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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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