A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review an effect similar to the GORGON simulations mentioned above; second a local depletion of line density due to the radial Hall current in the plasma adjacent to the cathode giving an axial flow of plasma towards the cathode surface; however, in a simulation using LASNEX it was shown that this effect could be swamped by gold vapour and plasma emitted from the cathode surface, [408], but the Hall effect would be the reason for the apparent necking-off at the anode seen by Sarkisov et al in his figure 7 [405] and also in images of early Z-pinches [300]. However when there is a hole in the bottom electrode this Hall-driven axial flow can occur unimpeded. It was in such an experiment that the unexpected axial asymmetry in radiated power was first identified [406]. This process is the basis of a plasma opening switch. A related paper by Bland et al [370] in which a wire array is extended by a factor of 3 in length by mounting it within holes in the cathode and anode, shows markedly different behaviour in the upper (anode) half where E r is negative compared with the lower half. There is no gradual transition as in [405] but a sharp change in behaviour occurs at the mid-point where E r reverses sign. The upper half has wire cores which have expanded to only 100 µmin diameter compared with the usual 300 µm in the lower half; the precursor plasma has a higher ablation velocity consistent with a lower mass ablation rate, perhaps caused by the smaller area of the core surface. As a result the precursor assembles on the axis earlier, at 75 ns, and is brighter in emission. The lower half (where E r > 0) precursor plasma stagnates at 90 ns, but the final implosion, which is dependent on the erosion of the core at axial positions associated with the filamentary instability, occurs first in the lower half and stagnates at 200 ns. The upper half stagnates at 230 ns. Thus there are two main x-ray pulses, which might be useful in profiling the x-rays for emission for ICF applications. This delay in implosion of the upper half requires ṁ to be halved, thus leading to a doubling of the precursor ablation velocity from 15 to 30 cm µs −1 . It is interesting that the wavelength of the axial instability is the same throughout, i.e. independent of the core diameter in contrast to [373]. Surprisingly it is the lower half (E r > 0) which behaves more closely like a standard short array fixed between anode and cathode (but where E r < 0) on MAGPIE. The reason for the sharp transition half-way along the wires is not understood, but it could be that the magnitude of the radial electric field is sufficiently large to cause plasma formation and current shunting uniformly in the upper half. At the transition region there could also be a local strengthening of the azimuthal magnetic field due to ∇T e ×∇n e magnetic field generation [75], which could tend to sharpen up the transition. The early concentration of current in the liquid-vapour core when E r is positive can also occur for E r < 0 when insulating coatings around each wire are employed [411]. The vapour formed around each wire core, and emitted from the electrodes could contain contaminants which might require conditioning techniques to remove [412]. At very early times, when the current associated magnetic field is small, there could be insufficient magnetic insulation, and electrons emitted by the wires with E r < 0, instead of forming an electron sheath as discussed above, will be accelerated to the return conductor. The critical value of the fields at which this occurs can be found as in a magnetron by examining the orbits of electrons leaving a cylindrical emitting surface. The equations of motion are m dv r dt m dv z dt =−eE r + ev z B θ , (5.21) =−ev r B θ . (5.22) For a large number of wires, again the near-field effects can be ignored and the global variation is dominant. Thus rE r and rB θ are independent of r, giving v 0 =−E r /B θ > 0 independent 99
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 68. Sketch of orbits of ions leaving a wire core with components of axial velocity positive or negative. of r and, with eB θ (r)/m = dτ/dt these give d 2 v r dτ =−v 2 r (5.23) and v r = v 0 sin τ, (5.24) where the boundary conditions v r = 0 = v z at τ = 0 are applied. Writing u ≡ (e/m)rB θ (r) equation (5.24) becomes u d(ln r) = sin τ, (5.25) v 0 dτ which integrates to give ( ) u r ln = 1 − cos τ. (5.26) v 0 R a The maximum radial excursion r max of an electron occurs at cos τ =−1, giving ( ) 2v0 r max = R a exp . (5.27) u Current shorting will occur when r max R c . The current density J will be given by Child’s law, which for a planar case is J = 4ε 0 (2e) 1/2 3/2 /(9me 1/2 d 2 ), (5.28) where is the potential drop over a distance d. In summary, at very early times electron shorting to the return conductor can occur until the azimuthal magnetic field builds up; such a radial current will cause axial plasma flows. As B θ increases the electrons are confined to an electron sheath, carrying a current described by equation (5.16). The radial electric field varies approximately linearly (equation (5.19)) from the anode to the cathode (where the feed point is at the cathode) and there is increased shunting of axial current from the plasma core to the electron sheath at the cathode end. The increased plasma current at the anode end is a reasonable explanation of the zippering of the implosion found by Sanford et al [407]. But there is another source of axial flow in addition to the J ×B force associated with radial current shorting to the walls. This is the effect of ion orbits as plasma ablates from the outer facing wires. See figure 68. It is assumed that ions leave the core equally in all directions, the ions with a component of velocity in the z-direction will be deflected by the azimuthal magnetic field to impact the wire core after a small fraction of a Larmor period. In contrast, 100
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A review of the dense Z-pinch

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