High-Latitude Electrodynamics - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 424

magnetic perturbations, since they are a superposition of contributions from the
horizontal ionospheric currents, field-aligned currents, distant currents in the
magnetosphere, and currents induced in the earth's surface. For these reasons
the ground magnetic perturbations are usually expressed in terms of “equiva-
lent” ionospheric currents. The study of magnetic perturbations and their inter-
pretations as current systems in the earth and in space is extremely complex
and we will not discuss this topic in detail. However, magnetic perturbations
are used to describe phenomena such as magnetic storms and substorms and to
derive indices such as DST, K p , and AE that describe the magnetic activity in
the earth's environment. It is therefore necessary to discuss the meaning of these
indices and the nature of the magnetic measurements. This discussion is located in
Appendix B.
Essentially two techniques are utilized to derive the equivalent overhead hori-
zontal current flowing in a thin shell near 100 km altitude. One method calculates
the magnetic signature on the ground from a current flowing east-west in a small
horizontal cell which in turn flows into field-aligned currents at the edges of the
cell (Kisabeth and Rostoker, 1971). The field-aligned currents are assumed to
flow along dipole magnetic field lines and subsequently close in the equatorial
plane. By a “best-fit” process, this technique yields the total three-dimensional
current system (horizontal and field-aligned) that produces the measured ground
magnetic perturbations.
The other, more widely used method expresses the overhead current J in terms
of a divergence-free component sometimes called the equivalent current J e and
a curl-free component sometimes called the potential current J p . The potential
current can be viewed as the closing current for the field-aligned currents. For
vertical magnetic field lines it can be shown that the field-aligned and potential
current circuit produces no magnetic perturbation at the ground. Further, if the
ionospheric conductivity is uniform it can be shown that the potential current
is the Pedersen current. In this case, since other magnetic effects are small, the
equivalent current will largely represent the horizontal ionospheric Hall current.
Since it is divergence-free, the equivalent current can be expressed in terms of a
current function

such that

J e = r ×∇ (8.22)

where r is the unit radial vector in a coordinate system with the origin at the
center of the earth.
No current flows near the ground, so the magnetic perturbation there, δ B , can

be expressed in terms of a magnetic potential ϕ such that

δ B =−∇ ϕ (8.23)

There are straightforward mathematical relationships that uniquely relate

and ϕ so that the horizontal equivalent current can be derived from the potential