Geoscience Reference
In-Depth Information
β
α
−
X
α
1+
X
α
≈
β
α
1+
β
α
−
1
(1 +
β
α
)
2
iω
ω
cα
,
β
α
(1.90)
β
α
1+
β
α
2
β
α
(1 +
β
α
)
2
1
1+
X
α
≈
iω
ω
cα
,
+
α
=
e
or
i,
(1.91)
where dimensionless parameters
β
e
=
ω
ce
/ν
e
and
β
i
=
ω
ci
/ν
i
are called the
electron and ion magnetization , respectively. The components
σ
xx
and
σ
xy
can be expressed in terms of the Pedersen
σ
xx
(
ω
→
0) =
σ
P
and Hall
σ
xy
(
ω
→
0) =
−
σ
H
conductivities as
c
2
iω
4
π
σ
xx
=
σ
P
−
,
xy
=
−
σ
H
.
(1.92)
c
2
A
It is accounted here that in the ULF-range the imaginary part of
σ
xy
is small
compared to its real part. Substituting (1.90), (1.91) into (1.86) and (1.87),
we find the Pedersen conductivity
β
i
1+
β
i
σ
P
=
c
Ne
B
0
β
e
1+
β
e
+
(1.93)
and the Hall conductivity
β
e
1+
β
e
−
.
β
i
1+
β
i
σ
H
=
c
Ne
B
0
(1.94)
(
m
e
/m
i
)
1
/
2
, we obtain for the Alfven
Omitting the terms
∼
m
e
/m
i
and
∼
velocity modified by the ion collisions
c
A
and the collisionless Alfven velocity
c
A
:
(1 +
β
i
)
(
β
i
−
c
A
β
i
B
0
(4
πNm
i
)
1
/
2
.
c
A
≈
1)
1
/
2
,
A
=
(1.95)
Usually,
c
c
A
and, therefore, the term with the displacement current in
(1.80) is omitted.
It can be shown (e.g. [5]), that
ν
in
/ν
en
∼
(
m
e
/m
i
)
1
/
2
. Then
ν
e
+
ω
2
ν
i
1
/
2
m
e
m
i
1
/
2
X
e
X
i
=
m
e
m
i
1
.
(1.96)
+
ω
2
The longitudinal conductivity is given by (1.88) which, in view of (1.96), may
be written as
ω
pe
4
π
(
ν
e
−
σ
=
iω
)
.
(1.97)
Further, the longitudinal resistivity from (1.97) is
ω
pe
4
πν
e
.
iω
4
π
ω
pe
σ
−
1
=
σ
−
1
0
−
,
0
=
(1.98)
Search WWH ::
Custom Search