Geoscience Reference
In-Depth Information
derivative in (2.5) must be retained. The velocity of the
j
th ion species then can
be found from the hydrodynamic equation (2.22b):
ρ
j
(
·
∇
)
V
j
=−
∇
p
j
+
ρ
j
g
+
n
j
q
j
(
+
V
j
×
)
−
ρ
j
ν
jk
(
V
j
−
V
k
)
V
j
E
B
(9.1)
k
t
term under an assumption of steady-state flow conditions
in time. In the equation of motion for the electrons, the terms containing the
electron mass can be neglected, as can the acceleration and advective derivative
terms. Then, as shown in Chapter 5, the ambipolar electric field component
parallel to
B
is given by
We drop the
∂/∂
E
||
=
(
1
/
n
e
q
e
)
∇
||
p
e
(9.2)
Remember that since
q
e
is a negative number,
E
is opposite in direction
||
to
points upward in the topside ionosphere. If we now let
s
be measured opposite to the magnetic field, the component of the equation of
motion in the
s
direction for the
j
th ion species in the Northern Hemisphere is
V
js
∂
∇
||
p
e
; that is,
E
||
V
js
∂
n
e
m
j
∂
n
j
m
j
∂
p
j
∂
1
p
e
∂
1
B
+
+
−
(
·
)
+
k
ν
jk
(
V
js
−
V
ks
)
=
g
0 (9.3)
s
s
s
j
=
Inspection of (9.3) reveals the forces acting on a minor ion. From right to left
they arise from friction brought about by collisions with the other ion species,
the component of gravity along the magnetic field, the partial pressure gradient
in the minor ion species itself, and the ambipolar electric field generated between
the electrons and the major ions. The final term on the left is the advective
derivative. The upward electric force must ultimately become small as the light
ions become the dominant species. Then on closed magnetic field lines the light
ion plasma can approach or attain a state of quasi-diffusive and hydrodynamic
equilibrium. While this situation can occur along the relatively short, closed mag-
netic field lines that exist in the midlatitude ionosphere, it may not be achievable
along the open or highly distended field lines in the high-latitude ionosphere.
A continuous outward flow of the minor light ions, H
+
and He
+
, can therefore
occur at either subsonic or supersonic velocities in a manner consistent with the
forces.
A description of this ion motion can be obtained by considering the equations
of continuity, motion, and energy for each species. These equations make up a
closed system that is generally solved numerically (see Schunk, 1977, and Schunk
and Nagy, 2000, for a full description of these equations), given a boundary
condition at the top of the ionosphere near 3000 km. If we apply some simplifying
assumptions, however, a description of the fundamental physics that is operating
can be obtained. Consider the motion of a minor ion species,
j
, embedded in
a major ion species,
i
, where the
i
th species is in diffusive equilibrium in an
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