Environmental Engineering Reference
In-Depth Information
One can combine these oscillations and drift waves by inserting the electron drift
into the balance equation for the electron number density. Then instead of disper-
sion relations (5.128) and (5.149), we obtain the following dispersion relation:
r
k
2
w
2
4
kw
2
2
E
,
ω
D
C
C
ω
(5.150)
which transforms to (5.128) or (5.149) in the appropriate limits.
5.5.6
Current Convective Instability
Just as there are a variety of mechanisms that can alter the manner in which un-
bound electrons can be introduced into and removed from a plasma, so also there
are different ways in which ionization instabilities can develop. In addition to the
above-described mechanisms for destabilizing a plasma, we now analyze one more
case that involves a plasma in a strong magnetic field. The configuration we explore
is one in which electric current flows in a cylindrical tube so that the electric and
magnetic fields are directed along the axis of this plasma column. The magnetic
field is so strong that electrons and ions in the plasma are magnetized, that is, the
Larmor radii of electrons and ions are small compared with the mean free paths for
these particles. Since charged particles recombine on the walls of the tube, the plas-
ma is uniform across the diameter of the tube. These are the conditions in which
an ionization instability called the current convective instability may occur [78, 79].
In the environment described, electrons will rotate with respect to ions in any
cross section of the plasma column. This gives rise to an azimuthal electric field
that enhances this motion. The development of the instability is slow if plasma
quasineutrality is conserved. If a small inhomogeneity of the plasma is present
in the direction of the current, then the plasma density is different in neighbor-
ing cross sections of the column. Since the current in the plasma is conserved, a
decrease in the plasma density in any cross section of the column must be com-
pensated for by an additional electric field.
Now let us assume an “oblique” perturbation of the plasma density that is com-
pensated for by an electric field directed at an angle to the column axis. Then the
azimuthal component of the electric field will produce rotation of the electrons and
ions. This will enhance a suitably directed perturbation, leading to an instability. Let
us derive a dispersion relation for such perturbations. The plasma is quasineutral
and magnetized, so the velocity of electrons and ions is
c
E
H
0
H
0
w
D
,
(5.151)
where
H
0
is the magnetic field and
E
is the electric field of the wave. The magnetic
field of the wave can be ignored since the perturbation is slow. For the same reason,
the electric field can be described by the potential
'
,where
E
D
r
'
.
The current density is
i
z
D
Σ
E
z
,wherethe
z
-axisisalongtheaxisofthecol-
umn and
Σ
is the plasma conductivity. The electric field is the sum of the external
