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where we nowhave allowed for finite vertical velocities. If there is no field-aligned
current,
V
i
V
e
=
≡
W
D
, the vertical
plasma
velocity. Taking
n
i
=
n
e
=
n
,
m
M
and adding these two equations yields
z
nk
B
(
T
i
)
=−
∂
∂
T
e
+
nMg
−
nM
ν
in
W
D
Solving for
W
D
yields,
z
nk
B
(
T
i
)
−
1
nM
∂
∂
g
ν
in
W
D
=−
T
e
+
(5.12a)
ν
in
W
D
is often confusingly called the diffusion velocity, even though it includes a
gravitational term. The continuity equation is then
∂
n
∂
+∇·
(
nW
D
)
=
q
−
l
(5.12b)
t
=
constant and let the ion and neutral mass be identical. We also ignore the change
of the cross section of a magnetic flux tube with height. Then the neutral scale
height,
H
, is independent of height. Using
z
=
(
Before proceeding, some simplifications are useful. Let
T
e
=
T
i
=
T
n
=
T
z
−
z
0
/
H
)
, where
z
0
is a reference
height, we have
∂
∂
H
∂
1
z
=
z
∂
2
k
B
T
M
Using the ambipolar diffusion coefficient
D
a
=
in
, we obtain,
ν
D
a
1
1
nH
∂
n
1
2
H
D
a
H
n
∂
n
1
2
W
D
=−
z
+
=−
z
+
(5.13)
∂
∂
n
0
e
−
z
and so
D
a
D
0
e
+
z
and finally,
This follows, since
ν
in
∝
n
neutral
∝
=
∂
D
a
∂
z
=
D
a
. Evaluating the spatial derivative in (5.12b) we have
2
n
∂
∂
D
a
H
2
∂
1
2
∂
n
∂
n
n
2
z
(
nW
D
)
=−
z
2
+
z
+
z
+
∂
∂
∂
H
2
∂
2
, then the continu-
2
1
3
2
∂
1
If we define a diffusion operator,
D
≡
+
z
+
∂
∂
z
2
ity equation becomes,
q
z
−
l
z
∂
n
∂
=−
D
a
D
n
+
(5.14)
t
In the next sections we explore some solutions to (5.14).
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