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electric field must also decrease along the field lines. For a dipole field, Mozer
(1970) has shown that the two electric field components (meridional E MI and
zonal E ZI )
map from the ionosphere to the magnetosphere in the equatorial
plane as
2 L L
1 / 2
3
4
E MI =
E RM
(2.48a)
L 3 / 2 E ZM
E ZI =
(2.48b)
where the L value is the distance from the center of the earth to the equatorial
crossing point measured in earth radii ( R e
)
, E RM is the radial magnetospheric
component at the equatorial plane, and E ZM is the zonal magnetospheric com-
ponent there. The equation for the zonal component (2.48b) is in excellent agree-
ment with the corresponding data in Fig. 2.8. Notice that the zonal ionospheric
electric field component maps to a zonal field in the equatorial plane but that the
meridional component in the ionosphere, E MI , becomes radial in the equatorial
plane ( E RM )
. In particular, a poleward ionospheric electric field points radially
outward at the equatorial plane. Thus, large-scale electric fields generated in
the E and F regions of the ionosphere can map upward to the magnetosphere
and create motions there. Likewise, electric fields of magnetospheric and solar
wind origin can map from deep space to ionospheric heights and have even been
detected by balloons at stratospheric heights (Mozer and Serlin, 1969). For per-
fectly conducting field lines, the ratio E 2 / B is conserved, since B varies as the
area, while E varies as the linear distance along the magnetic field.
Farley (1959) studied the upward mapping process realistically by includ-
ing the z dependence of the conductivities in his analysis. The basic equations
∇·
E are the same used in deriving (2.47), and, with
the assumption that variations of
J
=
0, E
=−∇ φ
, and J
= σ ·
σ
occur only in the z direction, they yield
2
x 2
2
y 2
+ (
P ) ∂/∂
( σ 0 ∂φ/∂
) =
φ/∂
+
φ/∂
1
z
z
0
The same change of variables now yields
z ln
2
2
φ + ∂φ/∂
z ∂/∂
1
/
0 σ P )
=
0
(2.49a)
1
/
2
σ 0 σ P )
= σ m is termed the geometric mean conductivity. Furthermore,
where (
c 0 z )
σ m can be modeled in the form
σ m =
(
if
c exp
, then the equation simplifies to
2
c 0 ∂φ/∂
z =
φ +
0
(2.49b)
This differential equation has a straightforward analytical solution. By con-
sidering the solutions with different Fourier wave numbers in the source field,
 
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