Geoscience Reference
In-Depth Information
variable
w
=(1+
νr
)
1
/
2
=
x
=cos
θ
is, with that, real. If the oscillations are
excited by an external driver at a real frequency
ω
, then (6.79)-(6.80) leads to
the complex
ν
r
=
ν
r
+
iν
r
. In this case,
ν
=
ν
r
provides a non-trivial solution
of this boundary problem.
The variable
w
is then complex and (6.79) are integrated over the curve
w
=(1+
νr
)
1
/
2
in the complex plane. Let
x
=(1+
ν
r
)
1
/
2
, then
w
=
1+
i
ν
ν
x
2
1
/
2
i
ν
ν
−
.
It is convenient to use only the second-order equation instead of the system
(6.79). Let us eliminate, for example
e
1
from (6.79). From the second equation
of (6.79),
e
1
=
ν
2
/ik
0
(
de
2
/dw
). Substituting it into the first equation and
into the condition (6.80), we have
d
2
e
2
d
w
2
+
k
0
ν
2
(1 + 3
w
2
)
ε
⊥
e
2
=0
,
(6.81)
w
=
w
0
d
e
2
d
w
=0
.
(6.82)
At real
ν
, from the boundary problem (6.79), (6.80), or similar problem
(6.81), (6.82), it is possible to find the spectrum of the resonance frequencies
ω
j
(
ν
) and decrements
γ
j
(
ν
)
,j
=1
,
2
,
3
,...
. But, if we put real frequency
ω
,
then one can define the location of the resonance magnetic shell (Re
ν
j
)and
its half-width (Im
ν
j
).
Similarly to (6.26)
γ
j
d
ω
j
(
ν
=
ν
j
)
d
ν
−
1
ν
j
≈
,
(6.83)
which defines the half-width of the resonance shell with a beforehand known
decrement of the resonance oscillations.
'Thin' Ionosphere
Because the skin depth in the ionosphere at frequencies
f
0
.
1 Hz exceeds
the thickness of the ionospheric conductive layer, it is possible to integrate
(6.74) or (6.79) over the height and use the 'thin' ionosphere approximation
which is valid if
l
P
c
2
4
πΣ
P
l
P
,
ω
Σ
P
=
σ
P
sin
I
0
d
s,
(6.84)
0
where integration is to be performed from the pierce point of the field line with
the boundary
S
a
up to the upper boundary of the conductive layer,
I
0
is the
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