A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
kinetic energy per unit length just prior to stagnation is equal to the internal energy after one
ion–ion collision time, with the Bennett relation equation (2.7) (pressure balance) holding, i.e.
( )
1 da 2
2 M l (1 − f)= N ieT i
dt t=t s
Ɣ − 1 = µ 0I 2 (t s )
8π(Ɣ − 1) , (4.24)
where f is the fraction of kinetic energy lost by ionization and radiation at the moment of
stagnation. Using equation (4.20) to evaluate da/dt together with equations (4.18) and (4.22)
the ratio t s /t m is given by
t s
t m
=
[
5
2
1+ 6 5 (Ɣ − 1)(1 − f) ] 1/6
. (4.25)
From this it can be seen that the largest value of t s /t m represents the largest fractional time
delay of stagnation after peak current is (5/2) 1/6 = 1.165. For f = 0 and Ɣ = 5/3, the
value of t s /t m is 1.056. These results are typically within the range of experimental data. With
this model the generator essentially controls the initial rate of rise in current and the early
acceleration of the load. At later times the momentum of the load controls the current which
then falls prior to stagnation.
The radius at stagnation is sensitive to the value of f as well as Ɣ. Writing Ɣ = (n +2)/n
where n is the number of degrees of freedom, equation (4.20) with (4.22) yields
a s
a 0
= 1 −
1
1+ 12
(4.26)
(1 − f).
5n
For example, if f = 0.5 and n = 3,a s /a 0 = 2/7, while a s /a 0 → 0asf → 1; but if f = 0
and n = 3, a s /a 0 is in contrast large at 4/9. Metallic wire-array pinches tend to be excellent
radiators, so f can be large and experimentally a s /a 0 is typically 0.1. On the other hand
deuterium gas puffs have f → 0, and the final pinch radius is a significant fraction of the
initial radius. In order to obtain higher compression it would be necessary to dope the gas with
argon, krypton or xenon in order to enhance the radiation loss, or as Potter [261] suggested,
further raise the current in an adiabatic compression. Further terms could be added to equation
(4.20) leading to additional terms in equation (4.21) if a better fit to experiment is required. In
experiments with wire arrays the ratio a s /a 0 is 1/20 to 1/10 typically, indicative of an efficient
radiator, i.e. f is about 0.9 for n = 3. To obtain such a large and rapid conversion of energy
into radiation will however probably require an additional physical mechanism as discussed
in section 5.8.
4.5. Early work on numerical simulations of Z-pinches
Perhaps the earliest MHD numerical simulation was of the Z-pinch by Hain et al in 1960 [264]
which was a 1D model of compressional pinch. It was conventionally called the ‘Hain-
Roberts’ code. Ten years’ later Roberts and Potter published a more general account of
magnetohydrodynamic calculations [265]. At this time Potter [266] also published results
from the 2D, two-fluid MHD code of the plasma focus (see section 7.3). The final pinch was
found to be stable to short wavelength MHD modes due to damping by ion viscosity, as shown
analytically earlier by Tayler [267]. Potter also found that the ions were magnetized and finite
ion Larmor radius effects could be important. Viscous effects were found to be important
in Z-pinch implosions simulated by Hopkins et al [268] in a 1D code especially at low line
density and high ion temperature where the shock structure was broadened. Viscous effects
will be important in ion heating with short wavelength instabilities in section 5.8.
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