A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
resistance (in SI units)
l
( µ0
) 3/2 I
R Lovberg =
4(N i m i ) 1/2 π a . (7.2)
This determined the rate of transfer into ‘turbulent kinetic energy’ of the plasma. To get
agreement with HDZP-II deuterium fibre Z-pinch experiment this heating was added only to
the ion energy equation, and was described as a ‘boiling’ of the pinch. The inward radial
velocity of the bubbles is equated to the Alfvén speed. Lovberg states that only 10% of the
pinch length is affected in this way, and agreement on the low neutron yield is obtained because
of the drop in density resulting from a faster expansion of the corona.
In Rudakov et al [235, 331] there is a discussion of the inward velocity of bubbles based
on buoyancy leading to a velocity v bubble given by
(
v bubble = 2α π r )
b 1/2
vA , (7.3)
a
where r b is the radius of the bubble, v A the mean Alfvén speed and α an undetermined constant.
For ‘definiteness’ Rudakov et al take r b /a = 0.2 and α = 0.3 to give the effective additional
nonlinear resistance
l
( µ0
) 3/2 I
R Rudakov =
2 · (N i m i ) 1/2 π a . (7.4)
This mechanism is further explained and employed by Velikovich [329] to compare with
experiments. It is essentially identical with the earlier work by Lovberg, differing only by a
factor of 2. Both models have some arbitrariness in the numerical coefficient. The effective
resistance in Haines’ viscous heating model, equation (5.13) has no arbitrary constant and
employs well established MHD instability growth rates, saturation levels and stress tensors.
Taking the growth rates in Coppins [149] R visc becomes
R visc = Ɣ(1+Ɣ/2)1/2 l
( µ0
) 3/2 I
8(N i m i ) 1/2 π a . (7.5)
For the ratio of specific heats Ɣ of 5/3 the coefficient becomes 0.282 which is just 10% larger
than in R Lovberg . Considering the large physical differences in the two models this is remarkable.
One reason for this coincidence is that in choosing a viscous Lunqvist number of 2 the product
of wave number and viscosity is eliminated to give the left-hand side of equation (5.10) for the
local ion-viscous heating rate. The positive aspect is that the resistance formula enables 0D
and 1D models to agree with experiments.
In Haines’ model the wavelength is short and corresponds to the fastest growing mode.
This saturates at a low amplitude, but the viscous heating for high k is substantial. Depending
on the ratio of ion to electron viscosity, the heating will be distributed between ions and
electrons. There is no need to postulate turbulence and associated cascading. If Menikoff’s
theory [500] is adapted to the MHD instability the wavelength of the fastest growing mode
is increased. In contrast a ‘flux limit’ to the momentum transport caused by the mean-free
path being only slightly smaller than the wavelength is equivalent to an increase in k. A
full nonlinear Fokker–Planck stability analysis is needed in which electron viscosity is also
included, not only in the equation of motion but also in Ohm’s law.
In the Lovberg/Rudakov/Velikovich (LRV) model, the implied wavelength is long and in
the linear phase will exponentiate only once in one Alfvén transit time. Thus there is little time
at stagnation for it to grow unless it is already well established as a MRT mode. There is even
less chance for the cascading to shorter wavelengths implicit in establishing MHD turbulence.
In contrast Haines chooses the fastest growing mode and has no need for cascading. Indeed
δB/B ∼ (ka) −1 here, whilst LRV requires δB/B ∼ 1. In LRV there is a problem with current
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