Geoscience Reference
In-Depth Information
j
D O
E
0
(2.4)
where
E
0
D
E
C
V
B
is the electric field in a reference frame moving at the neutral
flow velocity
V
, and
E
is the electric field in a reference frame fixed at the earth.
Thus, the Ohm's law in a collisional magnetized plasma differs from its usual
form (
1.8
) in homogeneous conducting media since the plasma conductivity is not
a scalar. The tensor components depend on both the gyrofrequencies of the ionized
particles and the collision frequencies between the plasma constituents and neutrals.
This tensor can be deduced from the equations that describe the electrodynamics of
a partially ionized plasma immersed in the external magnetic field. If we use a local
coordinate system in which the Earth magnetic field is in the direction of the
z
axis,
the tensor of plasma conductivity can be written as
0
1
P
H
0
H
P
0
00
k
@
A
;
O D
(2.5)
where
k
is the field-aligned/parallel plasma conductivity,
P
and
H
are called
the Pedersen and Hall conductivities, respectively. So, the scalar conductivity in the
Ohm's law (
1.8
) should be replaced by the tensor of plasma conductivity (
2.5
). As a
result the Ohm's law (
2.4
) can be written as follows:
j
D
k
E
k
C
P
E
0
?
C
H
O
z
E
0
?
;
(2.6)
where
E
k
and
E
0
?
are the electric field components parallel and perpendicular to the
undisturbed magnetic field
B
, and
O
z
D
B
=B stands for the unit vector parallel to
B
.
The tensor components are given by (e.g., see texts by Ginzburg (
1970
) and
Ginzburg and Rukhadze (
1972
), Kelley (
1989
), and references therein)
k
D
e
2
n
1
;
1
m
i
.
i
i!/
m
e
.
e
i!/
C
(2.7)
2
3
e
i!
m
e
n
.
e
i!/
2
i
i!
m
i
n
.
i
i!/
2
4
5
;
P
D
e
2
n
C
!
H
o
C
C
2
H
o
(2.8)
2
3
!
H
m
e
n
.
e
i!/
2
H
m
i
n
.
i
i!/
2
4
5
;
H
D
e
2
n
C
!
H
o
C
2
H
o
(2.9)
Here n
D
n
e
D
n
i
is the number densities
of
electrons and ions, ! is wave
frequency, !
H
D
eB
0
=m
e
and
H
D
eB
0
=m
i
are gyrofrequencies/cyclotron
frequencies of the electrons and ions,
in
,
ie
,
en
, and
ei
are the ion-neutral,
ion-electron, electron-neutral, and electron-ion collision frequencies, respectively,
i
D
ie
C
in
and
e
D
ei
C
en
are the total ion and electron collision frequencies.
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