A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 30. Schematic drawing of the radial velocity associated with the nonlinear, algebraic phase of the instability as a function of ka [280, figure 4]. Copyright © 1998 IEEE. Assuming continuity of velocity at t = t 1 ˙ξ k (t 1 ) = (ka) 1/2 c s /(ka) = α 1 c s . (3.45) Hence α 1 is (ka) −1/2 , which shows that the short wavelength modes can have a smaller amplitude in the nonlinear phase. For long wavelengths, which saturate at a time t 2 , the linear displacement is ξ k (t 2 ) = ξ 0k exp[(ka) 1/2 c s t/a] = βa < k −1 (3.46) and for t>t 2 , similar arguments lead to a nonlinear displacement ξ k = α 2 c s (t − t 1 ) + βa. (3.47) Continuity of velocity at t = t 2 gives ˙ξ k (t 2 ) = (ka) 1/2 c s β = α 2 c s . (3.48) Hence, in the long wavelength case α 2 is β(ka) 1/2 , and the amplitude increases as (ka) 1/2 . Similar arguments will apply to the RT instability (section 4.8). As a result the maximum amplitude occurs at the boundary between these two regimes, i.e. at ka = β −1 ∼ = 2, or λ ∼ = πa. This is illustrated in figure 30. This considerably oversimplified model ignores mode–mode coupling, but nevertheless describes the main trends of both experiment and simulation. The necking region of the m = 0 mode can eventually lead to a disruption with electron and ion beam formation, the former giving rise to hard x-rays and the latter, in the case of deuterium to a neutron burst. These phenomena are considered in section 3.14. So far we have concentrated on the m = 0 sausage instability which is the dominant mode in compression pinches and wire or fibre pinches. However, in gas-embedded Z-pinches where the initial current channel is triggered (e.g. by laser) along the axis, a Kadomtsev type of profile develops and the m = 0 instability does not occur. Instead the m = 1 kink mode is excited. An analytic ideal MHD model of a helical perturbation in a Z-pinch by Psimopoulos and Haines [199] showed that to fourth order in time no harmonic generation occurs. In another paper [200] the inclusion of the response of an external circuit on the nonlinear development shows that for ka above a critical value of 0.802 there is an abrupt transition to a rapidly expanding plasma, as seen in experiments. The formation of a helical pinch with associated axial magnetic field can be considered as an example of a plasma evolving towards a minimum energy state [201]. The nonlinear development of an m = 0 instability under large ion Larmor radius conditions has been studied by Arber [202] in a hybrid model of particle-in-cell ions in a 49
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review cold collisionless electron fluid background. As discussed in section 3.8 the linear growth rate can be reduced by 70–80%. However, nonlinearly these instabilities continue to grow exponentially without saturation until the plasma column is disrupted. These results are sensitive to the plasma vacuum boundary conditions, and indeed in this low density region other physical processes such as the lower hybrid instability might occur due to the high local electron drift velocity. Further work is needed here, but at the present time it would seem that the hoped for saturation of the instability under LLR conditions does not occur. The nonlinear development of a Z-pinch with a sheared axial flow has been studied [203]. Related to this is the stability of cosmic jets [204–206]. In Bell and Lucek it was shown that the m = 0 instability saturates leading to a Bennett-like structure. This is seen in some jets; indeed extra galactic jets are observed to be stable. In a 3D MHD simulation Lucek and Bell [207, 208] showed that with no additional axial magnetic field sheared, supersonic jets are indeed unstable to m = 1 helical modes. In a Z-pinch with sheared axial flow [209] it was shown how both m = 0 and m = 1 instabilities can saturate in amplitude. How to employ this phenomenon in a practical experiment is however difficult. An apparent reduction in instabilities was observed by Wessel et al [56] in wire surrounded by a flowing aluminium plasma. However a velocity of the order of the sound speed is theoretically required, and it is required to maintain this flow around the pinch for an axial transit time. Thus the apparent saturation may be a feature of the convection of the mode in a finite size system, i.e. due to the flow rather than the shear in the flow. 3.12. Electrothermal instability and the heat-flow instability The current-driven and heat-flow driven electrothermal instability can be important in relatively dense and cold plasma carrying a current and subject to rapid Joule heating. The essential mechanism is that a spatially varying perturbation in electron temperature, T e , orthogonal to the direction of the current density will lead to perturbed current density and Joule heating. This is due to the Te 3/2 power law for the electrical conductivity which will lead to perturbed Joule heating that will enhance the perturbed temperature. Such an instability is called an overheating instability in a tokamak [210]. When the electron temperature is decoupled from the ion temperature, it is named the electrothermal instability and was initially studied in the context of non-equilibrium plasmas for closed-cycle MHD power generation [211, 212]. Its application to a fully ionized plasma [158] led to an explanation of early occurring instabilities in a theta pinch. Here the k vector of the perturbation is in the direction of the magnetic field, and with ion motion also included it is found that there are two essential conditions for growth of the instability. One is that the Joule heating of the electrons as measured by the equipartition rate to ions (of fixed temperature in the model) must be sufficient to lead to a ratio of electron to ion temperature given by T e > 3+√ 57 ≈ 1.32. (3.49) T i 8 The second necessary condition is that the electron–ion mean-free path should be less than the collisionless skin depth, or more precisely ( ) 1/2 4α0 c λ mfp < , (3.50) 3γ 0 ω pe where in the notation of Braginskii [98] or Epperlein and Haines [100] the dimensionless factor (4α 0 /3γ 0 ) 1/2 has values ranging from 0.459 for Z = 1 to 0.170 for Z =∞. In terms of density (m −3 ) and temperature (eV), this condition is T e
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A review of the dense Z-pinch

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