Ionospheric Alfvén Resonator (IAR) - Ultra and Extremely Low Frequency Electromagnetic Fields - page 173

In a similar fashion we get

0 MF .!/
2rd

ıB ' .!;r; d/ D Mg ' : (5.61)

Here we have set h D 0. To illustrate the approximate solution given by Eqs. ( 5.59 )
and ( 5.60 ), a plot of the solution is shown in Fig. 5.12 with dotted line 2 0 that
corresponds to the distance r D 300 km. This dependence approximates the
numerical solution shown with line 2 to an accuracy of tens percentages.
Some understanding of the above approximation can be achieved by comparing
Eq. ( 5.61 ) with Eq. ( 4.40 ) for quasi-steady magnetic field of CG lightning occurring
between the perfectly conducting ionosphere and the Earth. In the case of small
distances, if r R e , we can simplify Eq. ( 4.40 ) with taking into account that
R e = cot .=2/ R e =2 r=2. In such a case Eq. ( 4.40 ) becomes identical to
Eq. ( 5.61 ), that is, the solution of the spherical-symmetric problem transforms to
the solution of plane problem. This implies that the approximate solution ( 5.61 )
for the component ıB ' can be derivable from the simple model of the perfectly
conducting ionosphere and the Earth.
In conclusion we discuss the assumption of vertical geomagnetic field used in
this study. At first we note that the primary electromagnetic field, that is, the TM
mode excited by a CG discharge is practically independent of both the state of
the ionosphere and the dip angle of the geomagnetic field. For example, when
deriving the perpendicular component ıB ' given by Eq. ( 5.61 ), the ionosphere in
the first approximation can be considered as a perfect conductor, so that the above
assumption is of minor importance.
On the other hand, the transformation of the TM-polarization into the TE-
polarization is caused by the mode coupling through the Hall currents in the bottom
ionosphere. The basic equations describing this coupling depend on inclination of
the geomagnetic field and local time may greatly affect the ionospheric parameters.
This means that the magnitude and resonant frequencies of the TE mode are
sensitive to the local dip angle of the geomagnetic field in contrast to the TM
mode. Since the IAR is referred to as the class of field-line resonances, the IAR
eigenfrequencies depend on the length L of the field line segment confined by the
IAR boundaries. Taking Eq. ( 5.42 ) for the rough estimate of the nighttime resonant
frequencies, that is, f n D V AI n=.2L/, where n is an integer value, we can conclude
that f n can increase with magnetic latitude because of the decrease in L. Despite
we have ignored the actual variations of V AI with altitude, the above conclusion is
in accordance with observations. For example, Bösinger et al. ( 2002 ) have reported
that the IAR resonant frequencies at low-latitude station are on average smaller by
a factor of 2-3 than those at high latitude. In particular, the fundamental resonant
frequency varies from 0:4 to 1:0 Hz with increase in latitude.
The amplitudes of the resonances seem to be not so sensitive to the local dip
angle of the geomagnetic field. In this notation, the most important factors are the
intensity of local lightning activity and the distance from the observation site to the
World thunderstorm centers. However, it appears that the primary TM mode excited