Geoscience Reference
In-Depth Information
By substituting (7.40) for
z
=
0 into (7.31)-(7.32) we can
express the ground magnetic components and vertical electric field in terms
of admittances:
−
h
and
z
=
−
Y
(
e
)
g
ζ
1
b
(
g
)(
e
)
b
(
e
)
τ
b
(
e
τ
(
≡
(
−
h
)=
−
−
0)
,
(7.41)
Y
(
m
)
g
ζ
4
b
(
g
)(
m
)
b
(
m
)
τ
b
(
m
)
τ
≡
(
−
h
)=
−
(
−
0)
,
(7.42)
Y
(
e
)
ε
a
(
−
0)
g
ζ
1
E
(
g
)
z
=
E
z
(
−
h
)=
−
E
z
(
−
0)
,
(7.43)
ε
a
(
−
h
)
h
)=
b
z
(
−
0)
b
(
g
)
z
=
b
z
(
−
.
(7.44)
ζ
3
For the homogeneous atmosphere
ε
a
(
z
)=
ε
a
=const
,
we have
cos
κ
a
(
z
+
h
)+
ik
0
ε
a
κ
a
Y
(
e
)
g
ζ
1
(
z
)=
−
sin
κ
a
(
z
+
h
)
,
κ
a
ik
0
ε
a
Y
(
e
)
ζ
2
(
z
)=
sin
κ
a
(
z
+
h
) + cos
κ
a
(
z
+
h
)
,
g
ik
0
κ
a
Y
(
m
)
ζ
3
(
z
) = cos
κ
a
(
z
+
h
)
−
sin
κ
a
(
z
+
h
)
,
g
κ
a
ik
0
sin
κ
a
(
z
+
h
)
Y
(
m
)
g
ζ
4
(
z
)=
−
−
cos
κ
a
(
z
+
h
)
,
(7.45)
where
κ
a
=
k
0
ε
a
−
k
τ
.
Equations (7.45) are obtained by taking into account the vertical displacement
currents and of atmospheric conductivity. It can be assumed that
σ
a
= 0. Then
only the condition for the magnetic mode needs to be retained in boundary
condition (7.11) while the boundary condition for the electric mode is replaced
by the condition of zero electric current from the ionosphere to the atmosphere
j
z
z
=
−
0
4
π
c
=
∂b
y
∂b
x
∂y
∂x
−
=0
,
(7.46)
where
j
z
is the vertical current component normal to the boundary between
the atmosphere and the ionosphere.
7.4 'Thin' Ionosphere
In the general case of ionospheric MHD-wave propagation, it is necessary to
solve the full wave equations (7.7)-(7.10) with the permeability tensor de-
pendent on altitude. Obviously, that problem can be solved only numerically.
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