Geoscience Reference
In-Depth Information
For the electrons
κ
e
is so large that the second term dominates in (4.1) and
m
eB
2
k
B
T
e
/
eB
2
n
B
2
V
e
=
E
×
B
/
−
/
(
g
×
B
)
+
(
∇
n
×
B
)
(4.3)
1 in the F region, but it is not so great that the first term in
(4.1) can be ignored, as was the case for the electrons. For the F region that term
can be written as
W
i
⊥
/κ
For the ions
κ
i
2
i
, and we have
b
i
/κ
i
E
g
i
D
i
/κ
i
2
2
2
B
2
V
i
⊥
=
+
ν
in
κ
−
(
∇
n
/
n
)
+
E
×
B
/
M
eB
2
k
B
T
i
/
eB
2
n
+
/
(
g
×
B
)
−
(
∇
n
×
B
)
(4.4)
×
B
term in (4.3) can be dropped. The
second and third terms in the right-hand side of (4.4) are small due to the fact
that
The electron mass is so small that the
g
B
term is identical for
ions and electrons, no current flows due to those terms.
J
is given by
κ
i
1, and we take
n
i
=
n
e
=
n
. Now, since the
E
×
J
=
ne
(
V
i
−
V
e
)
k
B
/
B
2
×
B
=
σ
P
E
+
(
ne
/
i
)
g
−
(
T
i
+
T
e
) (
∇
n
×
B
)
(4.5)
2
i
Here we have used the fact that for large
κ
i
,
σ
P
=
neb
i
/κ
. The gravity term
also is rewritten in terms of
i
. Notice that the gravitational current flows even
in a collisionless plasma, while the electric field term exists only if
σ
P
=
0—that
is, in a collisional plasma.
We now study the linear stability of a vertically stratified equatorial F layer
under only the influence of gravity—that is, the pure Rayleigh-Taylor case.
We set
E
0
=
E
.
The continuity and current divergence equations from Chapter 2 will be used in
the analysis. Ignoring production and loss, which is reasonable in the postsunset
time period when the F layer is very high, the continuity equation is
0 for now but retain a first-order electric field perturbation
δ
∂
n
/∂
t
+
V
·∇
n
+
n
(
∇·
V
)
=
0
(4.6)
where for
M
m
the plasma velocity
V
may be approximated by
V
i
. First,
consider the “compressibility” term
(
∇·
V
)
. From (4.4) with
E
=
0 and
κ
i
large,
M
eB
2
g
k
B
T
i
/
enB
2
B
−
∇·
V
=∇·
/
×
(
∇
n
×
B
)
(4.7)
Since
g
and
B
do not vary in the
g
×
B
direction, the first term vanishes. Since
we also have
0, Eq. (4.7) vanishes
and the plasma flow is incompressible. This is usually a good approximation
∇·
(
∇
n
×
B
)
=
0 and
(
∇
n
×
B
)
·∇
n
=
Search WWH ::
Custom Search