Inhomogeneous Ionosphere - Hydromagnetic Waves in the Magnetosphere and the Ionosphere - page 279

The above condition is too rigid a restriction for the pulsation range. This
is why in constructing models of interaction between MHD-waves and the
ionosphere, the following approaches are more preferable. In one approach,
already used in this topic, we set the incident MHD-wave above the ionosphere
and use it to determine the ground electromagnetic field and field line currents.
In another approach we proceed from a given system of longitudinal currents
and use (9.7). A third approach is possible, justified only for perturbations
that change quickly enough in the transverse direction. More precisely, the
transverse scale of perturbations k − 1
⊥

satisfies

k − 1
⊥

L ⊥ ,

where L ⊥ is the scale of change of magnetosphere and ionosphere parameters
in the transverse direction (see Chapter 10).
The equation for the potential (9.7) is ([9], [11])

Σ ( − ∇ ⊥ Φ 0 + E 0 )= − ∇ ⊥ F + ∇ × Ψ . (9.23)

Here F is a scalar determining the potential part of the horizontal current
and linked with the field-aligned currents by

2
⊥

∇ F = j .

Ψ provides the vortex part of the horizontal current. E 0 denotes external
sources of the electric field not connected with field-aligned currents, for in-
stance, E 0 can be transported from the magnetosphere to the ionosphere by
an incident FMS-wave or generated by neutral wind due to dynamo action.
In the latter case

1
c [ v n ×

E 0 = − B 0 ] .

When condition (9.3) holds it is possible to neglect electric fields and
corresponding currents of the secondary Alfven waves caused by the FMS-
wave (see (9.4)). Let L ⊥ ∼ ×

10 8 km / s for the dayside ionosphere. In the night, the conductivity is smaller
at least by a factor of 10 . Then, (9.4) gives

1000 km and T ∼ 100 s ,Σ H ∼ Σ P =0 . 7

E (1)
A
E (0)
A

∼ 0 . 2 day, (9.24)

E (1)
A

E (0 A ∼

0 . 02 night. (9.25)

The resulting estimations (9.24) and (9.25) may be considered as a foun-
dation for the investigation of geomagnetic variations using the mathematical
formalism suggested in the theory of long-period geomagnetic variations [5].