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and is often taken to be valid a priori for any ionospheric F-region calculation.
However, care must be taken in applying this result since it is not a fundamental
principle and must be checked in each case. It is certainly not true in the E region,
where compressibility plays an important role in the formation of images of
F-region phenomena (see Chapter 10). Setting
(
∇·
V
)
=
0, the equations we shall
linearize are
∂
n
/∂
t
+
V
·∇
n
=
0
(4.8a)
∇·
=
J
0
(4.8b)
To study the electrostatic instability of these equations in the presence of a
vertical zero-order density gradient, we can write the electric potential and the
plasma density as
e
i
(ω
t
−
kx
)
φ
=
δφ
(4.9a)
ne
i
(ω
t
−
kx
)
n
=
n
0
(
z
)
+
δ
(4.9b)
where the initial perturbation propagates in the
x
direction. Note that we have
already assumed charge neutrality so that
n
e
=
n
i
=
n
. Using (4.5) with
E
replaced
by
δ
E
, (4.8b) becomes
(
ne
2
i
E
×
B
2
∇·
ne
/
i
)
g
+
ν
in
/
M
δ
=
0
(4.10)
∇·
(
∇
×
)
=
where again we have used the fact that
n
B
0 to set the divergence
δ
of the pressure-driven current equal to zero. Here
E
is the perturbation electric
φ
σ
P
in the
field associated with the potential
, and we have substituted (2.40b) for
F region. Since
, the vector inside the square bracket in the preceding
equation only has an
x
component and taking the
x
derivative yields
δ
E
=−∇
φ
e
2
i
ne
2
i
x
2
eg
/
i
(∂
2
2
2
n
/∂
x
)
−
ν
in
/
M
(∂
n
/∂
x
)(∂φ/∂
x
)
−
ν
in
/
M
∂
φ/∂
=
0
The second term is of second order, and thus the linear form of this equation is
ne
2
i
x
2
eg
/
i
(∂
2
2
/∂
)
−
ν
in
/
∂
φ/∂
=
n
x
M
0
(4.11a)
Making the substitutions
B
2
,
Q
P
=
M
ν
in
/
=
Mg
/
B
(4.11a) becomes
Pn
x
2
2
(∂
/∂
)
−
∂
φ/∂
=
Q
n
x
0
(4.11b)
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