Inhomogeneous Ionosphere - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 280

The above can also be used to study quasi-stationary currents and electric
fields ([9], [11]). These estimations allow us to find the equivalent current
systems associated with Alfven and FMS-waves separately.
The distribution of field-aligned current j ( θ ,λ ) associated with Alfven
waves is connected with the horizontal ionospheric current I τ through the
two-dimensional divergence operator

( Σ − ∇ ⊥ · ∇ ⊥ · ∇ ⊥ Φ 0 )= j ,

I τ = (9.26)

where Φ 0 is a scalar electrostatic potential of the polarization electric fields
∇ ⊥ Φ 0 ; θ and λ are latitude and longitude, respectively. Solving (9.26) one
can find Φ 0 ( θ ,λ ) and currents

Σ

∇ ⊥ Φ 0 induced by the Alfven wave.
In the case of FMS-waves we put j = 0 in the right-hand side of (9.26)
and take into consideration the external electric field E 0 as the main source
of ionospheric electric fields and currents:

−

( Σ ( ∇ ⊥ · − ∇ ⊥ Φ 0 + E 0 )) =0 . (9.27)

To solve (9.26)-(9.27) the following boundary conditions for electrostatic
potential on the poles can be used:

Φ 0 =0 . (9.28)

In order to describe the geomagnetic variations produced by Alfven and
FMS-waves we use their representation in terms of equivalent current sys-
tems. These are two-dimensional solenoidal currents in the ionosphere, with
a magnetic effect equal to the effect observed on the ground. Such fictitious
currents are defined by

j eq = ∇ ⊥ Ψ × n (9.29)

where j eq is the equivalent current density; n is a unit vector of the normal
to the ionospheric surface (positive upward); Ψ is called the current function
and the current is parallel to the lines Ψ =const.
If there are no field-aligned currents, the equivalent current is considered
to be the real ionospheric current. In a three-dimensional case, the equivalent
and real currents are different because the magnetic effect on the ground is
produced by field-aligned currents as well. To calculate the equivalent currents
in this case we can use the following algorithm.

Alfven Wave

Let us assume that the solenoidal ionospheric current system is generated by a
localized field-aligned current j which flows into a homogeneous high-latitude
ionosphere with nonzero Pedersen and Hall conductivities. It is easy to show
that Ψ sol of this current system depends on the horizontal coordinate R as
ln R and that the current density and magnetic field are proportional to R − 1 .