A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 63. Radial optical streak photographs and comparison of the inferred radial positions of the inner and outer arrays with the 0D model. Reprinted figure with permission from [325]. Copyright 2000 by the American Physical Society. It is only when the current-carrying final implosion (of the inner array for a nested array operating in a transparent mode) stagnates on axis that the large x-ray pulse occurs. For small ion–ion mean-free paths a shock will recompress the precursor plasma on the axis, and there will be conversion of directed ion energy into thermal ion energy on the τ ii time scale. This is followed by equipartition on the longer τ eq time scale accompanied by soft x-ray radiation as proposed in the heuristic model [280]. Indeed for low wire number Al implosion it is τ eq 91
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review that determines the x-ray rise time, while, when this is shorter than the Alfvén transit time τ A ≡ a/c A , which determines the bounce time and subsequent plasma expansion, it is the latter which determines the pulse rise time and width. Another phenomenon which broadens the x-ray rise time and pulse width is axial nonuniformity associated with zippering which in turn can be associated with the axially varying radial electric field. This will be considered in section 5.10. The pinch radius, or rather the FWHM density profile of the hot plasma (excluding trailing mass), is probably determined by several physical mechanisms; the radial amplitude of the RT instabilities; the radius of the imploding current shell when it responds to the reflected shock from the axis; if the implosion is very three dimensional with kink modes and local angular momentum (see [324]) it is the angular momentum or rather the centrifugal force together with the generated axial magnetic field pressure which prevents further compression. In 1D and 2D simulations a problem arises when the current greatly exceeds the Pease– Braginskii current [8, 9] in that a radiative collapse will occur as the radiation loss at pressure balance exceeds the Ohmic (or Joule) heating. Since experimentally collapse does not occur, it has to be prevented artificially in the codes. In section 5.8 we will find that inclusion of m = 0 modes and ion viscosity provide a natural physical process for preventing radiative collapse. 5.8. m = 0 instabilities and ion-viscous heating In some experiments the radiated energy could be 3 or 4 times the kinetic energy [1, 365, 386–388], and, if it were assumed that the ion temperature is equal to the electron temperature the plasma pressure was too low to balance or overshoot the magnetic pressure at stagnation. A resolution of these two mysteries was proposed [130] in which the fast growing, short wavelength m = 0 interchange MHD instabilities would grow and nonlinearly saturate. Then ion-viscous heating associated with the velocity shear and compression is sufficient to raise the ion temperature T i in a few nanoseconds to a very high temperature, sufficient for T i ≫ ZT e (in a low mass array) and for pressure balance. The equipartition of energy to the electrons leads to the elevated radiation levels. For the particular stainless-steel array experiments on Z employing a relatively low line density record ion temperatures of 200 to 300 keV (over 2 billion degrees Kelvin) were predicted and measured from Doppler broadening of optically thin lines [130]. The choice of wavelength of the instability was based on a viscous Lundqvist number of 2, to give a maximum linear growth rate and maximum viscous heating, this wavelength being in the range 10–100 µm, just a little greater than the ion mean-free path. It would be difficult to detect these instabilities because of both the fine scale and the small amplitude at saturation, the amplitude of the perturbed velocity being only c A / √ ka where k is the wave number. At this ion temperature pressure balance, i.e. the Bennett relation, equation (2.7) holds at stagnation. Ideal MHD instabilities grow at a rate γ MHD ∼ = cA (k/a) 1/2 , i.e. increasing as k 1/2 . The critical damping of Alfvén waves by viscosity occurs at a viscous Lundqvist number L µ of 1 where L µ = 2ρ(c2 A + c2 s )1/2 (µ ‖ /3+ν 1 )k , (5.8) c s is the sound speed, µ ‖ is the parallel ion viscosity equal to p i τ ii and proportional to T 5/2 i and ν 1 is the perpendicular ion viscosity given approximately by µ ‖ /(1+4 2 i τ ii 2) where i and τ i are the ion cyclotron frequency and ion–ion collision time, respectively. The saturation amplitude of the perturbed velocity ṽ can be found by equating (ṽ ·∇)ṽ to γ MHD ṽ giving an eddy velocity amplitude of c A /(ka) 1/2 . Since in the example considered 92
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A review of the dense Z-pinch

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