A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
in [130] ka ≃ 200, it would be very difficult to detect these perturbations. Nevertheless the
viscous heating rate arising from the fine scale of the mode gives a relatively high value of ∇·ṽ.
For µ ‖ ≫ 3ν 1 (since i τ ii is ∼3) and recalling that the m = 0 MHD mode is a compressible
interchange mode, i.e. ∇·ṽ ≠ 0 a value for the heating water per unit volume is approximately
1
3 µ ‖|∇ · ṽ| 2 ∼ = µ‖ cA
2 k
(5.9)
a
and the linear dependence on k should be noted. As a result, by fixing the maximum k such
that L µ equals 2, then gives
ρ c2 A
a (c2 A + c2 s )1/2 = 3n ee
(T i − T e ), (5.10)
2τ eq
where the lhs is the viscous heating rate and has been equated to the equipartition rate where
the equipartition time τ eq is defined by
1
= 8(2πm e) 1/2 e 5/2 Z 2 n i ln ei
τ eq 3m i (4πε 0 ) 2
T 3/2
e
(5.11)
and T e ≫ (m e /m i )T i . Thus equation (5.10) shows that the viscous heating rate is essentially an
Alfvén transit time a/c A which for the example is 1–2 ns. This is consistent with the measured
ion temperature rise shown in figure 64, where τ eq is over 5 ns, and so the ion temperature T i
continues to be much larger than the electron temperature T e . In contrast the ion–ion collision
time τ ii is 37 ps, allowing rapid thermalization of the ions. In terms of global quantities equation
(5.10) can be combined with the Bennett relation to give
(T i − T e ) = 2.1 × 10 36 aI3 Te 3/2 A 1/2
Z 3 N 5/2 , (5.12)
i
ln ei
where A is the atomic number (55.8 for iron). For the measured current I and estimated line
density N i the Bennett relation gives T i of 219 keV while equation (5.12) gives a pinch radius
a of 3.6 mm. Including the viscous heating associated with ν 1 and other k-modes reduces a
to ∼1 mm. This formula is not very sensitive to the effects of trailing mass (see section 5.11)
since there will be comparable reductions in both I and N i .
It would be expected that at the start of stagnation there would be a skin current, and the
instabilities would lead to the rapid diffusion towards a Kadomtsev profile (see section 3.2),
converting magnetic energy to ion vortex motion but conserving magnetic flux. Assuming
that the electrodes are perfect conductors and that at this time there∫is a current short across
the AK gap the magnetic flux can only decay if the line integral K A Ez dz along the axis is
non-zero. If only axisymmetric m = 0 modes are present, and if the resistivity is negligible
then this can only be achieved if there are many hot spots formed on the axis with a component
of number density variation out of phase with the temperature variation [389]. A contribution
from electron viscosity through the electron stress in Ohm’s law might be important. All
experiments reveal the occurrence of many bright spots on the axis. Figure 65 illustrates this
from Z, [388]. Indeed the voltage drop can be roughly estimated as T e (in eV) times the number
of bright spots. Anomalous resistivity, as suggested in section 3.14, applies at a disruption and
can also dissipate magnetic flux.
In comparing the current and voltage waveforms with a lumped circuit model, e.g.
Waisman [390], it would be necessary in order to model this viscous heating process to equate
πa 2 l (i.e. the pinch volume) times the lhs of equation (5.10)toR vis I 2 . Then R vis is rather like
the inductance divided by the Alfvén transit time; in fact
R vis = µ ol
4πa · (c2 A + c2 s )1/2 . (5.13)
93