A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review where the lower figure refers to Z →∞and the higher to Z = 1. The fastest growing mode has a wavelength λ of [ λ ∼ σκmi ] 1/2 = 2π = (1.52–0.56) × 10 20 T 2 A 1/2 (3.52) 3n 2 e 3 n e Z ln ei and growth rate of γ ∼ m e α 0 = 1.2 = (1.03–1.77) × 10 −12 m e n e Z ln ei m i τ ei m i Te 3/2 (3.53) This theory satisfactorily explained the occurrence of an instability in a theta pinch during the early plasma formation [213], agreeing well on the variation of wavelength with density. Here the electron density at very early times was in the range 5 × 10 20 to 3 × 10 21 m −3 , the electron temperature 2 to 3 eV, the growth rates 5 × 10 5 to 8 × 10 7 s −1 and wavelengths 7.6 to 0.75 cm. A similar break-up of the current into filaments would be expected in the gaseous compressional pinch or plasma focus very soon after plasma formation if condition (3.51) holds. Such an instability however could also occur at stagnation of a highly radiating wirearray Z-pinch when the conditions ((3.50), (3.51)) can sometimes be satisfied, because of the high electron density achieved, leading to azimuthal variation with m ∼ 5–9. However care must be exercised in this interpretation because such structure might be triggered by the slots in the return current conductor. The reason for the optimum wavelength is that short wavelengths are damped by thermal conduction, κ, while for long wavelengths the combined effects of Faraday’s law and Ampère’s law ensure that an electric field perturbation will depend on k −2 , and this field will oppose the change in current density. This theory has been extended to the case of nonlinear heat flow [214], particularly as it applies to the heat flux in a laser-driven implosion for inertial confinement. Here the heat flow reaches about 10% of the free-streaming limit. Indeed this is necessarily the case because the inward heat flow, the source of energy for the implosion, must exceed the outward enthalpy flux associated with the ablation process. This strong heat flux can be approximated to a relatively collisionless inward flow of hot electrons (the mean-free path varies as V 4 ) and an equal and opposite current of cold collisional electrons. This cold return current is driven by an inward electric field—the thermoelectric field, which only arises because of the velocity dependence of the collision frequency. Indeed the outward acceleration of ions—the ablation process—is impeded by this electric field which is of the wrong sign to accelerate the ions, and it is the frictional force between the cold return current and the ions which causes the ablation [215]. J · E for the hot electrons is negative and is equal and opposite to the J · E for the cold return current. Thus, if the conditions (3.49) and (3.51) are satisfied, the Joule heating associated with the cold return current will cause an electrothermal instability associated now with heat flux [214]. This instability is the likely cause of fine-scale filamentary structures observed at first by laser shadowgraphy [216] and recently by MeV proton probing [217] of plasmas produced by laser–solid interactions. The latter demonstrated that the instability originates at the ablation surface. In the theory [214] the hot electrons exhibit a Weibel-like response to the perturbing electric and magnetic fields, especially if they are beam-like, but if their temperature is comparable to their directed energy they have no major effect on the electrothermal instability. Similar filamentary structures are observed on the surface of the cores of wires in wirearray Z-pinches [218, 219]. An interesting fact is that when experiments try to stimulate a different wavelength, the natural wavelength for a particular material still persists [220]. These experiments will be discussed later in section 5.4. 51
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review 3.13. Micro-instabilities and anomalous resistivity Some form of micro- or velocity-space-instability will be triggered when the drift velocity v d (≈ J/n e e) of the current-carrying electron exceeds some critical velocity v crit , which is typically of order the sound speed, c s (= √ [(ZT e + T i )/m i ]). The instability will rapidly grow in time and saturate in amplitude. The resulting electrostatic or electromagnetic turbulent wave structure will interact with the drifting electrons, causing them to be scattered. This scattering is usually much larger than that caused by Coulomb collisions with ions; hence an anomalous resistivity will be generated. The microinstability that is perhaps most relevant in the Z-pinch configuration is the lower hybrid drift instability, because the current and drift velocity are orthogonal to the magnetic field. It is related to the modified two-stream instability studied earlier by Buneman [221] and Ashby and Paton [222]. Krall and Liewer [223] derived a linear electrostatic kinetic model of the lower hybrid instability. It does not have a critical velocity threshold, but nonlinearly its effective collision frequency varies as the square of the electron drift velocity. A quasi-linear model to find a value for the anomalous resistivity was developed by Davidson and Gladd [224]. Huba and Papdopoulos [225] considered electron resonance broadening as a saturation mechanism especially in finite β plasmas. Particle simulations by Winske and Liewer [226] in contrast find that ion trapping is a saturating mechanism in the high drift velocity regime, while Davidson [227] found current relaxation and plateau formation can cause saturation at low drift velocity. Probably the two most relevant papers to yield in 2D nonlinear modelling a value for the anomalous collision frequency at saturation are Drake et al [228] and Brackbill et al [229]. The former considered nonlinear mode–mode coupling of the electrostatic waves showing that energy is transferred from long wavelength modes to short wavelength modes which are Landau damped. The anomalous collision frequency ν anom so found from this model is ν anom = 2.4( e i ) 1/2 (v d /v i ) 2 . (3.54) Here ( e i ) 1/2 , where e and i are the cyclotron angular frequencies of the electrons and ions, respectively, is the lower hybrid resonance frequency, while v d and v i are the electron drift velocity and the ion thermal speed, respectively. Brackbill et al [229] employed in contrast a 2D implicit e.m. particle-in-cell code, VENUS, initially with a Harris equilibrium. The saturation mechanism was considered to be electron kinetic dissipation with wavelengths greater than the electron Larmor radius. The resulting anomalous collision frequency found in this numerical simulation differs only slightly from equation (3.43), namely ν anom = C ( e i ) 1/2 (v d /v i ) 2 , (3.55) 4πβ i where C is 0.13–0.38 and β i is the ratio of ion pressure to magnetic pressure. Before applying these results to the Z-pinch it is perhaps worthwhile to recall the basic physics underlying the lower hybrid resonance. Although the ion cyclotron frequency is present in the formula, the ions are in fact to a good approximation unmagnetized. Figure 31 illustrates how the lower hybrid resonance arises when the ion plasma frequency is much higher. Consider a plane e.s. wave propagating in the z-direction. The unmagnetized ions will oscillate to and fro in the ±z-direction. With a steady magnetic field in the y-direction, the magnetized electrons will have an oscillatory Ẽ/B drift in the ±x-direction. Superposed on this is the polarization drift, Ė/(B e ) in the ±z-direction, which will convert the electron guiding-centre trajectory into an ellipse. The resonance occurs, i.e. a tendency for zero charge separation when the spatial amplitude of the ion oscillation eẼ/m i ω 2 equals the width of the ellipse Ẽ/(B e ); this gives ω = ( e i ) v 2 52
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A review of the dense Z-pinch

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