Geoscience Reference
In-Depth Information
The corresponding equations for the ionized species are
+∇·
ρ
j
V
j
=
P
j
−
L
j
M
j
∂ρ
j
/∂
t
(2.22a)
n
j
q
j
E
B
ρ
j
d
V
j
/
dt
=−∇
p
j
+
ρ
j
g
+
+
V
j
×
k
ρ
j
v
jk
V
j
−
V
k
−
k
j
(2.22b)
=
p
j
=
ρ
j
k
B
T
j
/
M
j
=
n
j
k
B
T
j
(2.22c)
Owing to the complexity of the equation sets (towhichMaxwell's equations must
yet be added), we will not yet consider the heat equations. This is equivalent to
treating temperature profiles as given quantities. In this topic we do not attempt
to include a thermal analysis. (We refer the reader to the text by Schunk and
Nagy, 2002.) This simplification would be very poor if our main interest was
thermospheric neutral gas dynamics or the topside ionosphere. However, many
interesting phenomena may be studied without including temperature changes
self-consistently.
To treat the electric and magnetic fields, Maxwell's electrodynamic equations
are needed, which, in their full form, are given by
∇×
E
=−
∂
B
/∂
t
(2.23a)
∇·
E
=
ρ
c
/ε
0
(2.23b)
=
μ
0
J
t
∇×
B
+
ε
0
∂
E
/∂
(2.23c)
∇·
=
B
0
(2.23d)
ρ
ρ
=
j
n
j
q
j
)
=
where
c
is the charge density (
and
J
is the current density (
J
c
j
n
j
q
j
V
j
). To this must be added the principle of conservation of charge
∂
∂
ρ
c
dV
=−
J
·
d
a
t
v
S
This equation states that the buildup or decay of charge inside a volume
V
is
determined by the net electric conduction current across the surface
. Note the
similarity to (2.1) for the conservation of mass. In differential form this may be
written
∇·
J
=−
∂ρ
c
/∂
t
(2.24)
Neither production nor recombination affects this equation because the net
charge does not change in either process. In the ionosphere the conduction
current is larger than the vacuum displacement current
t
in (2.23c) for
all wave frequencies of interest here, and the displacement current is there-
fore dropped. Furthermore, since the largest changes in the magnetic field at
ε
0
∂
E
/∂
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