Geoscience Reference
In-Depth Information
The refractive index
n
of a medium containing free electrons with the
external magnetic field
B
0
is determined by the Appleton-Hartree formula
[20]
X
n
2
±
=1
−
+
Y
L
1
/
2
,
(12.6)
Y
T
/
4
1
Y
T
/
2
1+
iZ
−
X
+
iZ
±
1
−
−
X
+
iZ
where
1
ω
,
and
B
L
,
B
T
are projections of the vector
B
on the longitudinal and transverse
to the radio-wave propagation direction;
Y
L
=
eB
L
mc
1
ω
Y
T
=
eB
T
mc
and
X
=
ω
pe
ω
r
ν
e
ω
r
,
,
Z
=
r
=2
πf
r
,
ω
pe
10
9
N
e
is square of the electron plasma frequency;
ν
e
is the
collision frequency of an electron with all the other particles.
The magnetic field
=3
.
18
×
B
(
t
)=
B
0
+
b
(
t
)
,
where
B
0
is the background geomagnetic field and
b
(
t
) is the pulsations
magnetic field. Let us rewrite (12.5) in the form
z
0
d
dt
V
∗
=
µ
(
B
L
(
z, t
)
,
B
T
(
z, t
)
,N
e
(
z, t
))
dz.
(12.7)
0
As the refractive index turns into zero at the reflection point, the last equation
can be written
∂µ
∂
B
L
dz
+
µ
(
z
0
,t
)
dz
0
dt
z
0
∂
B
L
∂t
∂µ
∂
B
T
∂
B
T
∂t
∂µ
∂N
e
∂N
e
∂t
V
∗
=
+
+
0
∂µ
∂
B
L
dz.
z
0
∂
B
L
∂t
∂µ
∂
B
T
∂
B
T
∂t
∂µ
∂N
e
∂N
e
∂t
=
+
+
(12.8)
0
The electron concentration change in time is found from the continuity equa-
tion
∂N
e
∂t
=
−
∇
·
(
N
e
v
e
)+
Q
−
L,
(12.9)
where
v
e
is the electron drift velocity in the MHD-wave field,
Q
and
L
are
the production and loss rates, respectively. Combining (12.8) and (12.9),
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