A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
The ratio of the convective to diffusive terms in the bracket of equation (3.1) gives the
dimensionless magnetic Reynolds’ number R m = µ o σvL; where σ(= η −1 ) is the electrical
conductivity, v is the plasma flow velocity and L is the characteristic length over which the
magnetic field varies ( [137]). When L is replaced by the wavelength of an Alfvén wave and
v by the Alfvén speed it becomes the Lundqvist [138] number S relevant to the damping of
the Alfvén waves. For the Z-pinch we define v A as the mean Alfvén speed,
v A = B θ(r = a)
, (3.2)
2(µ o ρ)
1/2
where a is the effective plasma radius and ρ = N i m i /πa 2 the mean density. σ is taken from
Spitzer and Härm [139] in the plane perpendicular to the magnetic field
σ = 3 (4πε 0 ) 2 (k B T e ) 3/2
4 (2πm e ) 1/2 e 2 Z ln = 9.7 × 103 Te
3/2 (eV)
, (3.3)
α c Z ln ei
where α c (Z) is given by Epperlein and Haines [100]. For a dense Z-pinch, ln can be quite
low, and when a numerical value is needed in this section, a value of 5.36 is chosen [136].
Using the Bennett relation (equation (2.7)) B θ (r = a) = µ o I/2πa, and assuming that T e = T i
(see equation (2.52)), S can be expressed as
3µ o εo
2 S =
64 √ 2 3/4 I 4 a
2πme
1/2 e 2 ln Z(1+Z) 3/2 N → 3.86 × I 4 a
2 1023 (3.4)
N 2
for m i = m D , Z = 1. More generally S can be written as
8.215 × 10 24 I 4 a
S =
Z(Z + T i /T e ) 3/2 A 1/2 Ni 2 . (3.5)
ln ei
The case of S = 100, Z = 1 is shown in figure 16 in a logarithmic plot of I 4 a versus N, this
being the value below which significant resistive effects occur.
The viscous damping of Alfvén waves depends on a Reynolds number R in which the
fluid velocity is replaced by the Alfvén speed,
R = ρv Aa
, (3.6)
µ ‖
where the parallel ion viscosity is n i k B T i τ i and τ i the ion–ion collision time is given by
τ i = 3 ( mi
) 1/2 (k B T i ) 3/2 (4πε o ) 2
. (3.7)
4 2π n i Z 2 e 4 ln ii
For values of Z greater than 5 and for T e = T i the electron viscosity, which is p e τ ei [98],
or rather p e /(τ −1
ei
+ τee −1)−1
, will exceed the ion viscosity, and in general the sum of the two
viscous coefficients should be used. Indeed the ratio of the parallel viscosities is
µ ‖e
µ ‖i
= R ‖i
= 0.0233 T e
5/2
R ‖e
T 5/2
i
Z 3
A 1/2
ln ii
ln ei + Z −1 ln ee
. (3.8)
Then, in terms of I, a and N, and for T e = T i , R (= (1/R ‖i +1/R ‖e ) −1 is given by
R = 16
3 √ e 4 ln Z 4 (1+Z) 5/2 N 3
π εo 2µ2 o 1+(m e /m i ) 1/2 Z 4 I 4 a . (3.9)
Like the form of S, R depends only on I 4 a and N.
Related to this the condition that the perturbed ion pressure should remain isotropic during
the growth of an MHD instability is given by γτ i < 1 where γ = v A /a is the characteristic
growth rate for an ideal MHD instability. Using equation (2.7) γτ i becomes
γτ i = 3√ π
64 √ 2
32
ε 2 0 µ2 0
e 4 ln . 2 3/2 I 4 a
Z 4 (1+Z) 3/2 N 3
⇒ 2.07 × 1039 I 4 a
N 3
= 1 R
for Z = 1 (3.10)