Earth-Ionosphere Cavity Resonator - Ultra and Extremely Low Frequency Electromagnetic Fields - page 109

and the ionosphere. The mode coupling to the shear Alfvén and FMS wave mode in
the bottom E region ionosphere is due to the Hall conductivity of the ionospheric
plasma. In our model the ionospheric conductivity is assumed to be a scalar, so
the TE mode cannot be excited since we have ignored the gyrotropic properties
of the ionosphere. On account of symmetry of the problem, all the quantities are
independent of the coordinate '. In such a case Eqs. ( 4.1 ) and ( 4.2 ) for TM mode
reduce to the following form
@ B ' sin
r sin D

i!
c 2 E r C 0 j s ;

(4.3)

r @ r rB ' D

1 i!

c 2 E ;

(4.4)

@ r .rE / @ E r D i!rB ' : (4.5)

where the symbols @ r and @ stand for partial derivatives with respect to r and .
Following Wait ( 1962 ) we now introduce the scalar potential, U , which corresponds
to the radial component of the Hertz vector. The field components can be expressed
via the potential U as follows:

i!
c 2 @ U;

B ' D (4.6)

E r D @ r C k 2 .Ur/;E D r 1 @ r .Ur/;

(4.7)

where k D !=c is the wave number. Substituting these quantities into the set of
Eqs. ( 4.3 )-( 4.5 ) one can check that Eqs. ( 4.4 ) and ( 4.5 ) become the identities, while
Eq. ( 4.3 ) yields

@ r C k 2 .Ur/ C

@ .sin @ U/
r sin D

j s
i!" 0 :

(4.8)

To proceed analytically, it is necessary at this point to construct a suitably idealized
model of the source that is a reasonable approximation to the lightning discharge
parameters.

4.1.2 Model of Lightning Discharge and Boundary Conditions

For now, we approximate the actual CG return stroke by a lumped source with the
current moment m.t/ D Il, where I .t/ is the lightning current and l .t/ denotes
the length of the current channel. Let m.!/ be a Fourier transform of the current
moment. The CG current is upward directed. In standard meteorological practice