A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
that determines the x-ray rise time, while, when this is shorter than the Alfvén transit time
τ A ≡ a/c A , which determines the bounce time and subsequent plasma expansion, it is the
latter which determines the pulse rise time and width.
Another phenomenon which broadens the x-ray rise time and pulse width is axial nonuniformity
associated with zippering which in turn can be associated with the axially varying
radial electric field. This will be considered in section 5.10.
The pinch radius, or rather the FWHM density profile of the hot plasma (excluding trailing
mass), is probably determined by several physical mechanisms; the radial amplitude of the RT
instabilities; the radius of the imploding current shell when it responds to the reflected shock
from the axis; if the implosion is very three dimensional with kink modes and local angular
momentum (see [324]) it is the angular momentum or rather the centrifugal force together with
the generated axial magnetic field pressure which prevents further compression.
In 1D and 2D simulations a problem arises when the current greatly exceeds the Pease–
Braginskii current [8, 9] in that a radiative collapse will occur as the radiation loss at pressure
balance exceeds the Ohmic (or Joule) heating. Since experimentally collapse does not occur, it
has to be prevented artificially in the codes. In section 5.8 we will find that inclusion of m = 0
modes and ion viscosity provide a natural physical process for preventing radiative collapse.
5.8. m = 0 instabilities and ion-viscous heating
In some experiments the radiated energy could be 3 or 4 times the kinetic energy
[1, 365, 386–388], and, if it were assumed that the ion temperature is equal to the electron
temperature the plasma pressure was too low to balance or overshoot the magnetic pressure at
stagnation. A resolution of these two mysteries was proposed [130] in which the fast growing,
short wavelength m = 0 interchange MHD instabilities would grow and nonlinearly saturate.
Then ion-viscous heating associated with the velocity shear and compression is sufficient to
raise the ion temperature T i in a few nanoseconds to a very high temperature, sufficient for
T i ≫ ZT e (in a low mass array) and for pressure balance. The equipartition of energy to the
electrons leads to the elevated radiation levels.
For the particular stainless-steel array experiments on Z employing a relatively low line
density record ion temperatures of 200 to 300 keV (over 2 billion degrees Kelvin) were
predicted and measured from Doppler broadening of optically thin lines [130]. The choice
of wavelength of the instability was based on a viscous Lundqvist number of 2, to give a
maximum linear growth rate and maximum viscous heating, this wavelength being in the
range 10–100 µm, just a little greater than the ion mean-free path. It would be difficult to
detect these instabilities because of both the fine scale and the small amplitude at saturation,
the amplitude of the perturbed velocity being only c A / √ ka where k is the wave number.
At this ion temperature pressure balance, i.e. the Bennett relation, equation (2.7) holds at
stagnation. Ideal MHD instabilities grow at a rate γ MHD
∼ = cA (k/a) 1/2 , i.e. increasing as k 1/2 .
The critical damping of Alfvén waves by viscosity occurs at a viscous Lundqvist number L µ
of 1 where
L µ = 2ρ(c2 A + c2 s )1/2
(µ ‖ /3+ν 1 )k , (5.8)
c s is the sound speed, µ ‖ is the parallel ion viscosity equal to p i τ ii and proportional to T 5/2
i
and
ν 1 is the perpendicular ion viscosity given approximately by µ ‖ /(1+4 2 i τ ii 2)
where i and τ i
are the ion cyclotron frequency and ion–ion collision time, respectively.
The saturation amplitude of the perturbed velocity ṽ can be found by equating (ṽ ·∇)ṽ
to γ MHD ṽ giving an eddy velocity amplitude of c A /(ka) 1/2 . Since in the example considered
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