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where the
denote height-integrated layer conductivities,
u
and
v
are the zonal
and meridional wind components, and
R
e
is the radius of the earth. It is impor-
tant to note that assuming
E
=
E
+
U
×
B
to be independent of height implies
that
u
,
are all independent of height, which is not very realistic.
Nonetheless, comparisons between the electrojet currents obtained using thin-
shell theory and experimentally measured currents have been made by treating
the zonal electric field as a free parameter (e.g., Sugiura and Cain, 1966; Untiedt,
1967) and by solving the dynamo equation by setting the zonal derivatives equal
to zero. However, the peak current density seems to be underestimated in these
models, and details concerning the vertical and latitudinal extent of the electro-
jet are not reproduced. More seriously, without vertical currents at the equator,
which are suppressed by the thin-shell model, it is very hard to satisfy the diver-
gence requirement
ν
, and
−∇
φ
∂
+
∂
J
J
λ
∂λ
θ
∂θ
=
0
since the electrojet varies rapidly in latitude but only slowly in longitude. Forbes
and Lindzen (1976) pointed out that allowing a vertical current could actually
increase the electrojet intensity predicted by the models. For example, returning
to the current equation at the equator and again ignoring the neutral wind,
J
x
=
σ
P
E
x
+
σ
H
E
z
J
z
=−
σ
H
E
x
+
σ
P
E
z
Eliminating
E
z
yields
J
x
=
σ
c
E
x
+
(σ
H
/σ
P
)
J
z
(3.20)
Thus, if
J
z
is nonzero and positive at the equator,
J
x
would exceed the Cowling
current and the electrojet current would be stronger than
σ
c
E
x
.
More realistic models that include such possibilities have been developed by
Richmond (1973a, b; see Fig. 3.17a) and Forbes and Lindzen (1976). Richmond's
improvement over the thin-shell model involved integrating the divergence equa-
tion along a magnetic field line as discussed above for the F-layer dynamo. The
zonal derivatives were all set to zero. The vertical electric field at the equator then
becomes
E
x
; that is, the integrated conductivi-
ties along the field line are used rather than the local values. Above 100 km the
integrated conductivity ratio exceeds the local value so
J
z
=
(
H
/
P
)
E
x
rather than
(σ
H
/σ
P
)
0 and both the east-
ward current and the vertical electric field are greater than the thin-shell model.
The result seems to overestimate the current (see Fig. 3.17a). Forbes and Lindzen
(1976) relaxed the requirement of zonal invariance in their solution of the three-
dimensional
∇·
J
=
0 equation, but they did not include integration along
B
.
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