A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review 3.7. Hall fluid model of stability Ideal MHD employs the Ohm’s law E + v × B = 0, and as discussed in section 3.1 and in particular in figure 16 it ceases to be valid at low line density. When the Hall effect and ∇p e term are introduced into a collisionless Ohm’s law to give E + v × B − J × B n e e + ∇p e n e e = 0, (3.39) these extra terms can be considered to be part of the FLR corrections. Indeed some authors incorrectly consider that these terms are the sole FLR terms (e.g. [87]). The connection between FLR for the ions and the Hall terms in the electron equation of motion can be seen by considering the equilibrium pressure balance for the ions, which if stationary as a fluid, is E =−∇p i /Zn i e or E/B ∼ = mi vi 2/ZeBL = v ia i /a where a i /L ≡ ε is the ratio of ion Larmor radius to the pinch radius or pressure scale length L. It follows that when the Hall terms are added to Ohm’s law a new component of electric field arises and all are of order εv i . However in the Hall fluid model it is assumed that the ions are collisional, i.e. i τ i < 1, and the FLR terms in the stress tensor are ignored. The resistivity is also ignored. Thus the range of parameter space for the validity of the Hall model is very limited. An important feature of the Hall fluid model for stability is that the growth rate is complex. This can be understood because Ohm’s law now has the magnetic field frozen to the moving electrons in the equilibrium and perturbation. Tayler [175] and Schaper [176, 177] first considered the Hall fluid model for the Z-pinch, though Schaper limited his theory of the m = 0 mode to the incompressible limit and constant density, finding a stabilizing effect. In contrast, using a linearized initial value code Coppins et al [178] showed that by increasing ε faster growth could occur for a Bennett profile. For a Kadomtsev profile, i.e. equation (3.27) (i.e. stable to ideal MHD), new instabilities could be triggered. Using one of Schaper’s profiles and near incompressibility, stabilization is found but new unstable modes can be triggered. 3.8. Large ion Larmor radius (LLR) effects For much of the parameter space in figure 16 at high I 4 a and low N the LLR regime applies. This is also the regime for a deuterium–tritium Z-pinch under fusion conditions or a hydrogen pinch prior to radiative collapse. It is therefore important to understand both the linear and nonlinear stability in this regime. As discussed earlier the standard FLR approach will fail for even ε ∼ = 0.1 because of the strong curvature of the magnetic field (section 3.5) and the absence of singular particle orbits (section 2.2) and resonant particles [179]. Therefore numerical modelling of the Vlasov equation for ions [180] with a cold electron fluid background to maintain quasi-neutrality is employed. The electron fluid includes the Hall term and so is frozen to the magnetic field. Like the Hall fluid model we find that growth rates are complex. The Vlasov model naturally allows anisotropic pressure to arise and can be used for arbitrarily large ion Larmor radius. Whilst for most MHD equilibria, formally the threshold for instability is the same as for ideal MHD [180], this is not the case for the m = 0 mode for a Z-pinch; indeed as ε → 0 the Vlasov model should and does yield the same as the CGL ordering [11]. In figure 24 the real part of the normalized growth is plotted versus ε using both the linearized initial value code FIGARO and a variational code for two equilibria: a parabolic density profile and the Bennett equilibrium (equation (2.10)). For the former case it can be seen that the growth rate at ε = 0.2 is a minimum of about 0.3 times the ε = 0 case. The growth rate is very sensitive to the equilibrium profile, and indeed large ion Larmor radius (LLR) destabilizes the Bennett equilibrium. 43
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 24. Real part of the normalized growth rate for m = 0 is plotted versus ε using the linearized initial value code FIGARO and a variational code for a parabolic density profile and the Bennett equilibrium. Reprinted figure 1 from [11]. Copyright 1994 by the American Physical Society. The reason for the increasing growth for ε>0.2 for the parabolic profile is not fully understood but it could be that as the ion Larmor radius increases there is less time for kinetic smoothing of the MHD mode because the ion cyclotron period approaches the characteristic growth time a/c A . Forafixedε the growth rate peaks at ka = 5 and then falls off, as expected, for larger ka. For a skin-current pinch Arber and Coppins [181] had earlier shown that the growth rate saturated at ka = 5. The variational code has also been used to explore the m = 1 mode [12]. Figure 25 shows the normalized growth rate for three profiles. LLR has a more marked stabilizing effect here, especially on the parabolic density profile, and there is an 80% reduction in growth. Experiments on the compressional pinch have indicated a reduction in growth rate under large Larmor radius conditions [131, 132, 182] but it is difficult to be more precise because of the time limitations of the experiments. Compression Z-pinch experiments can be devised such that at first compression i τ i is larger than 1 and a i /a is optimal at ∼0.1 [183]. Indeed a recent experiment by Davies et al [14] has shown a marked reduction by a factor of 2 for the m = 0 mode at ε = 0.1. Figure 26 shows the measured growth rate as a function of ε. 3.9. Effect of sheared axial flow An intuitive view of the effect of a sheared axial flow velocity v z (r) would suggest that if a significant velocity shear were to occur over the region occupied by the zero-velocity ideal MHD eigenfunction ξ r (r) for the perturbed displacement, then the plasma, being frozen to the 44
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A review of the dense Z-pinch

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