A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
(v) The mean electron drift velocity is 0.96 times the ion thermal speed. This should be too
low to cause ion-acoustic turbulence, particularly since T e ≈ T i , but lower hybrid drift
turbulence remains a possibility in the low density corona. See section 3.13.
(vi) The time taken to heat the Z-pinch ohmically with pressure balance is three times τ E
when following the so-called Haines–Hammel curve [103, 104] (see section 2.5). If a
faster I˙
is employed the heating time could be comparable to τ E and is accompanied by
compression.
2.5. Self-similar solutions and radial heat loss
We consider next the class of Z-pinch equilibria (i.e. profiles satisfying radial pressure balance)
which are time dependent, but in which we neglect any axial or azimuthal variation. All plasma
parameters are functions of radius r and time t only. The earliest example that led to a selfsimilar
solution, i.e. one for which a profile shape is maintained in time and is a function of a
similarity variable (in this case radius), is when Joule heating occurs in a stationary Z-pinch.
Then the current I follows the so-called Haines–Hammel curve of I ∝ t 1/3 ([103, 104]; also
found independently by Braginskii and Shafranov [105]).
This example assumed spatially uniform temperature (infinite thermal conductivity) and
found a self-similar current density profile proportional to I 0 (ξr/a) where ξ is the root of
I0 2(ξ) = 5 2 I 1 2 (ξ), i.e. ξ ≈ 1.656, i.e. a weak skin effect.
More general cases have been considered by Coppins et al [106, 107]. The importance of
self-similar solutions to nonlinear coupled partial differential equations is that they can act as
attractors in more general situations. The set of nonlinear equations are equation (2.1) radial
pressure balance, equation (2.2) Ampère’s law, together with Ohm’s law for, in general, a
non-stationary pinch
E z = ηJ z − v r B θ , (2.53)
where η −1 = αT 3/2 is the Spitzer resistivity, and an energy equation
n γ d
( p
)
= ηJ 2
γ − 1 dt n γ z − 1 r
∂
∂r (rq r) − β b n 2 T 1/2 . (2.54)
Self-similar behaviour is not necessarily obtainable when there are more than two terms in an
equation. The energy equation can therefore be problematical, and so far the bremsstrahlung
loss term has been neglected in searching for self-similar solutions. The radial heat flow can
be fitted into a self-similar scheme in the limit of strong ion magnetization, ω i τ ii ≫ 1. Then
q r is given by
∂T
q r =−κ ⊥i
∂r =− α 0n 2 ∂T
(2.55)
B 2 T 1/2 ∂r
and each of the three remaining terms in equation (2.54) has the same time dependence.
In [106] two cases have been studied: the generalization of the stationary pinch with
I ∝ t 1/3 , and the expanding pinch with the inertial term strictly zero, i.e.
dv r
= ∂v r ∂v r
+ v r = 0, (2.56)
dt ∂t ∂r
where all functions are now separable in v r and t, with v r = r/t ≡ u. In the first case there
is one free parameter, designated , which represents the ratio of the Ohmic heating time to
the characteristic time for current or pressure change. It is found that there is a critical value
of , crit ≈ 3.42, at which both the number density n(r) and heat flux q(r) tend to zero
at r = r 0 , the pinch radius. This represents a thermally isolated pinch, and has J z (r), n(r)
and T(r)profiles as shown in figure 8(a). For < crit there is a minimum in n(r) which
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