Geoscience Reference
In-Depth Information
or of the ion gas. Our calculations are in the neutral gas frame (or, equivalently,
we can assume
U
0), and we take the applied perpendicular electric field to be
zero for now. Parallel to
B
the velocity for each species is given by (5.1). Taking
T
i
=
=
T
e
=
T
for algebraic ease,
n
i
=
n
e
=
n
, and setting both velocities equal to
zero gives for the Northern Hemisphere
b
e
E
k
B
T
ne
mg
||
||
+
∇
||
−
=
n
0
(5.3a)
e
b
i
E
k
B
T
ne
Mg
||
||
−
∇
||
n
+
=
0
(5.3b)
e
Remember that the mobility parameter
b
j
carries the sign of the charge and that
e
is a positive number. Dividing each equation by the appropriate mobility, taking
m
M
, and adding yields
||
=
(
−
/
)
E
M
2
e
g
||
Likewise, setting the perpendicular components of velocity equal to zero in
(2.36b) and using the large-
κ
case, which is suitable for the upper F region (recall
that
κ
j
=
j
/ν
j
), we have
E
b
e
κ
e
k
B
T
ne
m
g
e
×
B
0
=
+
∇
n
−
(5.4a)
E
b
i
κ
i
k
B
T
ne
M
g
e
×
B
0
=
+
∇
n
+
(5.4b)
We can solve these two equations for the perpendicular electric field to find
E
⊥
=−
M
g
⊥
/
2
e
, and thus combining both components,
E
=−
(
M
/
2
e
)
g
Finally, substituting this result into (5.4b) yields
=
M
2
k
B
T
g
∇
n
/
n
/
This shows that the equilibrium density gradient is vertically downward, even
though the magnetic field is inclined at an arbitrary angle. This particular case is
an example of a general theorem from statistical mechanics which states that in
thermal equilibrium a magnetic field cannot affect the distribution of any fluid,
ionized or not. The plasma scale height is equal to 2
k
B
T
/
Mg
or more generally
k
B
(
T
i
. The plasma acts like a neutral gas in a gravitational
field with mean mass equal to the average of
m
and
M
. The electric field is
M
g
T
i
+
T
e
)/
Mg
if
T
e
=
/
2
e
,
minanO
+
plasma. This value is quite important
and thus has the value 0
.
8
μ
V
/
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