Geoscience Reference
In-Depth Information
Potentials
Let us present the total field as a sum of electric (
e
) and magnetic (
m
) modes
E
=
E
(
e
)
+
E
(
m
)
.
b
=
b
(
e
)
+
b
(
m
)
,
(7.12)
Electric
E
(
e
)
and magnetic
b
(
e
)
components of the electric mode (
e
-mode) is
also called the transverse magnetic mode (
TM
-mode). The vertical magnetic
component
b
(
e
)
of the
TM
-mode is
z
b
(
e
)
z
=0
.
and the horizontal magnetic components are 2D solenoidal, i.e.
∂b
(
e
)
+
∂b
(
e
)
x
∂x
y
∂y
=0
.
(7.13)
Magnetic
b
(
m
)
and electric
E
(
m
)
components relate to the magnetic mode (
H
-
mode) or transverse electric mode (
TE
-mode). In the
TE
-mode the vertical
component of the electric field vanishes, while the horizontal magnetic field is
2D-potential
∂b
(
m
)
∂b
(
m
)
y
∂x
−
x
∂y
=0
.
(7.14)
For finding the fields in the atmosphere let us take advantage of Debye's
potentials [6]. For example, the fields of the electric mode according to (7.13)
canbepresentedas
1
ε
(
z
)
∂
2
Φ
(
r
)
∂x ∂z
,
ik
0
∂Φ
(
r
)
∂y
E
(
e
x
(
r
)=
b
(
e
x
(
r
)=
−
,
∂
2
Φ
(
r
)
∂y ∂z
,
1
ε
(
z
)
(
r
)=
ik
0
∂Φ
(
r
)
∂x
E
(
e
)
y
b
(
e
)
y
(
r
)=
,
1
ε
(
z
)
∇
E
(
e
z
(
r
)=
2
⊥
(
e
z
(
r
)=0
,
−
Φ
(
r
)
,
(7.15)
and for the magnetic mode from (7.14)
(
r
)=
ik
0
∂Ψ
(
r
)
∂y
(
r
)=
∂
2
Ψ
(
r
)
∂x∂z
E
(
m
)
x
b
(
m
)
x
,
,
(
r
)=
∂
2
Ψ
(
r
)
∂y∂z
ik
0
∂Ψ
(
r
)
∂x
E
(
m
)
y
(
m
)
y
(
r
)=
−
,
,
2
⊥
E
(
m
)
z
b
(
m
)
z
(
r
)=0
,
(
r
)=
−
∇
Ψ
(
r
)
,
(7.16)
∂
2
∂x
2
+
∂
2
∂y
2
.
2
⊥
where
∇
=
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