A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
of the final piston radius r p to initial radius r 0 is found as a function of Ɣ and r s /r p to be
[
] Ɣ/Ɣ−1
r p Ɣ
=
(4.8)
r 0 Ɣ +1− rs 2/r2 p
and at r s = 0givesr p /a equal to 0.0263, 0.1516 and 0.3088 for Ɣ = 1.2, 1.4 and 5/3,
respectively. To obtain greater compression Potter proposes adiabatic compression of a
pinch by a carefully programmed current waveform, analogous to compression in inertial
confinement fusion.
The gas between the shock and piston behaves adiabatically, i.e. for a fixed mass, pV Ɣ is
a constant in time where V is the volume per unit length while the uniform pressure is p s and
exactly equal to the magnetic pressure, i.e.
p s = B2
= µ 0I 2
. (4.9)
2µ 0 8π 2 rp
2
There is a subtlety in the calculation of dV and Potter shows that for an instantaneous fixed
mass
dV = 2π(r p dr p − r s dR s ), (4.10)
where dR s /dt is equal to u s evaluated at the shock and is given by equation (4.4). The rate
of change in pressure is related to the rate of change in the shock velocity by differentiating
equation (4.6) to give, with equation (4.10),
dp s
= 4p s dv s
=−Ɣ P s dV
dt (Ɣ +1)v s dt V dt . (4.11)
Figure 34 is calculated for constant current [261] to consider also the case of linearly rising
current so that direct comparison can be made with the snowplough and shell models. Indeed
from equations (4.6), (4.9) and (4.11) a coupled pair of nonlinear equations for r p (t) and r s (t)
can be derived for any given I(t), namely
( )
dr s (Ɣ 1/2
+1)1/2 µ0 I
= (4.12)
dt 4π ρ 0 r p
and
[
dr p
Ɣ r p − 2 ]
( )
dt Ɣ +1 r dr s
s = (rp 2 dt
− r2 s )r p d I
. (4.13)
I dt r p
For the particular case of I = At the numerical solution of equations (4.12) and (4.13) is
shown also in figure 33.
4.4. Accelerating hollow shell
A zero-dimensional analytic model is a good guide for designing experiments using wire
arrays, liners and hollow gas puffs. Here the mass involved in acceleration is fixed and the
equation of motion is simply written as
d 2 a
M l
dt =−µ 0I 2
2 4πa , (4.14)
where M l is the mass per unit length and I(t) is either given or calculated from a circuit
equation which includes the varying impedance of the load [263].
For direct comparison with the snowplough model figure 33 includes a plot of a(t) for the
case of a linearly rising current I = At where now the dimensionless equation is
d 2 x
dy 2 =−y2
(4.15)
x
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