A review of the dense Z-pinch

Plasma Phys. Control. Fusion 53 (2011) 093001
Topical Review
If the magnetic field lines were straight as in slab geometry, the magnetic field has a
neutral effect on the RT instability for the k · B = 0 mode. To derive the RT growth rate for
this case the approach of Rosenbluth and Longmire [281] can be extended to finite β (β being
the ratio of plasma to magnetic pressure). For a distributed density gradient dρ 0 /dx >0in
the x-direction, opposing g in the −x-direction with an axial magnetic field B z0 (x) pressure
balance is given by
0 =−ρ 0 g − T dρ 0 /dx + J y0 B z0 . (4.30)
For a perturbation of the form f(x)exp(γ t − iky) the xand y components of the equation of
motion give
ρ 0 γv x =−ρ 1 g − T dρ 1 /dx + J y1 B z0 + J y0 B z1 , (4.31)
ρ 0 γv y = ikTρ 1 − J x1 B z0 , (4.32)
while incompressibility is a reasonable assumption for RT, giving
∂v x /∂x − ikv y = 0 (4.33)
and
γρ 1 + v x dρ 0 /dx = 0. (4.34)
Ampère’s law gives
µ 0 J y0 =−∂B z0 /∂x; µ 0 J y1 =−∂B z1 /∂x; µ 0 J x1 =−ikB z1 . (4.35)
Employing the ideal Ohm’s law E + v × B = 0, Faraday’s law gives, with equation (4.33),
γB z1 =−v x dB zo /dx. (4.36)
Equation (4.32) can be simplified using equations (4.33), (4.20) and (4.34) togive
ρ 0 γ 2 ∂v x
∂x =−k2 ρ 0 gv x , (4.37)
which can be integrated to give
( )
v x = v 0 exp − k2 gx
, (4.38)
γ 2
where v 0 is the constant of integration. (This eigenfunction needs to be modified to satisfy
boundary conditions at two finite values of x, but gives adequate results for short wavelengths.)
On substituting this into equation (4.31) and again using equation (4.30), the dispersion relation
γ 4 = k 2 g 2 (4.39)
is obtained. Thus γ 2 =±kg and the + sign is taken for real γ giving
γ = (kg) 1/2 . (4.40)
This is the same k 1/2 dependence as found for the m = 0 MHD instability in section 3.2, yet
it is independent of the magnetic field.
The RT instability can be mitigated by many effects:
(i) Ablation of plasma, which is important in ICF both during compression [282, 283] and
the deceleration phase of the confining shell [284]. This changes the growth rate to
γ = 0.9
(kg)1/2
(1+kL) − 3.1kV a, (4.41)
1/2
where V a is the ablation velocity and L is the density gradient scale length. A more
complete formula is derived in [283] while a similar formula to equation (4.41) applies
in the deceleration phase [284]. This tends to stabilize short wavelength modes. It might
have application to the ablative phase of wire-array Z-pinches, and compete with the
nonlinear heat-flow instability [215] for the origin of filamentary structures.
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