High-Latitude Electrodynamics - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 414

geometry is related to the distribution of electrostatic potential around the polar
cap boundary and within the polar cap itself. These are the two fundamental
properties addressed by empirical models. To date, our description has been only
qualitative or empirical. There is, however, convincing evidence that merging
or direct connection with the solar wind dynamo via the IMF is an important
contributor to the flow in the polar cap. In this case we may reasonably ask what
effect interplanetary conditions have on the magnitude of the drift velocities or,
alternatively, on the size of the polar cap and the potential drop across it. In this
area an analytical model can be extremely useful.
Amodel developed by Siscoe (1982) describes the Region 1 and Region 2 field-
aligned currents introduced in Section 8.1 as two concentric rings. A potential

φ
is assumed to be distributed sinusoidally in local time around the region 1 ring.
Then consideration of the conservation of magnetic flux in the ionosphere and
the energy dissipated in the Region 2 current loop can be used to show that, in
an equilibrium situation, the radius of the Region 1 circle—that is, the polar cap
radius r —is related to the potential by the expression

3 / 16

=

r 0 φ

r (8.19)

Here r 0 depends on the ionospheric conductivity and the width of the auroral
zone, but these quantities may be considered as constants to first order. The
limited data that exist do not disagree with this relationship, but the exponent
on the potential is extremely small and thus, for potential differences exceeding
about 20 kV, the polar cap radius is almost independent of the magnitude of

.
Magnetic storms and substorms (discussed briefly in the next section) are known
to cause expansions of the polar cap and the entire convection pattern, but even
a qualitative description of this behavior has not yet been accomplished.
The polar cap potential drop is, however, found to be a strong function of
the interplanetary magnetic field magnitude and orientation. From rather simple
considerations, one might expect that the potential difference depends on the
area over which the IMF and geomagnetic field interconnect and the efficiency
with which the electric field in the interplanetary medium is transferred across
that area. Hill (1975) approached this problem by determining the dissipative
component of the solar wind electric field at the magnetopause (i.e., the com-
ponent parallel to the magnetopause current) that separates an internal earth's
magnetic field, B 2 , from an external field, B 1 (the IMF). He called this field the
merging region field and denoted it by E j . He then showed that this field and
the relative orientation of the magnetic fields on either side of the magnetopause
would determine the effective potential transmitted across the boundary to the
ionosphere.
The electric field in the merging region can be expressed as

φ

2 2

α (α − cos θ)

E j = E 0 (8.20)

2

1 + α − 2 α cos θ