Environmental Engineering Reference
In-Depth Information
5.3.6
Hydrodynamic Instabilities
The instabilities discussed above are so-called kinetic instabilities, for which the
amplification of oscillations is due to the differences in the character of the motion
of various groups of particles [59, 60]. The development of oscillations ultimately
results in a change of the velocity distribution function for the charged particles
of the plasma. Another class of instabilities is known as hydrodynamic instabil-
ities. The development of hydrodynamic instabilities involves a displacement of
the plasma regions and results, finally, in variation of the spatial configuration of
the plasma. We shall analyze the simplest type of hydrodynamic instability, pinch
instability.
We examine the stability of the pinch configuration with respect to variation of
a current radius at a given point where the current axial symmetry is conserved
at such a variation. The corresponding instability is called a sausage instability.
We have to find under what conditions an accidental distortion of the pinch will
not develop further. Let us assume that the distortion of the pinch results only
in a slight curving of the magnetic lines of force; that is, the radius of curvature
of the magnetic lines of force is considerably larger than the radius of the pinch.
According to (4.163), the relation [58]
H
2
8
p
C
π
D
const
is satisfied in the plasma region.
We must analyze the variation of the parameters of the pinch due to variation
of its radius. The total current and magnetic flux through the cross section of the
pinch will be conserved. The electric current is
I
z
D
caH
'
/2, where
a
is the pinch
radius and
H
'
is the axial magnetic field strength. The condition
δ
I
z
D
0yields
(
H
'
is
the variation of the axial magnetic field at the pinch surface outside the plasma. The
longitudinal magnetic field is frozen into the plasma, so a displacement of plasma
elements does not change the magnetic flux through them. The condition for con-
servation of the magnetic flux,
δ
a
/
a
)
C
(
δ
H
'
/
H
'
)
D
0, where
δ
a
is the variation of the pinch radius, and
δ
a
2
H
z
,yields(2
0,
where
H
z
is the longitudinal external magnetic field as it exists inside the plasma.
Hence, (
Φ
D
π
δ
a
/
a
)
C
(
δ
H
z
/
H
z
)
D
z
H
z
/
H
z
D
H
'
/
H
'
). The variation of the magnetic field pressure in-
δ
2(
δ
[
H
z
/(8
D
H
z
H
z
/(4
side the plasma is
δ
π
)]
δ
π
), and the variation of the magnetic
pressure outside the plasma is
H
'
δ
H
'
/(4
π
)
D
H
2
'
δ
H
z
/(8
π
H
z
). It can be seen
that if
H
2
'
2
H
z
(5.101)
holds true, the additional internal magnetic field pressure produced by the distor-
tion of the pinch is smaller than the additional external magnetic field pressure.
When condition (5.101) is satisfied, the pinch is stable with respect to displace-
ments of the sausage type.
