perhaps anomalously quickly by the vortices of the fine scale m = 0 instabilities discussed in
section 5.8) the electron distribution function could be very non-Maxwellian. In the presence of
fine-scale instabilities and viscous heating, the rapid conversion of magnetic energy to thermal
energy will be accompanied by a greatly enhanced E z field, and hence enhanced Ettingshausen
advection of hotter particles to the axis, leading to hot spots and runaway electrons [395].
Due to the limitations of resolution in x-ray images of the pinch at stagnation, the fastest
growing but small amplitude m = 0 MHD instabilities that are most effective for viscous
heating of the ions accompanied by rapid conversion of magnetic energy, have not yet been
detected. However the more slowly growing and larger wavelength modes will overtake these
in a time of order of the radial Alfvén time, and probably are the origin of the large amplitude
hot spots which radiate many of the soft x-rays. Axial flows away from the high pressure hot
spots will occur, and the line density at each spot could drop below the critical value leading
to anomalous resistivity and electron and ion beam formation as discussed in section 3.14.
Such a disruption would occur several Alfvén transit times after the main soft x-ray pulse has
peaked.
This tendency for runaways is greatly enhanced in vacuum spark experiments, and quite
hard x-rays are generated here. Bright spots in vacuum sparks have been modelled by Vikhrev
et al [396] and for a Z-pinch [328] and also Maxon et al [397]. Sotnikov et al [398] have
employed a hybrid 3D MHD code including the Hall term, modelling both m = 0 and
m = 1 modes. The development of an electromagnetic flute mode in a precursor plasma
carrying a current [399] showed the evolution of large scale structures and the generation of
shorter wavelength turbulence. But simulations would require 2 µm resolution in order to
model the fast-growing short wavelength modes and their associated viscous heating of [130].
There is clearly a need to understand the energy conversion process to x-rays. Whilst it is
difficult to calculate the ion kinetic energy of the implosion accurately, there is a consensus
that the energy radiated can be much greater than the kinetic energy [400]. Candidates for
this are reversible pdV compressional heating by the J × B forces [321]; magnetic bubbles
from m = 0 instabilities coupled to resistive dissipation [331, 332]; m = 1 instabilities
leading to helical structures and enhanced resistive losses [324] associated with the current
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