A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review However in many later simulations, single fluid MHD equations have been used, with an artificial viscosity replacing the real viscosity [269] and where the emphasis has been on solving compressible 3D MHD problems [270] for many Alfvén transit times. The recent work on simulating wire-array Z-pinches will be considered in section 5. 4.6. The skin effect and inverse skin effect One of the earliest theoretical problems in describing the dynamical behaviour of a Z-pinch was to calculate the radial current distribution. The total current is rising in time at first and then falls due to the increase in inductance as the pinch forms. For a cylindrical conductor of uniform conductivity the current density can be found knowing only the total current I(y)as an arbitrary function of dimensionless time, y = t/(µ 0 σa 2 ) [110], where a Laplace transform was employed. The formula for j z (x, y) where x = r/a is j z (r, t) = 1 2πi lim η→∞ ∫ ζ +iη ζ −iη e yp p 1/2 I 0 (p 1/2 x) 2πa 2 I 1 (p 1/2 ) ∫ ∞ 0 I(q)e −pq dq dp, (4.27) I 0 and I 1 being modified Bessel functions and where a Fourier–Mellin inversion is employed. The case of a current pulse as arising from an overdamped capacitor discharge is shown in figure 35. The early skin effect is clearly identified, while after peak current the radial gradient of current density at the conductor surface becomes negative. If the rate of fall of current is sufficiently high the current density reverses, leading to an outward J ×B force on the plasma. This was further explored theoretically by Culverwell et al [271]. Experiments have been carried out by Jones et al [272]onZ-pinches which have attributed a current density reversal to the inverse skin effect. In these experiments the outer layer of plasma is ejected by the reversed J ×B force and this, together with its current, was measured. More recently Lee et al [273] have run a 1D single fluid MHD simulation of a Z-pinch as used at CERN as a magnetic lens [274] (see also section 8.3). In this the authors find a reversed current density locally and internal to the pinch associated with the reflected shock moving radially outwards. The shock leads to compression of the azimuthal magnetic field, and so in the shock layer a large negative value of ∂B ϑ /∂r exists which through Ampère’s law leads to a local negative j z sheet. In this paper the authors criticize Kumpf et al [275] in their simulations and, more positively, draw attention to other experimental data [276, 277]. It should be noted however that many terms are omitted in this single fluid code which might significantly modify the results. 4.7. The Rayleigh–Taylor instability The RT instability is important in all dynamic pinches where the acceleration of the plasma or shell is caused by the J × B magnetomotive force in the outer, lower density plasma. This instability is also the most damaging phenomenon in capsule implosions and the later fuel expansion into the denser shell in inertial confinement fusion (ICF). Classical RT concerns the stability of the interface of a heavy incompressible fluid of density ρ 2 supported against gravity g by a lighter one of density ρ 1 . It was first considered theoretically by Lord Rayleigh [278] and experimentally verified by Taylor [59] sixty years’ later. The linear exponential growth rate γ is given by γ = (kgA) 1/2 , (4.28) where the Atwood number A is given by A = ρ 2 − ρ 1 (4.29) ρ 2 + ρ 1 63
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 35. Current density distributions for normalized total current I(t)−exp(−αy)−exp(−βy), showing early skin effect and later inverse skin and reversed surface current effects [110]. and k is the wave number orthogonal to the direction of g. Instead of gravity acting downwards the system could be accelerated upwards, with the same result. A sinusoidal perturbation at the interface will release potential energy, the heavier fluid falling and, π out of phase, the lighter fluid rising. Some ten years later Latham et al [279] attributed the surface instabilities found in a dynamic Z-pinch in argon following the first bounce on axis as being due to RT, and growth rates were measured for the wavelengths found. Reference [39] presents this work in more detail. In this case the light fluid can be considered to be the magnetic field and so A = 1. Of course the MHD m = 0 compressible mode is also unstable and the competition between these modes is discussed in [39]. Later an accelerating shell will be considered, and here the RT will dominate over MHD, by the square root of the ratio of the shell radius to its thickness [280]. In general the combined mode is termed MRT, and cannot easily be dissected in simulations. However an important effect of the azimuthal magnetic field is that the MRT is essentially confined to the r–z plane. 64
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