A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 25. Normalized growth rate for m = 1 for three equilibrium profiles. Reprinted figure 1 from [12]. Copyright 1995 by the American Physical Society. Figure 26. Experimentally measured growth rate and ka as a function of the fill-pressure p 0 showing a marked reduction to 0.21 of ideal MHD at ε = 0.15, corresponding to p 0 = 1 mbar. Reprinted figure 5 with permission from [14]. Copyright 2001 by the American Physical Society. magnetic field, would be strongly affected. Indeed, as shown by Arber and Howell [57], this is the case, and there is a marked stabilizing effect. For high ka where ξ r (r) is localized close to the surface, a sheared velocity close to r = a can reduce the growth rate (even to zero) and the eigenfunction becomes very narrow. Figure 27 shows the normalized displacement eigenfunction as a function of r for three values of the dimensionless axial velocity u o where v z (r) = v T i u o (1 − r 2 /a 2 ) (3.40) for m = 0, ka = 10, β = 1.1 and for an equilibrium pressure profile p = p 0 (1 − r 2 /a 2 β), β being a parameter greater than one which determines the pressure at r = a. This is for an internal mode. For free boundary eigenfunctions they are non-zero at r = a, and the plasma is more unstable. Figure 28 shows the normalized growth rate for the three fastest free boundary modes, nevertheless showing stabilization for increasing u o . But flow Mach numbers in excess of 1–5 are required. However at even larger values of u o the Kelvin–Hemholtz instability can be triggered. The effect of sheared rotation, which changes the equilibrium pressure profiles, has also been studied. 45
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 27. Normalized displacement eigenfunctions as a function of r for three values of the dimensionless axial velocity. Reprinted with permission from [57]. Copyright 1996, American Institute of Physics. Figure 28. Normalized growth rate for the three fastest free boundary modes as a function of ka. Reprinted with permission from [57]. Copyright 1996, American Institute of Physics. A theory which predicts even greater stabilization with axial velocity shear was published by Shumlak and Hartman [135], but this was criticized for having a discontinuity in velocity gradient at r = 0[184]. This was not satisfactorily justified in [185]. It is probably this which led to the singularity in the eigenfunction at r = 0. Zhang and Ding [186, 187] recently studied the effects of compressibility and axial flow profiles on stability, though they seemed unaware of the earlier work of [53]. Experiments with sheared axial flow [188] are reported in section 7.7. 3.10. Addition of axial magnetic field It is well known that an axial magnetic field can have a stabilizing effect; indeed it was the addition of such a field that led to the development of the toroidal pinch, the reversed field 46
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