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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
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
θλ /
θ)(∂φ/∂λ) + 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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