A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review move in circular Larmor orbits. From the fluid point of view, if the ion centre-of-mass velocity is zero there is no net magnetic force on the ions per unit volume, only the electric field force. In general the azimuthal magnetic field is both curved and radially non-uniform. At the axis B θ is zero and here and nearby conventional guiding-centre theory breaks down. At larger radius where guiding-centre theory holds there are three types of ion guiding-centre drift: E r /B θ , i.e. the axial component of the E × B/B 2 drift; −(m i vi⊥ 2 /Z eBθ 2)(∂B θ/∂r), the gradient B drift which depends on the velocity component v i⊥ in the plane perpendicular to the magnetic field, i.e. the r–z plane; and m i vi‖ 2 /ZeB θr, the curvature drift associated with the velocity of the particle parallel to the magnetic field vi‖ 2 , and the curvature of the magnetic field 1/r. Following [45] the relationship between the mean values of these drifts, which are all parallel to the z-axis and the centre-of-mass ion velocity v iz in the z-direction is that they differ by the axial component of the diamagnetic velocity −(1/n i m i )curl ⊥ (P ⊥i i / 2 i ), where P ⊥ is the perpendicular pressure and i = ZeB/m i is the vector cyclotron frequency. This can be seen by generalizing the radial ion pressure balance equation (2.13) to include both a net c.m. axial velocity v zi and anisotropic pressure, 0 = Zn i eE r − Zn i ev iz B θ + P ‖i − P ⊥i − ∂P ⊥i (2.14) r ∂r and rearranging it to give an equation for the axial ion centre-of-mass velocity, v zi =− 1 ( ) ∂ rP⊥i − P ⊥i ∂B θ Zn i er ∂r B θ Zn i eBθ 2 ∂r + P ||i Zn i eB θ r + E r . (2.15) B θ The first term on the right-hand side is the diamagnetic velocity; the other terms can easily be recognized as the average gradient B drift and curvature drift plus the E × B/B 2 drift. It is perhaps useful to recall the meaning of the diamagnetic velocity. In the frame of reference in which the mean guiding-centre velocity is zero the Larmor (circular) orbits of charged particles of each species will lead to a contribution to the magnetization per unit volume of M v = nµ where µ is the magnetic moment − 1 2 mv2 ⊥ B/B2 and µ is the mean averaged over the distribution function. The value of curl M v gives the contribution of that species to the magnetization current density; (1/ne)(∇ ×M v ) ⊥ is therefore the diamagnetic velocity which becomes −(1/nm)(∇×(P ⊥ / 2 )) ⊥ . An alternative description can be given in terms of the contribution to momentum in a small element of volume of particles passing through this element as they execute circular motion (see [93]). Electron motion can be similarly described, their radial pressure balance, by analogy with equation (2.14) being 0 =−n e eE r + n e ev ez B θ + P ‖e − P ⊥e − ∂P ⊥e r ∂r . (2.16) Addition of equations (2.14) and (2.16) together with the assumption of quasi-neutrality yields J z B θ =− ∂P ⊥ + P ‖ − P ⊥ , (2.17) ∂r r where P ⊥ is P ⊥e + P ⊥i and P ‖ is P ‖i + P ‖e . Equation (2.17) represents MHD pressure balance for an anisotropic plasma, but with zero off-diagonal stresses, as the distribution functions in the rest frame of each species are bi-Maxwellian. Equation (2.15) contains what might seem a paradoxical result: namely that even if v zi is zero there can be a net flow of guiding centres of ions in the z-direction. It would appear that the particles flow on average from one end of the Z-pinch to the other, but their c.m. velocity could be zero. There therefore must be a return flow of ions somewhere. It will now be shown that this flow is associated with the singular ion orbits caused by snake-like orbits 13
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review Figure 4. A sketch of ion guiding-centre and singular orbits. Reprinted with permission from [349, figure 2]. Copyright 2000, American Institute of Physics. of ions within one Larmor radius of the axis as illustrated in figure 4. The net guiding-centre flow of ions is ∫ a ( Er 2πn i r − P ) ⊥i ∂B θ 0 B θ Zn i eBθ 2 ∂r + P ‖i dr. Zn i eB θ r Using equation (2.15) this becomes ∫ a [ 2πn i r v zi + ( )] [ ] 1 ∂ rP⊥i 2π rP r=a ⊥i dr = N i v zi + , (2.18) 0 Zn i er ∂r B θ Ze B θ r=0 where N i is the ion line density and v zi is the mean ion centre-of-mass velocity in the z-direction. At r = a, P ⊥i will vanish. The last term in equation (2.18) is therefore − 2π Ze ( rP⊥i B θ )r=0 =− 2π Ze ( 2P⊥i , (2.19) µ 0 J z )r=0 where equation (2.3) has been integrated for small r. Equation (2.19) represents the return singular flow of ions with their mean thermal speed within one Larmor radius of the z-axis where B θ vanishes. It corresponds to a singular ion current, I si [45], given by ( 4πP⊥i I si = , µ 0 J z )r=0 (2.20) = n i Zev ⊥i πRi 2 , (2.21) where R i is the distance from the axis at which the mean ion Larmor radius is also R i , i.e. R i = v ⊥i i (R i ) = 2mv ⊥i , (2.22) Zeµ 0 J z R i 14
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A review of the dense Z-pinch

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