Geoscience Reference
In-Depth Information
For the power plasma distribution (6.75) within the thin ionosphere ap-
proximation we get
i
ν
2
c
Ω
2
ω
1
w
2
6
−p
=0
,
ω
ν
2
c
y
2
y
+
i
−
−
(6.119)
ν
1+3
w
0
1
/
2
η
+
.
y
|
w
=
±w
0
=
−
(6.120)
w
0
with the initial values determined by (6.118) or (6.120) to
w
=
w
0
.
Then we
equate the obtained boundary values of
y
+
=
y
(
w
0
) to the boundary value of
the admittance at
w
=
w
0
.
In doing so we construct the admittance equation
written for the boundary conditions (6.120) in the form
Take the integral of (6.117) or (6.119) numerically beginning with
w
=
−
F
(
ω, ν
)=
y
+
+
ν
(1 + 3
w
0
)
1
/
2
η
+
.
(6.121)
Zeroes of the function
F
(
ω, ν
) determine the Alfven resonance frequencies.
Consider now the ionosphere of a finite thickness. The coecients of
(6.117) have different characteristic scales in the ionosphere and magne-
tosphere. Therefore, split the interval of integrating into two subintervals.
The first corresponds to the ionosphere and the second to the magnetosphere.
Introduce new variables for the ionospheric interval:
2
X
LR
E
u,
τ
=
(
r
−
R
I
)
l
P
y
=
,
with
X
=
4
πΣ
P
c
Σ
P
,
P
=
.
σ
P
Here
is the height averaged Pedersen conductivity,
l
P
is the thickness of
the ionospheric conductive layer. Then from (6.117) we have
σ
P
u
2
d
u
d
τ
σ
P
1
−
µτ /
4
−
1
/L
=
ik
0
l
P
X
µτ
)
1
/
2
∓
1
/L
)
1
/
2
,
(6.122)
σ
P
(1
−
(1
−
µτ
−
where the upper (lower) sign corresponds to the northern (southern)
ionosphere;
µ
=
l
P
/
(
LR
I
). The boundary conditions (6.118) are given by
u
|
τ
=0
=0
.
(6.123)
Integrating (6.122) with the boundary conditions (6.123) for both hemi-
spheres, we find
u
at
τ
= 1. Calculating the admittance
y
and returning to
(6.117) we construct the function
F
(
ω, L
) and obtain the dispersion equation
(6.121).
Roots of (6.121) can be found using, for instance, Newton's method. Let
ω
(0
j
(
L
) be an zero approximation of the
j
-th FLR frequency
ω
j
(
L
). Then the
next approximation is
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