Geoscience Reference
In-Depth Information
Equation (6.91) allows us to express
n
j
in terms of the longitudinal magnetic
component. Denote the physical component of the longitudinal magnetic field
as
b
.
Then field line distribution of the cold plasma expressed in terms of
b
can be written as
w
0
mc
L
d
ω
j
d
L
(1 + 3
u
2
)
1
/
2
b
(
u
)
e
2
j
(
u
)d
u,
n
j
=
(6.112)
−
w
0
where the integral is calculated over a contour in the complex plane.
From (6.89), (6.107), and (6.112) we have
sin
W
j
(
w
)
,
Λ
Bj
U
j
j
=2
n
+1
b
ϕj
=
,
(6.113)
−
cos
W
j
(
w
)
,
j
=2
n
(
L
−
L
j
)(1
−
w
2
)
3
/
2
cos
W
j
(
w
)
,
E
νj
=
Λ
Ej
U
j
(1 + 3
w
2
)
1
/
2
(
L
j
=2
n
+1
,
(6.114)
sin
W
j
(
w
)
,
j
=2
n
−
L
j
)(1
−
w
2
)
3
/
2
where
n
=0
,
1
,
2
,
3
,...
,
W
j
(
w
)=
j
π
iγ
j
)
w
2
(1
−
w
0
,
d
u b
1+3
u
2
1
/
2
cos
W
j
(
u
)
,
w
0
j
=2
n
+1
U
j
=
,
(6.115)
sin
W
j
(
u
)
,
j
=2
n
−
w
0
a
j
c
2
L
2
R
E
Ω
d
ω
j
a
j
c
LR
E
d
ω
j
d
L
Λ
Bj
=
im
,
Ej
=
m
.
(6.116)
d
L
6.5 Numerical Simulation
Dispersion Equation
Let us describe a method of derivation of the dispersion equation based
on the impedance successive sweep method. Introduce a partial admittance
y
=
e
1
/e
2
. Note that this admittance is equal to the ratio of covariant field
components contrary to the usual admittance determined in terms of physical
components. From (6.81) obtain a Riccati-type equation for
y
:
c
1+3
w
2
ε
⊥
=0
,
ω
ν
2
c
y
2
i
ω
y
+
i
−
(6.117)
with the boundary conditions
y
|
w
=
±w
0
=0
.
(6.118)
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