  We consider mathematical aspects of the axion electrodynamics in application
to the problem of evolution of geomagnetic and terrestrial electric fields,
which are coupled by relic axions born in the early Universe and
(hypothetically) forming now the cold dark matter. We find axionic analogs of
the Debye potentials, well-known in the standard Faraday - Maxwell
electrodynamics, and discuss exact solutions to the equations of the axion
electrodynamics describing the state of axionically coupled electric and
magnetic fields in a spherical resonator Earth-Ionosphere. We focus on the
properties of the specific electric and magnetic oscillations, which appeared
as a result of the axion-photon coupling in the dark matter environment. We
indicate such electric and magnetic field configurations as longitudinal
electro-magnetic clusters.
