Earth-Ionosphere Cavity Resonator - Ultra and Extremely Low Frequency Electromagnetic Fields - page 115

smaller than that predicted by Eq. ( 4.26 ). Numerous data obtained by a number
of researchers may be summarized to give the following mean annual values:
f 1 D 7:8, f 2 D 14:0-14:1, f 3 D 20:0-20:3, f 4 D 26:0-26:4, f 5 D 31:8-32:5 (in
Hz, Nickolaenko and Hayakawa 2002 ). More usually, the highest order harmonics
.n 6/ are hardly distinguished from the background since they may be below the
signal-to-noise threshold.
In the first place the small discrepancy between the theory and observations
results from the fact that we have used the idealized resonator model that ignores
actual ionospheric and atmospheric conductivity which in turn leads to the absorp-
tion of wave energy. The atmospheric conductivity becomes significant at high
altitude since it exponentially increases with altitude. More accurate models of
the conductivity profiles, including exponential and two-layer models have been
developed to account for actual distribution of the electromagnetic field at the
bottom ionosphere (Wait 1960 , 1962 ; Galejs 1961 ; Jones 1964 ). In this case some
complication arises due to the Earth magnetic field impact on conductivity of
the ionospheric plasma. Inside the gyrotropic E-layer the plasma conductivity
becomes anisotropic so that the proper boundary condition at the lower ionosphere
should be applied instead of the simple boundary condition of Eq. ( 4.13 ). Addition-
ally, the plasma conductivity tensor depends on both dip angle of the geomagnetic
field and the ionosphere status, which is highly dependent on the current time.
For instance, the conductivity of sunlit ionosphere is much greater than that of
nighttime ionosphere (see Fig. 2.5 ). This implies that the boundary conditions at
the ionosphere depend on both polar and azimuthal ' angles, and local time. The
reader is referred to the topics by Wait ( 1972 ), Bliokh et al. ( 1980 ), and Nickolaenko
and Hayakawa ( 2002 ) for details about more accurate calculation of the resonator
eigenfrequencies.
It is worth mentioning that another branch of Schumann resonances can be
excited in the Earth-Ionosphere cavity. These resonances are due to TE mode
propagation, which result in the formation of standing electromagnetic waves
perpendicular to the resonator sides, that is perpendicular to ground surface and
lower boundary of the ionosphere. Assuming that the integer number of electro-
magnetic wavelengths must keep within the atmospheric altitude d, we get the
rough estimate of the transverse normal mode eigenfrequencies f n D cn=.2d/,
where n D 1;2;3::: (Outsu 1960 ; Hayakawa et al. 1994 ; Shvets and Hayakawa
1998 ). This estimate shows that the corresponding values of eigenfrequencies must
be greater than 2 kHz. Due to strong damping of the electromagnetic waves in
this frequency range, such resonances are practically indistinguishable from the
background variations and thus they are of minor interest.

4.2.2 Quality/Energy-Factor

Further complexities of the theory of Schumann resonances arise from the attenu-
ation of electromagnetic waves due to Joule dissipation of the wave energy at the