Geoscience Reference
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assumption is that no vertical current flows anywhere in the system; that is, all
currents flow in a thin ionospheric shell. Setting
J
z
equal to zero (in geographic
coordinates now) yields
E
λ
+
σ
z
θ
E
θ
+
σ
zz
E
z
=
σ
z
λ
0
or
E
z
=−
E
y
+
σ
z
θ
E
θ
σ
z
λ
/σ
zz
For
I
0 this reduces to (3.13) at the equator. Using this expression to eliminate
E
z
,
J
R
can now be written as a two-dimensional vector and the dynamo equation
as a function of
=
θ
and
λ
only (Forbes, 1981),
J
ξ
λλ
E
E
θ
ξ
λθ
J
=
·
ξ
θλ
ξ
θθ
θ
where
σ
H
cos
I
2
+
σ
0
sin
2
I
σ
P
cos
2
I
ξ
λλ
=
σ
P
+
/
−
σ
0
σ
H
sin
I
+
σ
0
sin
2
I
σ
P
cos
2
I
ξ
λθ
=−
ξ
θλ
=
/
σ
0
σ
P
+
σ
0
sin
2
I
σ
P
cos
2
I
ξ
θθ
=
/
are the so-called layer conductivities. Insertion into
∇·
J
=
0 in geographic coor-
dinates yields
(
∂/∂λ
)
(
ξ
λλ
/
θ
)(
∂φ/∂λ
)
+
(
ξ
λθ
/
R
e
)(
∂φ/∂θ
)
[
R
e
sin
]
+
(∂/∂θ)
{
θ
(ξ
θλ
/
θ)(∂φ/∂λ)
+
(ξ
θθ
/
R
e
)(∂φ/∂θ)
}
sin
[
R
e
sin
]
(3.18)
=
(∂/∂λ)
−
ξ
λλ
uB
z
+
ξ
λθ
vB
z
]
+
(∂/∂θ)
{
sin
θ
[
−
ξ
θλ
uB
z
+
ξ
θθ
vB
z
]
}
[
Assuming
E
λ
and
E
θ
to be independent of height, (3.18) can be integrated over
height to give the thin-shell dynamo equation
∂φ/∂λ
R
e
(∂/∂λ)
λλ
/
R
e
sin
θ
+
λθ
/
∂φ/∂θ
sin
R
e
+
∂/∂θ
θ
θλ
/
R
e
sin
θ
∂φ/∂λ
+
θθ
/
∂φ/∂θ
(3.19)
−
ξ
λλ
sin
v
B
z
dh
v
B
z
dh
−
ξ
θλ
∂
∂λ
∂
∂θ
=
u
+
ξ
λθ
+
θ
u
+
ξ
θθ
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