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q
j
(
j
=1
,
2
,
3
,
4) and the eigenvectors of the matrix
T
which meet the charac-
teristic equation
det(
T
−
q
1
)=0
,
(1.116)
where
1
is the unit 4
×
4 matrix. Each root
q
j
has a corresponding eigenvector
s
j
of the matrix
T
which satisfies
Ts
j
=
q
j
s
j
.
(1.117)
The longitudinal component of the wavenumber is now defined as
k
(
j
)
=
k
(
j
)
=
k
0
q
j
. Then (1.114) has four solutions
s
(
j
)
exp(
ik
(
j
)
z
) corresponding
to two wave modes each propagating into the positive and the negative
z
-
direction.
From (1.116) we get
z
k
2
k
0
ε
⊥
(
ω
)
k
0
ε
⊥
(
ω
)
ε
⊥
(
ω
)
ε
(
ω
)
k
2
k
2
−
−
⊥
−
+
k
0
ε
(
ω
)
k
0
ε
(
ω
)
=0
,
ε
⊥
(
ω
)
ε
(
ω
)
k
2
−
(1.118)
⊥
where
ε
α
(
ω
)=
i
4
πσ
α
(
ω
)
/ω, α
=
⊥
,
,
. Solve the biquadratic equation
(1.118) with respect to
k
2
:
1+
σ
⊥
(
ω
)
σ
(
ω
)
k
2
2
k
2
=
k
0
ε
⊥
(
ω
)
−
1
1
/
2
2
k
4
4
−
σ
⊥
(
ω
)
σ
(
ω
)
(
ω
)+
σ
⊥
(
ω
)
k
0
ε
2
σ
(
ω
)
k
0
ε
(
ω
)
k
2
±
−
(1.119)
⊥
Equations (1.118) and (1.119) are meaningful only if an explicit depen-
dence of the dielectric permeability on frequency is known. For a cold plasma,
σ
α
(
ω
) is determined by (1.86)-(1.88). In a general case of a partially ionized
plasma, these equations are rather cumbersome. However, the simple equa-
tions for the tensor of dielectric permeability obtained in the previous section
allow us to study dispersion equations for almost all the cases in the ULF
frequency range.
Equation (1.118) may be written more explicitly for the specific regions
of the magnetosphere and the ionosphere. But note once again that we limit
our consideration to the cold plasma and neglect kinetic effects, which can be
important in the remote regions for the small-scale disturbances.
In the ionospheric
D
-layer the medium is the isotropic conductor with the
conductivity
σ
0
(1.99). Substituting
ε
⊥
≈
ε
≈
4
πσ
0
/ω
, where
σ
0
is defined
in (1.99), and
ε
≈
0, into (1.118), we get
k
=
k
2
⊥
1
/
2
c
2
4
πσ
0
k
2
,
+
k
2
iω
=
.
(1.120)
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