A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
The first and second models are given in detail in [97] in which the transport coefficients in
a magnetized plasma of Marshall [99] were employed. An asymmetry in heat flow is introduced
by the inclusion of the enthalpy flow in the heat-flow equation and the thermoelectric field in
Ohm’s law, which are
q z =−κ ∂T
∂z + ψJ z (2.31)
J z = σE z + φ ∂T
(2.32)
∂z
when the coefficients are κ = λT 5/2 , ψ =−ζT, σ = αT 3/2 and φ = αγT 3/2 , λ, ζ , α
and γ being constant apart from a weak ln dependence. The energy balance equation with
bremsstrahlung loss included is
J · E = ∂q z
∂z + β bn 2 e T 1/2 + 1 ∂
r ∂r (rq r), (2.33)
where β b is the bremsstrahlung coefficient.
On averaging equation (2.33) over radius, i.e. integrating ∫ a
0
2πr dr to the outer radius a
at which the radial heat flux q r is assumed to be zero, it becomes
N 2
πa 2 ∂q z
∂z + 4 3 β b
πa T 1/2 = IE 2 z (2.34)
On introducing equations (2.7), (2.31) and (2.32) to eliminate N(z), E z (z) and q z (z) a second
order, nonlinear ordinary differential equation for T(z)is found
πa 2 λ d ( )
5/2
dT
T + I(ζ − γ) dT
dz dz
dz + I 2 (
1 − I 2 )
πa 2 aT 3/2 IPB
2 = 0, (2.35)
where the Pease–Braginskii current I PB [8, 9] is defined by
( ) 192 1/2
πk
I PB =
= 0.433(ln ) 1/2 × 10 6 A (2.36)
αβ b µ 0
and is derived later in section 2.8.
The variables can be changed to ξ and x, defined by
ξ = πa2 T 5/2
, (2.37)
z 0 I
x = z (2.38)
z 0
so that equation (2.35) becomes
2
5 αλξ d2 ξ
dx 2 + 4 25 αλ ( dξ
dx
) 2
+ 2 α(ζ − γ)dξ
5 dx +1− I 2
IPB
2 = 0. (2.39)
When Marshall’s transport coefficients are employed it is found there is a limiting negative
value of dξ/dx in the domain of interest. This can be seen in a plot of the first term of equation
(2.39) versus dξ/dx in figure 6 where (d 2 ξ/dx 2 ) → 0as(dξ/dx) →−(5/4λ)(ζ −γ)(1−s) =
ξ ′ 1
where s is given by
[
(
4λ
s = 1 −
1 − I 2 )] 1/2
α(ζ − γ) 2 IPB
2 (2.40)
the coefficient 4λ/α(ζ − γ) 2 having a value of 1/1.47. This limitation means that the heat flux
to the cathode is very small and negative, while all the Joule heating apart from that radiated
is transferred to the anode. Equation (2.39) can be analytically integrated once to give ξ as a
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