A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review pinch and the tokamak. Two important points however need to be made. The first is that the addition of an axial magnetic field component B z (r) changes the nature of the MHD mode because k · B can be zero at, say, a radial position r s , leading to a regular singular point in the Sturm–Liouville equation; as a result adding B z does not necessarily stabilize it. The second is that for the dense Z-pinch B θ is typically of order 100 T (or 1 MG) and the only way in the laboratory to form a B z of similar or larger magnitude is through compressing a seed B z formed from a coil or perhaps dynamically creating a B z from a helical m = 1 mode (see the next section). It has recently been shown that an axial magnetic field of the required magnitude can be generated from photon spin [189, 190], using a high intensity laser with circularly polarized light. The combined effects of B θ and B z as first studied by Suydam [191] and extended by Newcomb [192]. The most significant result in the Suydam criterion which has been generalized for the case of anisotropic pressure [193] togive ⎛ ⎞ d ⎝ ∑ (p ( ) ‖ + p ⊥ ) ⎠ dµ/dr 2 + 4 dr species µ r s Bz 2/µ o − ∑ > 0, (3.41) (p ‖ − p ⊥ ) species where µ is B θ /(r s B z ). This is a necessary condition for ideal MHD stability. In this case it is shear of the equilibrium magnetic field that provides a stabilizing effect. In a pinch of finite length line-tying at the electrodes stabilizes long wavelength global modes [194]. In contrast for a torus of major radius R the stabilization of the m = 1, n = 1 kink is provided by the Kruskal–Shafranov limit that the safety factor q ≡ rB z /RB θ is greater than one [195]. However the introduction of even a small amount of resistivity which allows the field lines to tear and reconnect leads to unstable modes with eigenfunctions localized at the singular surfaces where k · B = 0, k being the mode wave-vector. In a hot fusion plasma where the Lunquist number might be as high as 10 9 , this results in the tearing layer being very narrow with a high local current density. Aparicio et al [196] have shown how magnetic reconnection can occur on a fast time scale due to the triggering of micro-instabilities such as the ion-acoustic drift mode in this narrow layer, i.e. the effective resistivity in the reconnecting layer becomes anomalously high. However this rich and interesting subject lies outside the scope of this paper. 3.11. Nonlinear growth of MHD modes The nonlinear evolution of instabilities has benefited from modern computers. In figure 29 a 2D numerical simulation of the evolution from a cold start of an exploding 33 µm diameter carbon fibre is compared with schlieren images [197] at various times. The fibre at first breaks down and the coronal plasma expands. At the same time the current is rising and in a 1D simulation it would at some point recompress because of the increasing relative strength of the J × B pinch force. However, due to the possibility of growth of a m = 0 MHD instability in a 2D simulation, this mode grows strongly at this time from (in this particular simulation) an initial random perturbation. In the necking regions bright spots of x-ray emission occur where the hot pinching plasma compresses onto the dense fibre [198]. These bright spots bifurcate axially both in the simulations and experiment as ionizing fronts propagate, driven by the J ×B forces associated with the radial component of current density. Similar simulations have been undertaken by Lindemuth [162] to model the evolution of cryogenic deuterium fibres. An interesting feature in both experiment and simulation is that a particular axial wavelength dominates nonlinearly, and indeed increases as the pinch radius increases. This 47
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 29. Comparison of 2D MHD simulations with schlieren images (left of each pair on top line). Lower line images are simulations at 33 ns. Reprinted with permission from [197]. Copyright 1997, American Institute of Physics. is despite the fact that linear stability theory states that the growth rate increases with k, approximately as k 1/2 . One reason for this can be deduced by considering the nonlinear ρ(v ·∇)v term in the fluid equation of motion. The m = 0 mode leads to the development of necking regimes where there is a high J × B pinching force and bulges which expand almost freely with a velocity related to the sound speed. For the bulges if we assume that the nonlinear convective term balances the pressure gradient and that the J × B force is much weaker, i.e. ρ(v ·∇)v ∼ = −∇p, (3.42) it follows that v ∼ = cs , the sound speed (Ɣp/ρ) 1/2 , and that the radial displacement will be proportional to c s t. The mode evolves to this nonlinear state when either ξ r is ∼βa or k −1 , whichever is the smaller. (The value of β depends on the radial profile of ξ r in the linear phase, e.g. in figure 27, and is approximately 0.5.) This means that the fastest growing short wavelength modes will become nonlinear earlier and will change from exponential growth to algebraic, in this case a linear dependence on t. If this occurs at time t 1 , and the linear growth rate γf is approximately (ka) 1/2 c s /a, we can write ξ k (t 1 ) = ξ 0k exp[(ka) 1/2 c s t 1 /a] = k −1 t 1 the displacement is given by ξ k = α 1 c s (t − t 1 ) + k −1 . (3.44) 48
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A review of the dense Z-pinch

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