A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review shunting [405]. Increasing the wire diameter reduced the asymmetries of polarity which is consistent with the following theory. Whilst the wire core is surrounded in general by a quasi-neutral coronal plasma of radius a p , it itself is surrounded by an electron sheath formed by emission of electrons by the negative radial electric field. The Maxwell equation ∇·D = ρ in the sheath gives 1 ∂ ε 0 r ∂r (rE r(r)) =−n e (r)e for r>a p . (5.14) This outer sheath shields the coronal plasma from the radial electric field and also leads to E r /B θ guiding-centre drift of the electrons, so causing an axial current density J z (r) and a magnetic field from Ampère’s law given by E r (r) J z (r) =−en e B θ (r) = 1 ∂ µ 0 r ∂r (rB 0(r)). (5.15) On integrating this with the boundary conditions that at r = a p , E r = 0 and rB θ = µ 0 I p /2π where I p is the current flowing in both the coronal plasma and core the fundamental sheath relation 1 > 1 c 2 ( Er B θ ) 2 ∣ ∣as = 1 − I 2 p I 2 T > 0 (5.16) is obtained where a s >a p is the outer radius of the sheath and I T is the total current in the sheath, coronal plasma and core. It should be noted that significant shunting of current into the sheath can only occur at early times when B θ is small, and E r /B θ approaches the velocity of light. The theory is relativistically correct for |E r /B θ | cno guidingcentre drift in the frame moving with E r /B θ will occur; instead there is a local inertial frame in which B θ = 0 and free acceleration of the electron in the negative radial electric field will occur, causing the electron to hit the return conductor. To find what determines the value of the radial electric field at the sheath edge, a closed line integral ∫ E · dl is taken along the (assumed perfectly conducting) return conductor and anode, along a distance z outside the sheath of the wire, and across the vacuum gap to the return conductor as illustrated in figure 67. There will be a resistive contribution to this along the wire proportional to z where Ohm’s law E z = ηJ z ≈ η pI p (5.17) nπap 2 holds, and the plasma current I p is now distributed among nwires, The corresponding inductive contribution from Faraday’s law is also proportional to z and is the time derivative of the magnetic flux enclosed, zµ 0 I T ln(R c /a s ), R c being the radius of the return conductor. Thus the radial electric field E r is proportional to z, the distance from the anode, where for the usual case the power feed is at the cathode. In addition in the vacuum outside the sheath rE r is essentially independent of r. Thus the charge per unit length in the sheath Q l to yield this electric field is Q l = 2πε 0 rE r . (5.18) For a wire array as opposed to a single wire an approximation has to be made. Without going into detail, it can be shown that for a large number of wires radial electric field Er 1 at the outer surface of the sheath on each of the n wires is approximately [ Er 1 (z) ∼ z η p I p = − 2na s nπap 2 ln R + µ 0 ˙ ] I T , (5.19) c/R a 2π 97
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 67. Diagram of path of the closed line integral ∮ E·dl. where R a is the radius of the wire array. It can further be shown that at early times and during the current prepulse when a s and a p are small and close to the original radius of each wire a w the resistive contribution is dominant, but later the inductive term is more important. For the former case −Er 1/E z → Z/(2na s ln R 0 /R a ); and close to the cathode at early times is typically ∼10 to 10 2 . This formula approaches that of Sarkisov [405] when ln(R c /R a ) → /R a where R c = R a + and ≪ R a . Another dimensionless number (somewhat like an inverse Reynolds number) − E1 r ηl cBθ 1 (z = l) = cnµ 0 ap 2 ln(R (5.20) c/R a ) describes the relative importance of the radial electric field, in particular the electron drift velocity compared with c as in equation (5.16). Here l is the anode to cathode separation, and Er 1 and B1 θ are the fields at the surface of each sheath, the sheath radius a s being assumed small compared with πR a /n. In Sanford et al [407] it is shown that the axial asymmetry of the imploding wire array is sensitive to the value of −Er 1/E z, small values leading to the least asymmetry in the axial x-radiation power between top and bottom. The asymmetry is also manifested in a zippering of the main implosion with stagnation occurring first near the anode. As argued in an earlier paper by Sanford et al [408], the wire core receives greater heating near the anode because here the radial electric field is zero, and no current shunting occurs. In [407] a simulation using the RMHD code GORGON [324] showed that a hotter core had less precursor plasma and a faster implosion (nearer the anode), while a colder core had significant precursor and a later implosion. However, very close to the cathode there is even earlier stagnation and x-ray emission. This could be caused by several effects; first, a local heating of the core due perhaps to a poor wire-cathode contact [409, 410] resulting in 98
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A review of the dense Z-pinch

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