A review of the dense Z-pinch

More documents

Recommendations

Info

Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review larger than in current direct-drive laser-capsule experiments elsewhere or in dynamic hohlraum experiments on Z, this being mainly due to the higher implosion velocities. Recently Welch et al [495] have produced kinetic particle-in-cell (PIC) simulations of the Sandia deuterium experiment [492] and have found both thermonuclear and beam–plasma contributions to the neutron yield. More needs to be done with this type of modelling. Can ion viscosity or even electron viscosity [496] be included in such a model? However, as in nearly all high current Z-pinches it appears that the final thermal energy of the pinch is over double that arising from the kinetic energy. Thus in [494] to obtain agreement with the experimental neutron yield, noting that the experimentally measured electron temperature is 2.2 keV and that the calculated equipartition time exceeds twice the Alfvén transit time τ A (i.e. the total bounce time of the plasma column) an ion temperature of 10 keV (i.e. greatly exceeding the Bennett value) and an ion density n D of 3 × 10 20 cm −3 (compared with the measured 2.4 × 10 20 cm −3 [492]) is required. In contrast a new calculation based on the Bennett pressure balance at stagnation gives T i = 5.5 keV. Using the experimental density for two ion transit times (each of 8 ns) and σv = 2.3 × 10 −19 cm 3 s −1 the neutron yield is 4 × 10 13 , close to experiment. Here we have used the formula Y n = 1 4 πa2 z 0 n 2 D σv × (2τ A) (7.1) where a is the pinch radius and z 0 its length. The kinetic energy prior to stagnation, based on the ion radial velocity of 70 cm µs −1 [492] is 5.1 keV per ion with very low electron temperature. The equipartition time is 25 ns (≫τ A ), allowing the ion temperature to exceed the electron temperature over the period of neutron emission. In this model it is assumed that the increase in ion thermal energy is the result of ion-viscous heating associated with small amplitude, short wavelength m = 0 MHD instabilities [130] rather than just reversible P dV heating. The latter in any case would be included in current 2D MHD simulations, and is usually insufficient to account for preferred ion heating. As noted in [494] these simulations generally include fudge factors to prevent radiative collapse, which can be physically prevented by ion-viscous heating [130] or some extra phenomenological ion heating source [329–332]. A puzzling result is that of Kroupp et al [497] who, from detailed time-resolved Doppler shifted spectroscopic measurements of optically thin lines in neon, showed that there is in this experiment a balance between the ion kinetic energy and the radiated energy, via electron energy through equipartition. There is no rise in T i at stagnation in contrast to that measured by LePell in [130]. However it can be shown that at this relatively low current (320 kA at stagnation) gas-puff Z-pinch there is negligible viscous heating and all short wavelength MHD modes are essentially resistively damped, i.e. the magnetic Prandtl number is much less than one [253]. There are several ways of arriving at this conclusion. One is to examine where in the I 4 a–N i stability regimes diagram, namely figure 16 (but altered for the higher Z value according to equations (3.4), to (3.13), this experiment at stagnation lies. I 4 a is found to be 4.2×10 18 (A 4 m) at N i = 1.27 × 10 19 (m −1 ). The electrons are weakly magnetized with e τ ei = 4.6 (using the spatial average of B θ ) while the ions are unmagnetized at i τ i = 2.4 × 10 −3 making the ratio a i /a = 0.025 irrelevant. The magnetic Reynolds number S is 91, i.e. ∼100, on the border of resistive stabilization effects [153, 154] while the Reynolds number R is 1.7 × 10 4 taking into account the contribution from electron viscosity. γτ i = 1.2 × 10 −4 ≪ 1 is closely related to R −1 , and shows that viscous effects are negligible. In another approach considering the relative damping of Alfvén waves by resistivity and viscosity it is found that the competing diffusivities are η/µ 0 = (µ 0 σ) −1 and ν (the effective kinematic viscosity), respectively. For [497] η/µ 0 is 0.386 m 2 s −1 while ν is 2.0 × 10 −3 m 2 s −1 showing that resistive damping dominates over viscous damping. Thus 109
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 73. (a) A magnetic bubble is formed as a m = 0 RT or MHD perturbation at the pinch surface. The plasma closes back behind it, producing a flux tube. (b) The flux tube is driven to the axis due to both curvature stress and buoyancy force. (c) Near the axis, it still carries the same magnetic flux, but most of its energy has been dissipated in the plasma. Reprinted figure with permission from [235, figure 1]. Copyright 2000 by the American Physical Society. in this experiment, in contrast to [130] and many high current implosion experiments, there is essentially no extra viscous conversion of magnetic energy to thermal energy but only resistive heating of electrons and compressional heating. On the other hand, Coverdale et al [492]’s gas-puff experiment has parameters I 4 a = 1.97 × 10 26 A 4 m, N i = 1.21 × 10 22 m −1 , e τ e = 10 3 , i τ i = 76, a i /a = 7.3 × 10 −3 , S = 9.1×10 5 (≫10 2 ), R = 5.7, γτ i = 0.45, η/µ 0 = 4.5×10 −3 m 2 s −1 , ν = 3.0×10 2 m 2 s −1 , showing that viscous MHD is dominant, and resistive effects are negligible. Interestingly both experiments have relatively long equipartition times, γτ eq being 3.5 for Coverdale’s and 0.76 for Kroupp’s experiment. Thus in Coverdale T e is expected to be much lower than T i during the stagnation period whereas in Kroupp T e will converge towards T i which itself will fall as there is essentially no viscous heating, just equipartition from the thermalized ion kinetic energy. There are many other experiments which have displayed higher T i than can be explained by resistive MHD with gas puffs. One of these by Wong et al [498] has measured an ion temperature of 36 keV in neon (90%) + argon (10%) while T e is 1.25 keV with a current of 8 MA (using Saturn at Sandia). To satisfy the Bennett relation the effective line density N i has to be half the nominal annular fill, namely 4 × 10 20 m −1 . From the measured electron density the ion density is 7.6 × 10 25 m −3 leading to a pinch radius of 1.3 mm, consistent with x-ray profiles. From this R is 114 while S is 9×10 4 . Thus viscosity dominates over resistivity (S ≫ R) in contrast to Kroupp where R(∼ 10 4 ) ≫ S(∼ 90). A discussion of the distinction between viscous and resistive Z-pinches and the role of numerical viscosity in simulations is to be found in [395]. A detailed analysis for neon, argon and krypton gas puff Z-pinches by Labetsky et al [499] claimed that the additional heating was due to toroidal magnetic bubbles associated with m = 0 instabilities, penetrating to the axis. A model in Rudakov et al [235] suggests that the extra energy is not due to pdV work, nor Ohmic dissipation nor anomalous resistivity but that magnetic energy in the bubbles is somehow converted into thermal energy leading to an effective resistance. Figure 73 taken from [235] illustrates the mechanism. This idea was first published in 1993 in a little quoted paper by Lovberg et al [329] who calculated an effective 110
Page 1 and 2:
Home Search Collections Journals Ab
Page 3 and 4:
Plasma Phys. Control. Fusion 53 (20
Page 5 and 6:
Page 7 and 8:
Page 9 and 10:
Page 11 and 12:
Page 13 and 14:
Page 15 and 16:
Page 17 and 18:
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
Page 55 and 56:
Page 57 and 58:
Page 59 and 60: Plasma Phys. Control. Fusion 53 (20
Page 109: Plasma Phys. Control. Fusion 53 (20
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169:
show all

A review of the dense Z-pinch

Create successful ePaper yourself

Delete template?

Save as template?