A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 22. The stabilizing effect of resistivity is illustrated by the delayed growth of the MHD mode until a critical S is reached through Joule heating [133] or[153]. In a footnote Neudachin and Sasarov [156] state that they could not reproduce the stabilizing effect of resistivity found by Cochran and Robson. In their paper the stability of various heterogeneous resistive equilibria was studied with varying thermal conduction added, and thermal instabilities can exist in equilibria of high dissipation (see section 3.11). For heterogeneous equilibria with a cold dense core, it is the corona that is subject to MHD instabilities. Finally, a more careful analysis by Coppins and Culverwell [157] reveals that at early times the pinch is not stable but a thermal instability [158] driven by Joule heating occurs. This is out of phase with the MHD mode which at low R m is more slowly growing. 3.4. Visco-resistive models of stability Several authors have included viscosity in addition to resistivity to the fluid model of stability. It might be considered that such a model is of more hypothetical interest, as can be seen from figure 16, the only regime where both effects are simultaneously important is where the ion Larmor radius is about one tenth of the pinch radius. The Hall effect and the FLR terms in the stress tensor at least should therefore be included, though the conventional stress tensor does not correctly include the effect of singular orbits (section 2.2) nor of strong curvature of the magnetic field (see section 3.5). Furthermore for Z>5 it is necessary to include electron viscosity in the equation of motion, and more generally in Ohm’s law. The triggering of some anomalous resistivity such as that caused by hybrid turbulence should also be included, particularly in the low density regimes (section 3.12). Mathematically the damping of Alfvén waves in an homogeneous plasma by resistivity is identical to the damping by isotropic viscosity. Spies [159] employed this and studied the combined effect of scalar viscosity and resistivity to short wavelength modes in the incompressible limit. He found that for ka > (SR) 1/4 the Z-pinch was stable. However the product SR is equal to 2.86 × 10 −16 N for Z = 1or122a 2 /ai 2 . Thus a small value of SR implies large ion Larmor radius and Hall effects, which are not included in this model. Cox [160] considered the m = 0 and m = 1 linearized modes without resistive or viscous heating, extending the work of Spies to longer wavelength. He found that taking the full 39
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 23. Normalised growth-rate variation with (SR)* illustrating the simultaneous stabilizing effect of viscosity and resistivity. Reprinted with permission from [161]. Copyright 1993, American Institute of Physics. anisotropic viscosity in a magnetic field, as i τ i became ≫1 the solutions essentially reverted to ideal MHD unstable behaviour, the fluid-like FLR terms having only a minor effect. Cochran and Robson [161] added scalar viscosity to the 2D resistive MHD model to study the m = 0 growth in a time-evolving equilibrium. For the range of ka explored they found the critical value (SR) ∗ shown in figure 23 which the pinch is unstable, considering the two extreme viscous models of i τ i = 0 and ∞. All of these papers were motivated by trying to explain the apparent anomalous stability of the early cryogenic deuterium fibre experiments, but it turns out that the critical line density implied by (SR) ∗ is some 10 3 smaller than that in the experiments. Nonlinear 2D simulations of Lindemuth et al [162] show that a fibre develops into a hot unstable plasma surrounding a cold core which is largely unaffected by instabilities. There appears also to be a nonlinear saturation of the coronal MHD instability. This is similar to what has been found in later work on single fibre experiments and simulations (see sections 7.5 and 7.2). Some mention should be made of papers by Gomberoff and co-workers [163, 164] who in various models with resistivity, thermal conduction and viscosity but also with an additional small axial component of magnetic field, predict the onset of convective cells. Further analysis of the effect of viscosity and viscous heating is deferred to section 5.8. 3.5. The stress tensor; axial differential flow and strong curvature Using the notation of Robinson and Bernstein [165] and taking the magnetic field to be only in the θ direction the components of the stress tensor are P rr = p ⊥ − 1 ( 2 τ ∂vz θθ + v 1 ∂z − ∂v ) ( r ∂vz + v 2 ∂r ∂r + ∂v ) r , ∂z P zz = p ⊥ − 1 ( 2 τ ∂vz θθ − v 1 ∂z − ∂v ) ( r ∂vz − v 2 ∂r ∂r + ∂v ) r , ∂z 40 P θθ = p || + τ θθ = p || − µ || ( 2 r ∂v θ ∂θ − 2 3 ∇·v + 2v r r ) ,
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A review of the dense Z-pinch

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