Geoscience Reference
In-Depth Information
Equating Eqs. (8.8) and (8.9) yields
B sw δ
V sw
δ
P L pc E pc =
(
V A δ
t
)(
V sw δ
t
).
t
We should note that by letting dt in the time derivative equal
δ
t , we ar e assum-
ing the slowdown occurs at a large spatial scale. Now, using V A = B sw 0 ρ
and letting
η A = μ 0 V A be the intrinsic impedance of a magnetized plasma at low
frequencies, we can write
E pc
B pc
.
δ
V sw
V sw = P η A
B sw
E sw
However, if magnetic field lines are equipotentials, then E 2
/
B is a constant
l 2 , where
anywhere along the field line, since E
l and B
1
is the
constant potential and
l is the distance between two field lines. Thus, the term
in brackets goes to 1 and we get
δ
δ
V sw
V sw = P η A .
(8.10a)
This simple yet elegant result demonstrates how a planetary magneto-
spheric/ionospheric system interacts with a stellar wind and the controlling influ-
ence of the polar cap conductivity. For the summer polar cap,
1 and,
P =
5
taking
, we find a 50% slowdown. This equation implies that the
winter polar cap electric field should be larger than the summer field. This impli-
cation is suggested by data such as that in Fig. 8.1b in which the polar cap in
winter exceeds the summer value in agreement with Eq. (8.10a).
More sophisticated studies by Siscoe et al. (2002) and Hill (1984) find
the following relationship between the polar cap potential and interplanetary
parameters:
η A =
0
.
1
6 E sw P 1 / 3
sw D 4 / 3 F
57
.
(θ)
pc =
(8.10b)
P 1 / 2
sw D
+
0
.
0125
ξ P E sw F
(θ)
where
pc is the polar cap potential drop, E sw is the electric field in the upstream
solar wind
(
E sw =|
V sw ×
B sw | )
, P sw is the ram pressure exerted by the solar wind
P sw = ρ sw V sw )
(
, D is the earth's dipole field normalized to 1 for the present-day
value, F
is a function of the clock angle of the IMF to account for the geometry
of reconnection (here F
(θ)
sin n
(θ) =
(θ/
2
)
,so F
(θ) =
0
/
1 for IMF pure northward
or southward),
is a dimensionless coefficient between 3 and 4 that depends on
the geometry of currents in the ionosphere (Crooker and Siscoe, 1981), and
ξ
P is
the height-integrated Pedersen ionospheric conductivity (assumed to be uniform
 
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