A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review the interface grows into a bubble and spike. The bubble is rising low density fluid driven by buoyancy, while the spike is high density fluid falling almost freely under gravity. In turn the sides of the spike can be unstable to KH instabilities leading to a characteristic roll-up as the heavier fluid passes through the light fluid. In the case where the light fluid is a magnetic field, no KH instabilities will occur. An overview of classical hydrodynamic RT in the nonlinear state has been given by Sharp [304]. 2D simulations of the nonlinear growth leading to turbulent mixing has been reviewed by Youngs [305, 306] while experiments using solid fuel rocket motors to accelerate downwards rectangular tanks of two immiscible fluids have been reported by Read [307] with accelerations of 750 m s −2 over distances of 1.25 m. The main conclusion of the experiments is that the denser fluid penetrates the lighter fluid at a rate 0.07Agt 2 in 3D. Narrow tanks were also employed to give comparison with 2D simulations where the coefficient of 0.07 is replaced by 0.06 in experiment compared with 0.04 to 0.05 in simulations. The bubbles of lighter fluid had displacements also proportional to Agt 2 , with a constant of 1/2 to 1/3 that of the spike growth. Following the bubble and spike formation, the next stages are bubble amalgamation, spike break-up and turbulent mixing, discussed briefly in [304]. Of course bubbles and spikes are purely 2D: in 3D spherical implosions Town and Bell [308] have shown that, dependent on the nature of the initial perturbation, either spikes and (connecting) valleys, or bubbles and (connecting) ridges are formed. Such spherical simulations are important for ICF capsule modelling, ideally with thermal conduction and ablation also present. A 2D representation of a spherical shell implosion with these processes present is given by Henshaw et al [309], where the bubble evolution is particularly important for shell integrity. The nonlinear growth of bubbles is shown to occur immediately for a perturbation amplitude greater than 0.04 times the wavelength. 2D nonlinear RT has been explored in ablation-driven shells by McCrory et al [310], who showed spike amplitude saturation due to ablation while the distortion of the remaining shell by the bubble development poses a limit to 3D spherical compression. Some interesting analytic approaches to nonlinear RT in 2D are given by Ott [311], Basko [312] and by Infeld and Rowlands [313]. These have been used by Haines [280] in developing a heuristic model of the wire-array pinch. The MRT relevant to Z-pinch implosions has mainly been studied in the r–z plane as a 2D problem, assuming that the unperturbed plasma is a uniform shell. This will be discussed in more detail in section 5.5. 5. The wire-array Z-pinch 5.1. Overview It might seem rather odd that the dynamic Z-pinch with the highest and most reproducible performance starts off as a cylindrical array of many wires stretched between two electrodes only 1 or 2 cm apart. Figure 36 is a photograph of nested arrays as typically used in high performance shots at Sandia. There is however great precision in the mounting and spacing of these wires. Even though each wire plasma at early times develops axial instabilities, the wavelength of the mode appears to only be material dependent, but their phases on different wires are uncorrelated. Thus as proposed in [280] the resulting amplitude of the seed perturbations at merger of the wire plasmas varies as n −1/2 where n is the number of wires, a result verified by experiments at Imperial College [314]. The physical nature of this early instability which leads to many separate inward flowing plasma jets is will be discussed later in section 5.4. The single most important experimental discovery was by Sanford et al [20] who showed that above a critical wire number, or, rather below a critical size of gap of the order 1 mm, the resulting x-ray power increases dramatically as shown in figure 37. This is 67
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 36. Photograph of nested arrays of tungsten wires strung between gold plated electrodes. Courtesy of Sandia National Laboratory. Figure 37. X-ray power versus wire spacing for two array radii. Reprinted figure 4 with permission from [20]. Copyright 1996 by the American Physical Society. illustrated also in figure 38 where going to 90 Aluminium wires on the Saturn generator at 8 MA demonstrated not only increased power but also a sharpening up of the pulse. As a result of this, Sandia reoriented their research programme, and converted the PBFA-II generator for use as a Z-pinch driver of 11 MJ, 20 MA in 100 ns. A typical current pulse and radiated power [323]is shown in figure 39 showing a 5 ns FWHM x-ray pulse of ∼120 TW peak power. At first it was thought that the critical gap size was associated with an early merger of wire plasmas to form a shell [20, 280] which then continues accelerating towards the axis, undergoing RT instabilities. Indeed most numerical modelling at that time, e.g. [21, 234] studied the evolution of the MRT instability nonlinearly for a shell in the r–z plane until stagnation. Then the ion kinetic energy of tens of keV is converted to thermal energy by ion–ion collisions partly through shock waves the amplitude of the RT determining the final pinch radius. Subsequently by equipartition to electrons, the electron temperature increases from tens of eV to ∼500 eV or more with associated ionization and radiation. Indeed for aluminium arrays, the equipartition of energy to the electrons can govern the radiation pulse [280] while for tungsten equipartition is typically faster and the Alfvén transit time at stagnation appears to be the controlling factor [315]. 68
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A review of the dense Z-pinch

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