Geoscience Reference
In-Depth Information
•
The half-wave mode,
R
S
R
N
>
0 . In this case, an integer number of half-
waves lies on a field-lines. The resonance frequencies are
ω
n
ω
A
i
γ
n
=
n
−
ω
A
,
n
=0
,
±
1
,
±
2
,....
(5.20)
The relative decrement is independent of the harmonic number
n
and
γ
n
ω
A
1
2
π
log (
R
S
R
N
)
.
=
−
(5.21)
•
The quarter-wave mode,
R
S
R
N
<
0
.
The field-lines length is 1
/
4
,
3
/
4
,
5
/
4
,...
of the wavelength. The resonance frequencies are
ω
n
ω
A
1
2
−
i
γ
n
=
n
−
ω
A
,
n
=0
,
±
1
,
±
2
,
±
3
,...
(5.22)
and the relative decrement is
γ
n
ω
A
1
2
π
log
=
−
|
R
S
R
N
|
.
(5.23)
In the two limiting cases:
Σ
P
1and
Σ
P
•
Small conductivities,
1, the decrement is
π
Σ
P
+
Σ
P
.
γ
n
ω
A
≈
1
(5.24)
Σ
P
1and
Σ
P
•
Large conductivities,
1
,
the decrement is
1
Σ
P
.
γ
n
ω
A
≈
1
π
1
Σ
P
+
(5.25)
Σ
P
=
Σ
P
=
Σ
P
,and
R
=
R
S
=
R
N
=
In the symmetrical case at
1
Σ
P
/
1+
Σ
P
, we find from (5.21) that
−
π
log
Σ
P
1+
Σ
P
γ
n
ω
A
1
1
−
ω
A
=
πc
A
l
z
=
−
with
.
The dependence of the decrement
γ
n
and the Q-factor
Q
n
=
ω
A
/
2
γ
n
of the
fundamental mode on
Σ
P
are presented in Fig. 5.3. For dayside conditions at
moderate solar activity the normalized
Σ
P
≈
10, and for nighttime conditions
Σ
P
≈
0
.
5. It can be seen from Fig. 5.3 that the Q-factor varies from 9
.
8at
the daytime to 1
.
4 at nighttime.
Below, we shall present fields expanded by eigenfunctions (5.12). However,
it is cumbersome to find a series the eigenfunctions of the generalized boundary
problem. We therefore prefer to regard (5.9)-(5.10) as a problem on wave
eigennumbers, keeping for it the designation of 'Dungey's problem'.
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