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isothermal topside ionosphere. In such a case the electron and major ion gas is
distributed along the magnetic field tube according to the expression
n
e
0
exp
−
(
H
p
,
n
e
=
s
−
s
0
)/
(9.4)
where
n
e
0
is the electron density at some reference altitude
s
0
,
H
p
is the plasma
scale height, and the variation of gravity with altitude has been neglected
(Rishbeth and Garriott, 1969). The pressure for each species may be written
as
p
j
=
n
j
k
B
T
j
. If we also assume that the altitude is sufficiently high for neutral
collisions to be neglected, the minor ions collide only with the major ions and
(9.3) can be rewritten as
V
js
∂
k
B
T
e
m
j
H
p
k
B
T
j
m
j
1
n
j
∂
V
js
∂
n
j
∂
·
B
ν
ji
V
js
+
=
(
g
)
+
−
(9.5)
s
s
where we allow
T
e
T
i
or
T
j
but do not allow variationwith altitude. Neglecting
the perpendicular motion of the plasma, the steady-state continuity equation for
species
j
can be written from (2.22a) in the form
=
A
(
P
j
−
L
j
)
=
(∂/∂
s
)(
An
j
V
js
)
(9.6)
where
A
is the cross-sectional area of a magnetic flux tube,
P
j
is the production
term, and
L
j
is the loss term. Integrating this equation along the magnetic flux
tube from the reference altitude to any point
s
,wefind
(
1
/
Q
j
)(∂
Q
j
/∂
s
)
=
(
1
/
n
j
)(∂
n
j
/∂
s
)
+
(
1
/
V
js
)(∂
V
js
/∂
s
)
+
(
1
/
A
)(∂
A
/∂
s
)
(9.7)
where
s
Q
j
=
A
(
P
j
−
L
j
)
ds
(9.8)
s
0
Substituting for
(
1
/
n
j
)(∂
n
j
/∂
s
)
in (9.5), we arrive finally at the equation
V
js
−
V
t
B
(
/
V
js
)(∂
V
js
/∂
)
=−
ν
ji
V
js
+
·
−
(
k
B
T
e
/
m
j
H
p
)
1
s
g
V
t
(
)
,
(9.9)
−
1
/
Q
j
)(∂
Q
j
/∂
s
)
−
(
1
/
A
)(∂
A
/∂
s
where the thermal speed of the
j
th species is given by
1
/
2
V
t
=
(
k
B
T
j
/
m
j
)
.
(9.10)
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