A review of the dense Z-pinch

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Plasma Phys. Control. Fusion 53 (2011) 093001 ( ∂vr P zr = P rz =−v 1 ∂z + ∂v ) ( z ∂vz + v 2 ∂r ( ∂vθ P zθ = P θz =−µ 1 ∂z + 1 r ( 1 P rθ = P θr =−µ 1 r where for ions we approximately can write v 1 ∼ = µ ‖ 1+4 2 i τ 2 i v 2 ∼ = p ⊥i 2 i · µ 1 ∼ µ ‖ = 1+ 2 i τ i 2 , µ 2 ∼ = p ⊥i i · while, from Haines [93] [ ∇·P = r ∧ 1 ∂ r ∂r (rP rr) + 1 r [ + ∧ 1 z r , 4 2 i τ 2 i 1+4 2 i τ 2 i 2 i τ 2 i , ∂v z ∂θ ∂v r ∂θ + ∂v θ ∂r − v θ r , ∂z − ∂v r ∂r ) ( 1 − µ 2 r ) , ∂v r ∂θ + ∂v θ ∂r − v θ r ) + µ 2 ( ∂vθ ∂z + 1 r ∂v z ∂θ Topical Review ) , ) , (3.29) 1+ 2 i τ i 2 µ ‖ = 1.1p ‖ τ i , (3.30) ∂ ∂θ P θr + ∂ ∂z P zr − P θθ r ∂ ∂r (rP rz) + 1 r ∂ ∂θ P θz+ ∂P zz ∂z ] [ + θ ∧ 1 ∂ r 2 ∂r (r2 P rθ ) + 1 ∂ r ∂θ P θθ + ∂ ] ∂z P zθ ] . (3.31) The conventional finite ion Larmor radius (FLR) terms have the coefficients v 2 and µ 2 and persist in the collisionless limit (τ i →∞), but do not enter the irreversible viscous heating rate which is given by − τ : ∇v = µ ‖ 3 ( 2 r +µ 1 [ (1 r ) 2 + v 1 [ (∂vr ∂v θ ∂θ + 2v r − ∂v r r ∂r − ∂v z ∂z ∂v r ∂θ + ∂v θ ∂r − v ) 2 ( θ 1 ∂v z + r r ∂θ + ∂v θ ∂z ∂r − ∂v ) 2 ( z ∂vz + ∂z ∂r + ∂v ) ] 2 r ∂z ) ] 2 , (3.32) which is positive definite, demonstrating irreversible entropy production consistent with the second law of thermodynamics. Here the traceless stress is τ = P − p ⊥ (ˆr ˆr + ẑẑ) − p ‖ ˆθ ˆθ. The papers of Cox [160] and of Cochran and Robson [161] use either this or the isotropic (collisional or unmagnetized) limit ( i → 0). The FLR terms can lead to the spontaneous development of sheared axial flow. This is analogous to the differential rotation of θ-pinches proposed by Velikhov [166] and contained in a review [93]. It occurs even in 1D, i.e. with only radial dependence. Taking the collisionless limit, the axial component of the equation of motion is ρ ( ∂vz ∂t + v r ∂v z ∂r ) = 1 r ∂ ∂r ( r p ⊥i ∂v r 2 i ∂r ) . (3.33) This indicates that, whilst the total axial momentum is conserved, a sheared axial velocity of order εv t can be generated through radial motion where v i is the ion thermal speed and ε is a i /a, the ratio of ion Larmor radius to pinch radius. The stress tensor in equation (3.29) was developed for straight magnetic lines of force. The effect of strong curvature such that (B θ /r) ≈ (∂B θ /∂r) is to increase the relative magnitude 41
Plasma Phys. Control. Fusion 53 (2011) 093001 Topical Review of the curvature drift. In the collisionless FLR ordering, as developed by Rosenbluth and Simon [167] and extended to β ≈ 1 by Bowers and Haines [168, 169], heat-flow terms also appear in the stress tensor, and indeed reduce the growth rate of instabilities because the higher energy ions have a higher guiding-centre drift velocity associated with the grad B and curvature drifts; hence leading to a collisionless heat flux and a spatial smoothing of inhomogeneities. To include strong curvature would at first sight appear to be contrary to the standard ordering scheme in which the growth rate γ should be of order ε 2 i . Such an inconsistency manifests itself as a violation of quasi-neutrality. To overcome this ∂/∂z can be ordered as ε 2 /r, i.e. only long axial wavelengths are included. Then, for example, P rr to second order becomes (omitting the subscript i) P rr = p ⊥ + 1 ∂ 4r ∂r (rθ z) − θ ‖z r + p ⊥ ∂v z 2 ∂r , (3.34) where the heat flux terms for a doubly Maxwellian distribution are θ z =− 2p ⊥ ∂ ∂r θ ‖z =− p ⊥ ∂ 2 ∂r ( p⊥ ρ ( p‖ ρ ) , (3.35) ) + p ‖ ρr (p ‖ − p ⊥ ). (3.36) These extra terms will modify both the stability and the development of sheared axial flow. However, because of the limitations of the FLR expansion near the axis as B → 0 and the need to consider singular ions, a more complete theory will be included when discussing large ion Larmor radius effects in section 3.8. 3.6. Anisotropic pressure effects In the limit of small Larmor radius the collisionless Chew, Goldberger and Low [140] (CGL) theory applies when the collision frequency is much less than the MHD growth time (see equation (3.10)). In this general CGL theory there is always a difficulty of prescribing the undefined heat-flow parallel to the magnetic field. However, as pointed out by Akerstadt [170] this is identically zero for the case of m = 0 stability of a Z-pinch, and the CGL theory exactly applies. The characteristics of the CGL theory are the doubly adiabatic laws d B (p 2 ) ‖ = 0 = d ( ) p⊥ (3.37) dt ρ 3 dt ρB which together with the equilibrium pressure balance equation (2.17) lead to a more optimistic condition for m = 0 stability [171]. For anisotropic equilibria the pinch is stable to m = 0if r − (p ⊥ + p ‖ ) (p ⊥ + p ‖ )< 3β ⊥ +4β ‖ + 5 2 β ⊥β ‖ + 1 8 (β ‖ − β ⊥ ) 2 , (3.38) (1+β ⊥ )(β ⊥ − β ‖ ) where β ⊥ is 2µ o p ⊥ /B 2 and β ‖ is 2µ o p ‖ /B 2 . For an isotropic equilibrium the rhs becomes (5β +14)/(4(1+β)) which is always larger than 2Ɣ/(1+Ɣβ/2), the corresponding (Kadomtsev) ideal MHD case (for Ɣ
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A review of the dense Z-pinch

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