A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 13. Analytic zero-dimensional model showing that radiative collapse is transient. The pinch radius a(t) and current I(t)are plotted [122, figure 1]. some additional physics such as electron degeneracy or opacity. Vikhrev [118] in a zerodimensional model of a dynamic pinch with an external circuit, anomalous resistivity, viscous heating, transition of bremsstrahlung to black-body radiation and degenerate electron pressure modelled the collapse of deuterium with 1% xenon added to enhance radiation loss. Initial currents of 1 or 10 MA were used, the latter leading to a radiative collapse despite the inclusion of axial outflow of plasma. This was extended by Vikhrev and Gureev [119] for an initial current of 100 MA in an MHD model with axial outflow, leading to a loss of plasma from the constriction at a degenerate density of 10 33 m −3 when the current is 78 MA. The inclusion of anomalous resistivity by Vikhrev et al [120] led to the expansion of the coronal plasma. With the quasi-static evolution of a Z-pinch from an initially cryogenic fibre in mind Robson [121] and Haines [122] developed semi-analytic 0D models of a radiative collapse. Robson and Haines both found that a collapse to infinite density would occur at a current of √ 3IPB . This arises from a solution of the energy equation giving a(t) for a given I(t)in the form ∫ I 3 a 2 = B (1 − I 2 /IPB 2 ) dt, (2.68) where the constant B is B = 8 ( ) 3/2 16πNkB . (2.69) 3µ 0 α ⊥ µ 0 By including the circuit of a pulse-forming line Haines showed that a finite collapse to 10 4 × solid density could transiently occur. It is interesting to see how the large voltage (perhaps 100 MV) needed to drive a mega-ampère through a very narrow collapsed Z-pinch is provided by a large value of L dI/dt as is shown in figure 13. By including the variation of ln , Chittenden et al [123] showed that the collapse was not limited by the dissipation and fall of the current flow below I PB because I PB itself also fell. Robson [124] showed that the lower initial value for I PB for Z = 2 fully ionized helium led to a weaker collapse since the stored magnetic energy at the time of collapse was smaller. In Robson [125] a 0D model in which anomalous resistivity was introduced in the outer corona gave an enhanced value of I PB while 29
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 14. Radius of the pinch and current as a function of time. Reprinted with permission from [129, figure 6]. Copyright 1990, American Institute of Physics. end losses [97] led to a smaller current in energy balance (section 2.3). Thornhill et al [126]in another 0D model included degeneracy pressure of the electrons bremsstrahlung and opacity effects, but maintained a Spitzer resistivity. Meirovich [127] in a broad ranging paper, applied also to astrophysical situations, considered radiative collapse to an electron degenerate state. In this and his later paper [128] he postulated a relative drift velocity much greater than the thermal speed. As can be shown a relativistic beam leads to a breakdown of quasi-neutrality, but it is likely to lead also to micro-instabilities and anomalous resistivity. In a full 1D time-dependent resistive MHD simulation Chittenden and Haines [129] employed a realistic external circuit, a radiation transport package including opacity, magnetized electron transport and equation of state. The limit of radiative collapse was found to be the onset of opacity effects and degeneracy pressure. In figure 14, the pinch radius and current are shown as a function of time, while figure 15 shows the radiative power and density. It is interesting to note that at the peak density of 10 5 ×solid the high plasma frequency prevents even propagation of x-rays of wavelengths longer than 10 Å from the central core. However recent work on nonlinear growth of m = 0 instabilities, to be discussed in section 5.8, indicates that the conversion of magnetic energy to ion thermal energy and pressure through viscous dissipation will prevent radiative collapse altogether [130]. Furthermore the 30
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A review of the dense Z-pinch

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