A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
Under pressure balance the ratio of ion Larmor radius a i to pinch radius, a, is
( )
a i
a = 2kB T 1/2 ( )
i m 1/2
i
m i ZeB θ
a = 2πmi 2 1/2
µ o e 2 N Z(1+Z) → 8.08 × 1/2 108 N −1/2 for D +
(3.15)
and depends only on ion line density. The lines a i /a = 0.1 (finite ion Larmor radius or FLR)
and a i /a = 1 (large ion radius or LLR) are shown in figure 16. However, FLR effects only
apply if the ions are magnetized, i.e. i τ i > 1 where i is the mean ion cyclotron frequency.
In terms of I, a and N under pressure balance, i τ i is given in terms of I 4 a and N by
i τ i = 3
64 √ 2
εo 2µ5/2
o
m 1/2
2 1/2 I 4 a
i
e 3 ln ii
Z 3 (1+ZT e /T i ) 3/2 N → 3.64 × I 4 a
5/2 1030 for D +
N 5/2 (3.16)
i τ i = 1 is plotted in figure 16, and it is interesting that it passes through the wedge-shaped
region in which ideal MHD is applicable, indicating that, strictly, anisotropic viscosity should
be employed in the R = 1–10 range. For completeness the e τ e = 1 line, with
e τ e = (m i /m e ) 1/2 Z 2 i τ i (3.17)
is plotted in figure 16, showing that magnetized electron transport is important for all but very
resistive plasmas.
We have thus constructed a universal diagram in I 4 a–N space, which determines the
appropriate theoretical model for an equilibrium Z-pinch. We now consider theory and
experiments in the various regimes.
3.2. Ideal MHD stability
The linear stability problem in ideal MHD has been shown by Bernstein et al [143] tobe
reduced to the eigenvalue equation
− ρ o ω 2 ξ = F (ξ), (3.18)
where ρ o is the equilibrium mass density, the eigenfunction ξ is the displacement of the fluid
plasma from its equilibrium position, ω 2 is the eigenvalue for a time dependence of ξ on e −iωt
and the perturbed force density F is in general
F (ξ) =∇(Ɣp o ∇·ξ + ξ ·∇p o ) + 1 µ o
[(∇ ×B o ) ×∇×(ξ × B o )]
+ 1 µ o
[∇×∇×(ξ × B o )] × B o , (3.19)
where p o and B o (r) are the equilibrium pressure and magnetic field, respectively, and Ɣ is
the principal ratio of the specific heats. The operator F has been shown for general boundary
conditions to be self-adjoint [83, 143, 144] and thus the eigenvalues ω 2 are real, positive values
corresponding to stable oscillations and negative values to growing instabilities.
For an infinitely long cylinder equation (3.19) can be reduced to a second order ordinary
differential equation [145] of the Stürm–Liouville form
d
dr
⌊
f(r) d dr (rξ r)
⌋
− g(r)rξ r = 0, (3.20)
where Fourier harmonics in the θ- and z-direction are considered for ξ of the form
34
ξ(r,t)= ξ(r)e i(mθ+kz−ωt) (3.21)