A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
and τ 0 is (4πM l a0 2/µ 0A 2 ) 1/4 as before. It can be seen that the implosion is slower because
all the mass is initially concentrated at the radius a 0 , and the shell arrives on the axis at the
dimensionless time of 1.752. This was found from a series solution in y up to power of y 20 ,
where in a series similar to equation (4.3) only terms in powers of y 4 exist, the coefficient of
y 4 being 1/12 and of y 8 1/672.
However the current waveform for Z at Sandia and for MAGPIE at Imperial College is
more closely of the form
( ) πt
I = I 0 sin 2 , (4.16)
2t 0
where the current rises to a maximum I 0 in a time t 0 , at least for early times. Employing the
identity
sin 4 z = 3 8 − 1 2 cos 2z + 1 8 cos 4z = z4 − 2 3 z6 + 1 5 z8 − 34
445 z10 + 62
14175 z12 ··· (4.17)
and the dimensionless parameters, x = a(t)/a 0 , y = t/t 0 and
α 2 = µ 0I0 2t 0
2
4πa0 2M (4.18)
l
a series solution of the form
x = 1 − a 6 y 6 + a 8 y 8 − a 10 y 10 + a 12 y 12 ··· (4.19)
can be found where a 6 = α 2 π 4 /480,a 8 = α 2 π 6 /5376 and a 10 = α 2 π 8 /115 200. The
nonlinear effect of α 2 enters for higher order terms, e.g. a 12 = 17α 2 (π/2) 10 /62 370 −
α 4 π 8 /1 013 760. This series converges only very slowly, if at all, for practical values of α 2 .
Furthermore the current from a pulse-power generator is strongly affected by the impedance
of the load especially due to strongly increasing inductance as the shell converges to the axis.
Coupling the equation of motion to a circuit equation (as was done for example in [122])
requires a numerical solution.
However a new approach to obtain an exact, closed, analytic solution of equation (4.14)
with a current waveform that peaks prior to final stagnation can be found as follows. For
example, if the leading term in the above solution (equation (4.19)) is taken, i.e.
a(t)
= 1 − α2 π 4
a 0 480
( t
t 0
) 6
(4.20)
and substituted into equation (4.14) to give the current I(t)as
( ) πt 2 (
I(t) = I 0 1 − α2 π 4 t 6 ) 1/2
2t 0 480 t0
6 . (4.21)
These values of a(t) and I(t)exactly satisfy equation (4.14). I has a maximum I m at t = t m
given by
( )
t m
= 121/6 2 2/3
= 1.12 , (4.22)
t 0 α 1/3 π α1/3 I m
= 35/6 π 2/3 2.40
= . (4.23)
I 0 5 1/2 α2/3 α2/3 It follows that t m = t 0 for α = 1.40 while I m = I 0 for α = 3.71. A smaller value of M l a0
2
results in larger values of α and hence smaller values of t m and I m .
To further guide the choice of experimental parameters for a given generator (or vice
versa) it is useful to consider the final stagnation. It is assumed to occur at a time t s when the
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