Geoscience Reference
In-Depth Information
Basic Equations and Boundary Conditions
The Maxwell's equations for a monochromatic wave with time dependence
exp(
−
iωt
)are
∇
×
b
(
r
)=
−
ik
0
ε
E
(
r
)
,
(7.5)
∇
×
E
(
r
)=
ik
0
b
(
r
)
.
(7.6)
Here
k
0
=
ω/c
is a wavenumber in the vacuum,
ε
is the complex permeability
tensor.
By virtue of horizontal homogeneity of the medium, solutions of these
equations can be found for Fourier harmonics. Then a solution is obtained for
an arbitrary distribution with the help of the inverse Fourier transformation.
Substitute into (7.5)-(7.6) harmonics in the form
E
(
x
1
,x
2
,x
3
)=
E
(
x
3
)exp(
ik
1
x
1
+
ik
2
x
2
)
,
b
(
x
1
,x
2
,x
3
)=
b
(
x
3
)exp(
ik
1
x
1
+
ik
2
x
2
)
.
Denote the amplitudes of spatial Fourier harmonics in the same way as the
initial fields, explicitly indicating whenever necessary the arguments (
x
1
,x
2
)
or
k
τ
=(
k
1
,k
2
). For instance, the Fourier transformation with respect to
x
1
,
x
2
from
E
1
(
x
1
,x
2
,x
3
) will be written as
E
1
(
x
3
;
k
τ
) or simply
E
1
,etc.
Now, consider the Maxwell's equations (7.5), (7.6) in the coordinates
x
1
,x
2
,x
3
. Let the longitudinal conductivities of the ionosphere and the
magnetosphere tend to infinity. In this case, the longitudinal electric field
vanishes, thus
E
3
= 0. If the variable
b
3
is eliminated from (7.5) and (7.6), we
obtain the equations for
b
1
,b
2
,E
1
,E
2
in the ionosphere and magnetosphere:
=
E
1
+
4
πσ
P
c
E
2
+
ik
1
b
1
cot
I,
+
i
k
1
k
0
∂b
1
∂x
3
4
πσ
H
c
sin
I
−
i
k
1
k
2
k
0
−
(7.7)
=
E
1
−
4
πσ
H
c
sin
I
−
E
2
+
ik
2
b
1
cot
I,
i
k
2
k
0
∂b
2
∂x
3
4
πσ
P
c
sin
2
I
−
i
k
1
k
2
k
0
−
(7.8)
∂E
1
∂x
3
=
ik
0
b
2
,
(7.9)
∂E
2
∂x
3
=
−
ik
2
E
1
cot
I
+
ik
1
E
2
cot
I
−
ik
0
b
1
.
(7.10)
Equations (7.7)-(7.10) are similar to the equations of the radio wave in the
ionosphere [5]. Equations in the atmosphere and on the ground can be ob-
tained in the same manner.
On the boundary between the atmosphere and the ionosphere (
x
3
=0)it
is necessary to demand continuity of horizontal components of
E
and
b
.It
is more convenient to split the problem solution into two parts. First, let us
find the fields reflected from the ionosphere back to the magnetosphere. Then,
from the known values of the initial incident field and the reflected field, we
can find the field penetrated to the ground surface.
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