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For the second mode
b
x
,E
y
,j
x
=0 and
∂b
x
∂z
=
4
π
c
∂E
y
∂z
∂b
x
∂t
=
1
c
j
y
,
,
y
=
σ
0
E
y
.
(15.12)
A wave propagating over the neutral atmosphere can only couple to the
first mode because the second mode is not connected with hydrodynamic
motions. For the first mode, we use the vector-potential
A
=
A
x
x
of the
electromagnetic field:
∂
2
A
x
∂z
2
b
y
=
∂A
x
∂z
1
c
∂A
x
∂t
c
4
π
,
E
x
=
−
,
x
=
−
.
(15.13)
Substituting (15.13) into the first and last equation of (15.11) we find the wave
equations for coupling of the neutral wave motions to electromagnetic waves.
For periodical small perturbations
A
x
,v
z
,
etc.
∝
exp (
−
iωt
), we obtain [1]
1+
i
ω
1
ω
v
z
+
ω
2
c
s
∂
2
v
z
∂z
2
−
+
ω
2
c
s
1
H
∂v
z
∂z
ω
1
B
0
A
x
=0
,
∂
2
A
x
∂z
2
+
iω
4
πσ
C
c
2
4
πσ
C
c
2
A
x
−
B
0
v
z
= 0
(15.14)
with
m
i
m
n
.
Equation (15.14) describes the process of the MHD-wave propagation in an
inhomogeneous medium.
Let us introduce dimensionless variables
ν
ni
1+(
β
e
β
i
)
−
1
,
ni
=
ν
in
N
i
N
n
ω
1
=
z
2
H
,
ω
ω
1
,
c
s
2
H
.
z
=
Ω
=
ω
∗
=
Then (15.14) becomes
∂
2
V
∂ z
2
−
2
∂V
∂ z
+
Ω
2
η
(
z
)
A
x
+
Ω
[
Ω
+
iη
(
z
)]
V
=0
,
∂
2
A
x
∂ z
2
+
iΩζ
(
z
)
A
z
−
ζ
(
z
)
V
=0
,
(15.15)
where the function
V
=
B
0
v
z
/ω
∗
. The coecients in (15.15) are expressed as
dimensionless functions:
ζ
(
z
)=
4
πσ
C
(
z
)
c
s
c
2
ω
∗
η
(
z
)=
ω
1
(
z
)
ω
∗
,
.
(15.16)
Figure 15.1 shows height dependencies of functions
ζ
and
η
with
H
=
10
6
s
−
1
. Here
η
(
z
)isthe
height distribution of the normalized
ν
ni
. The dimensionless conductivity
ζ
(
z
)
10
−
2
s
−
1
,ω
e
=9
10 km,
c
s
=0
.
4 km/s,
ω
∗
=2
×
×
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