A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 75. Contours of (a) density and (b) rB θ at stagnation, t = 230 ns. Reprinted with permission from [321]. Copyright 1998, American Institute of Physics. m = 1 activity in the simulation which itself is dependent on the initial, arbitrary azimuthal perturbations in the simulation. In ALEGRA simulations in order to obtain energy conservation when switching from Lagrangian to Eulerian meshes plus the numerical viscosity, Lemke [367] transfers the excess energy to ion thermal energy. This then appears equivalent in effect to the ion-viscous heating of Haines [130]. It is interesting that because the choice of k, the wave number of the fastest growing mode in equation (5.7) with L µ = 2, depends on the value of the viscosity, the resulting viscous heating is independent of both k and viscosity. Thus, provided S ≫ R holds, and provided the excess energy is the result of 2D or 3D activity, it is reasonable in simulations to deposit the excess energy into thermal energy. Early attempts to account for the extra heating included an arbitrary increase in all dissipative processes by some assumed turbulence [401]. The 3D modelling introduces an initial chosen azimuthal perturbation, and together with artificial viscosity and a coarse mesh leads to an axial magnetic field of the same order as the azimuthal field [443] much earlier than the experimental observation of any helical structure. This means that there is a generation ∫or injection of helicity to the pinch. Helicity per unit volume, A · B (or some authors use A · B dV ) where A is the magnetic vector potential is governed by the equation ∂ (A · B) + ∇·(E × A + B) =−2E · B, (7.6) ∂t where E is −∂A/∂t−∇. The source term, −2E·B is usually negligible in a highly conducting plasma and so the local change of A · B must be due to a surface flux of (E × A + B) which in simulations has to be prescribed in effect at the outer boundary. This implies the imposition of a radial magnetic field or a radial component of E × A at the outer boundary. 7.3. Plasma focus There are two main configurations of the plasma focus device; the ‘Mather type’ and the ‘Filippov type’, named after their inventors [40, 41] and shown in figure 76. The largest Mather-type device, the 1 MJ experiment at Frascati, produced 10 12 neutrons when optimized for 500 kJ stored energy at 5–8 Torr filling pressure [42]. The neutron yield has a I 4 scaling, as does the x-ray yield, and there are many mechanisms proposed, [241, 243, 246, 503], most of which are designed to account for the anisotropy of the neutron production through the onset of an intense ion beam followed by beam–plasma nuclear reactions. As pointed out in [243] and in section 3.14 ideal and resistive MHD cannot produce a uni-directional flow of ions 113
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 76. The plasma focus; (left) Mather type; (right) Filippov type [84, figure 1]. Reprinted with permission from The Royal Society. from a m = 0 MHD neck in a Z-pinch because this theoretical model is symmetric above and below the axial position of the minimum radius. To merely state that the axial electric field is large [35] is insufficient (a) because in ideal MHD with E + v × B = 0, in the rest frame of the moving plasma itself the electric field is zero, and (b) the only force per unit volume on the plasma due to an electric field is q v E where q v is the net charge density. Because q v = ε 0 ∇·E it follows that the electrostatic force per unit volume is ε 0 E∇·E which is usually negligible compared with the magnetic force and is ignored. In section 3.14 the mechanism of a disruption was discussed as it is a phenomenon common to all Z-pinches and not just the plasma focus. Here a model is outlined which gives a neutron yield from a single m = 0 neck which is close to that found in experiments. Ion acceleration towards the cathode will occur as the m = 0 neck forms [243] (with off-axis ions accelerated in the opposite direction to conserve total axial momentum). When the local line density drops below the critical value given by equation (3.58) anomalous resistivity with associated heating sets in, leading to a transient positive voltage spike. This is because the electron drift velocity exceeds the ion sound speed. Thereafter there is no pressure balance as the current is limited in the m = 0 neck and a fast, transient, and local outward flow of plasma occurs. The current and magnetic field are also swept out and there is a redistribution of magnetic field on the Alfvén transit time scale. Figure 77 illustrates this, and the associated axial electric field will have a large positive value near the axis which accelerates ions in the +z direction and an e-beam in the −z-direction with singular orbits. Further out the electric field reverses in sign, its value at the main pinch radius a z leading possibly to a negative voltage spike. Ions are accelerated in the −z-direction in this region while the axial electron current is driven by the v r B θ electric field associated with the outward flow of plasma. As mentioned earlier, neutrons produced off axis by this reversed ion flux have been measured by Tiseanu and Mandache [250]. By this means axial momentum is conserved. This can be confirmed by examining the induced electric field arising from the change in magnetic flux in the neck region. If the magnetic energy released is given mainly to the ions at a line density N crit an energy E I (eV) arises given by 2N crit leE i = µ 0I 2 ( ) l 4π ln a2 , (7.7) a 1 where a 1 is the initial radius of the neck and a 2 is the final radius (probably greater than the radius of the adjacent bulges). The length of the m = 0 neck, l, cancels in equation (7.7) but 114
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A review of the dense Z-pinch

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