Geoscience Reference
In-Depth Information
B
2
Carrying out the triple cross product, [
(
U
×
B
)
×
B
]
/
=−
U
and so
⊥
V
j
B
2
⊥
=
E
×
B
/
This equation shows that in the collisionless case the plasma moves at the
E
×
B
velocity in any reference frame, provided the electric field and velocity are
expressed in that reference frame. In the earth-fixed frame, then,
V
j
1
B
2
E
−
k
B
T
j
/
q
j
∇
+
M
j
/
q
j
g
×
⊥
=
/
n
/
n
B
(2.36d)
which is identical in form to (2.36c). In (2.36a) we have left the prime on
E
,
but it should be noted that the transformation (2.32a) leaves the component of
E
parallel to
B
invariant.
The solutions of (2.34) for intermediate values of
κ
are given by
V
j
W
j
||
=
(2.37a)
||
and
W
j
⊥
1
κ
j
W
j
⊥
1
j
×
B
V
j
⊥
=
j
+
(2.37b)
2
2
+
κ
+
κ
where again we have expressed the result in terms of the steady-state unmag-
netized velocity solution
W
j
⊥
. These expressions show explicitly that for small
V
j
tends toward
W
j
⊥
κ,
, while for large
κ
the motions tend to be perpendicular
to the forces. The absolute values of
κ
e
and
κ
i
are plotted in Fig. 2.5 for an
400
B
5
0.25 Gauss
300
i
200
e
100
0
0.01
0.10
1.0
10
100
1000
Ratio of gyro to collision frequencies
Figure 2.5
Typical values for
κ
e
and
κ
i
in the equatorial ionosphere for a magnetic field
10
−
5
tesla.
of 0
.
25G
=
2
.
5
×
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