A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 87. Schematic diagram of the inverse Z-pinch [611]. Copyright © 1963 Cambridge University Press. resulting configuration should be MHD stable because of the favourable magnetic curvature of the B ϑ field inside the plasma shell and the neutrally stable B z field outside [610]. An outer conducting wall together with the field coils can provide the means of attaining an equilibrium. This is illustrated in figure 87. In the formation of the plasma shell, the current forms a piston and a radially moving shock will move ahead. This is the subject of several papers by Vlases [611]. He finds that both theoretically and experimentally there is a big difference between assuming zero or finite electrical conductivity ahead of the shock. For zero conductivity the shock velocity is larger but the shock strength is weaker. Because of the need for applying an axial magnetic field it is difficult to match such a field with the MG azimuthal magnetic fields which can be produced with modern pulsed power. But closely related is the Z– pinch and work on liner compression of axial magnetic fields discussed in section 8.4. An inverse Z-pinch has been employed by Harvey-Thompson [612] in order to gain a greater understanding of the ablation process in a wire array. Interestingly when the technical design led to some precursor current through the wires, no axial instabilities occurred. This would be consistent with a temperature exceeding the critical value for electrothermal instabilities. This is a very recent finding, and further confirmation is needed. 8. Applications of Z-pinches 8.1. Early reactor concepts for the Z-pinch A Z-pinch has some peculiar advantages over other magnetic confinement schemes. The most important is that no external magnetic field coils are employed. Such coils limit the 131
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review magnetic field to ∼5 T, unless further compressed. Indeed the azimuthal magnetic field for 20 MA in 1 mm radius is as high as ∼4000 T, the local β is of order 1 and the plasma density at fusion temperatures (∼10 keV) is some 10 7 times that of a tokamak. The nuclear reaction rate depends on the square of the ion density, and so the confinement time becomes a critical factor, together with the instantaneous fusion power. If the confinement time is less than an ion transit time along its length, there will be no problems with impurities unlike in other magnetic confinement devices unless the plasma forms at first on an insulating wall. Wire-array and gas-puff Z-pinches do not interact with a wall, as, unlike other magnetic confinement devices, the currents for equilibria are not driven by outward diffusion of the plasma. For example the tokamak has radially outward Pfirsch–Schluter diffusion to generate the equilibrium currents. Thus in considering reactor configurations the ‘first wall’ can be far away, and indeed could be a liquid not limited in energy flux to 10 MW m −2 , and could also carry the return current [10]. The disadvantages of the Z-pinch for fusion are well documented in papers extolling the virtues of tokamaks, stellarators and other low density quasi-steady-state devices. The first is its instability to m = 0 (sausage) and m = 1 (kink) instabilities. The second is the heat loss to the two electrodes. In section 2.3 it was shown that the power loss is mainly to the anode and is of order 5 2 IT.ForI = 2 × 107 A and T = 10 4 eV this amounts to 0.5 TW. Clearly for this to be negligible a fusion power in the tens of TW at least must be considered and therefore for a reasonable size power station the Z-pinch must be pulsed. This indeed fits in with current knowledge of dynamic Z-pinches in which, for example the intense soft x-ray power occurs over 1 or 2 radial Alfvén transit times. For high Z radiating loads this is sufficient time to radiate a significant fraction of the stored energy, especially if the viscous heating mechanism (section 5.8) applies. Indeed this mechanism takes advantage of MHD instabilities and in this context instabilities are excellent for Z-pinches. The dynamic ‘bounce’ of the pinch is very non-adiabatic, and is accompanied by the onset of longer wavelength modes, and later, probably a disruption (section 3.14). The disruption is characterized by the creation of electron and ion beams, the latter which for deuterium will produce neutrons from a beam–target mechanism. To provide a longer confinement time various ideas have been employed. (a) The preparation of Kadomtsev current density profile (section 3.2) as in a gas-embedded pinch (section 7.4) can eliminate the m = 0 instability, or at least, for a plasma–vacuum boundary condition, reduce the growth rate. Instead the m = 1 instability dominates and indeed endeavours to form a helical minimum energy configuration, i.e. an axial magnetic field is spontaneously generated (presumably of arbitrary sign). (b) Sheared axial flow can also extend the stability time (sections 3.9 and 7.7) and further work at high currents is needed to explore this concept. (c) Large ion Larmor radius effects have been shown both experimentally [14] and theoretically to reduce the growth rate of the m = 0 instability (section 3.8). (d) An axial magnetic field could be applied and compressed by means of a liner to provide a more stable configuration (see section 7.10). Early ideas and concepts for a Z-pinch fusion reactor were summarized by Sethian [613], before some of the recent advances were known, and when it was considered that radiative collapse would limit the current dynamically to a few megampéres. Hartmann [614] envisaged a gas-puff or a gas-embedded Z-pinch (which with presentday knowledge is, in its hollow form, a strong candidate). Later he extended this to the continuous flow pinch [615] which had earlier been proposed by Morozov [616]. This relied on sheared axial flow to stabilize the pinch. Indeed Newton et al [617] did find in an experiment conducted in 1967 that a continuous flow pinch could form at the end of a Marshall gun’s central electrode at 0.5 MA. The pinch lasted 100 µs at several ×10 17 cm −3 electron density at 100– 200 eV. However reference to section 2.3 and [97] would indicate a heat loss of 2.5TI, equal to 0.25 GW to the electrodes even at this low power. At the temperature and current required 132
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A review of the dense Z-pinch

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