A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 31. Schematic drawing of trajectories of oscillating ions and guiding-centre electrons, showing equal amplitude in the z-direction at the lower hybrid resonance [253]. Copyright © 2001 Cambridge University Press. The lower hybrid instability arises when a density gradient is present in the −x-direction; then the electron elliptic motion leads to an increase in positive potential and charge at the positive peak of the wave and a similar amplification of the negative charge at the negative potential peak. The density gradient corresponds to a net diamagnetic drift of electrons in the −z-direction. Hence such a current is responsible for the growth in the wave amplitude. It is to be noted that nearly all the simulations and theory have been undertaken for a collisionless plasma. However the coronal plasma around each wire of a wire array is typically so collisional that the Hall parameter e τ ei calculated using linear classical transport theory is less than unity [230]. This is also true for single wires, and it is only in the very low density outer corona that the Hall parameter is large enough to be able to apply these models of lower hybrid turbulence. However, it is very useful (see [231, 232], to apply such an anomalous collision frequency as one approaches the plasma vacuum boundary, as it then leads to a more realistic reduction in current density. In particular, a non-singular electron drift velocity is obtained as the density approaches zero or a prescribed lower limit. When modelling the magneto-Rayleigh–Taylor (MRT) instability in the r–z plane, e.g. [233, 234], the inward moving bubbles of plasma leave behind extended spikes. A current reconnection across these spikes in the lower density ‘vacuum’ would generate lower hybrid turbulence. The large, anomalous resistivity would tend to restrict the current shorting despite the low inductance path. Current reconnection is an intrinsic feature of the model proposed by Rudakov et al [235] to explain the dissipation of magnetic energy at stagnation (see section 5.8). A compressible electromagnetic flute mode was considered by Sotnikov et al [236] inatwofluid description to explain enhanced transport across large scale structures. The model was based on a theta pinch with curvature of the magnetic field represented by a radial gravitational 53
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review force. Perhaps as a result the growth rate did not follow the usual k 1/2 ideal MHD dependence for a Z-pinch, but decreased with k. Introduction of lower hybrid anomalous resistivity will also change the formula for the Pease–Braginskii current, a point followed up by Robson [125]. In section 3.14 anomalous resistivity will be introduced in a model of disruption when gaps of low density plasma exist between density islands at the time of a m = 0 disruption and thereafter. It is to be hoped that with the new generation of computers further work, perhaps in 3D, full electromagnetic will be undertaken on lower hybrid turbulence, and the related modified two-stream instability. Meanwhile attention could be drawn also to the need for better diagnosed and specially designed experiments. A paper by Takeda and Imezuka [237] presents measurements of anomalous resistivity associated with large amplitude noise at the lower hybrid frequency. Here the ratio v d /v i was large (∼10 2 ) and the anomalous collision frequency was essentially e . This leads to Bohm diffusion. It is perhaps permissible to speculate that the formula for ν anom given by equation (3.54) or(3.55) is only valid up to a value of e , when it becomes independent of (v d /v i ). For a hydrogen plasma this will occur at v d /v i equal to 4.2. A suitable form for the effective collision frequency νanom ∗ as a function of (v d/v i ) 2 through equation (3.54) or(3.55) for e τ e > 1isgivenby νanom ∗ = ν anom/(1+ν anom / e ). (3.56) 3.14. Disruptions; ion beams and neutrons During the nonlinear evolution of the m = 0 instability of a stagnated compressional Z-pinch or plasma focus employing deuterium as the working gas, a significant pulse of neutrons appears, typically a yield of up to 10 12 neutrons [238, 239]. The origin of this has been a great source of debate and controversy which has still not been resolved. Some of the proposed mechanisms are fluid-like and others are kinetic in nature. An important test of any hypothetical theory is whether total axial momentum is conserved during the ion beam formation. In fact the normally principal pulse of neutrons at the time of m = 0 disruption is often the second pulse [240]. The likely explanation of the first pulse is essentially kinetic in origin. It was first hinted at in Trubnikov’s paper [241] and developed more realistically by Deutch and Kies [242]. Here it is considered that there is a sheet current piston which compresses the plasma as the first pinch is formed. Unlike the snowplough model, it is postulated that the ions on encountering this moving piston are reflected by it (in fact by the radial electric field in the current sheet) with twice the velocity of the piston. During this process they are essentially collisionless and also conserve their angular momentum (a point made by Trubnikov [241] and earlier by Haines [243]). On encountering the moving piston a second time, there is another gain of energy; indeed the faster ions have more reflections and gain even more energy, a feature of this Fermi acceleration mechanism. There will be, as a result, a high energy ion tail, and the ion–ion collisions which will grow in number when the piston reaches close to the axis and the density rises will also yield DD nuclear reactions, but of a suprathermal nature. If the incoming piston is conical in shape, it will also lead to a growth in axial motion. It should be noted that multiple collisions of essentially collisionless ions are needed for this, a condition which might also pertain in certain conditions in the precursor plasma of an imploding wire array. Turning to the principal pulse of neutrons, there are important features; (i) the neutrons occur at the time of an m = 0 instability, a point confirmed experimentally by Bernard et al [238], (ii) the neutrons have an anisotropic distribution consistent with originating from an ion beam in the z-direction: This was also confirmed by Bernard who employed a deuterated 54
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A review of the dense Z-pinch

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