A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
However in many later simulations, single fluid MHD equations have been used, with
an artificial viscosity replacing the real viscosity [269] and where the emphasis has been on
solving compressible 3D MHD problems [270] for many Alfvén transit times.
The recent work on simulating wire-array Z-pinches will be considered in section 5.
4.6. The skin effect and inverse skin effect
One of the earliest theoretical problems in describing the dynamical behaviour of a Z-pinch
was to calculate the radial current distribution. The total current is rising in time at first and
then falls due to the increase in inductance as the pinch forms. For a cylindrical conductor of
uniform conductivity the current density can be found knowing only the total current I(y)as
an arbitrary function of dimensionless time, y = t/(µ 0 σa 2 ) [110], where a Laplace transform
was employed. The formula for j z (x, y) where x = r/a is
j z (r, t) = 1
2πi lim η→∞
∫ ζ +iη
ζ −iη
e yp p 1/2 I 0 (p 1/2 x)
2πa 2 I 1 (p 1/2 )
∫ ∞
0
I(q)e −pq dq dp, (4.27)
I 0 and I 1 being modified Bessel functions and where a Fourier–Mellin inversion is employed.
The case of a current pulse as arising from an overdamped capacitor discharge is shown in
figure 35. The early skin effect is clearly identified, while after peak current the radial gradient
of current density at the conductor surface becomes negative. If the rate of fall of current is
sufficiently high the current density reverses, leading to an outward J ×B force on the plasma.
This was further explored theoretically by Culverwell et al [271].
Experiments have been carried out by Jones et al [272]onZ-pinches which have attributed
a current density reversal to the inverse skin effect. In these experiments the outer layer of
plasma is ejected by the reversed J ×B force and this, together with its current, was measured.
More recently Lee et al [273] have run a 1D single fluid MHD simulation of a Z-pinch
as used at CERN as a magnetic lens [274] (see also section 8.3). In this the authors find a
reversed current density locally and internal to the pinch associated with the reflected shock
moving radially outwards. The shock leads to compression of the azimuthal magnetic field,
and so in the shock layer a large negative value of ∂B ϑ /∂r exists which through Ampère’s
law leads to a local negative j z sheet. In this paper the authors criticize Kumpf et al [275] in
their simulations and, more positively, draw attention to other experimental data [276, 277].
It should be noted however that many terms are omitted in this single fluid code which might
significantly modify the results.
4.7. The Rayleigh–Taylor instability
The RT instability is important in all dynamic pinches where the acceleration of the plasma
or shell is caused by the J × B magnetomotive force in the outer, lower density plasma. This
instability is also the most damaging phenomenon in capsule implosions and the later fuel
expansion into the denser shell in inertial confinement fusion (ICF).
Classical RT concerns the stability of the interface of a heavy incompressible fluid of
density ρ 2 supported against gravity g by a lighter one of density ρ 1 . It was first considered
theoretically by Lord Rayleigh [278] and experimentally verified by Taylor [59] sixty years’
later. The linear exponential growth rate γ is given by
γ = (kgA) 1/2 , (4.28)
where the Atwood number A is given by
A = ρ 2 − ρ 1
(4.29)
ρ 2 + ρ 1
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