Electromagnetic Effects Resulted from Explosions - Ultra and Extremely Low Frequency Electromagnetic Fields - page 453

To study the ionospheric perturbations in a little more detail we consider first
a polar ionosphere. The geomagnetic field B 0 is therefore vertically parallel to z
axis. In this case Maxwell equations in the ionospheric E-layer can be written in
cylindrical coordinates
@ z B ' D 0 P E r H E ' C H V r B 0 ; (11.36)

@ z B r @ r B z D 0 P E ' C H E r P V r B 0 ; (11.37)

r 1 @ r .rB r / C @ z B z D 0; (11.38)

where B z , B r , and B ' are the components of the geomagnetic field perturbations,
E r and E ' are the components of the electric perturbations. Pedersen and Hall
conductivities of the ionospheric plasma, i.e. P and H , are considered as constant
values. The field-aligned ionospheric conductivity is assumed to be infinite and
hence E z D 0. At given orientation of the vector B 0 , only radial component V r
of the gas velocity enters the set of Eqs. ( 11.36 )-( 11.38 ).
The area of gas flow due to the acoustic wave is shown in Fig. 11.15 with three
arcs. This area is of a form of narrow band since the longitudinal size of the acoustic
wave is much smaller than a width l of the ionospheric E region. The upper arc
corresponds to the N-wave front. Because of the bipolar form of the acoustic N-
wave, at first the gas moves forward in radial directions and then it moves back.
The first area is bounded by the upper and middle arcs, while the next area is
restricted by the middle and lower arcs. The currents generated by the motion of
conductive media are oppositely directed in these areas. The changes in the current
direction occur in the middle portion of the wave where the gas velocity vanishes.
In Fig. 11.15 the opposite directions of the extrinsic currents are represented by the
circles with cross and point.
The total extrinsic current is proportional to the integral of the gas velocity
over the area covered by the wave. Now we divide this area into narrow sectors
in the z , r plane formed by rays diverging from the explosion point. Notice that the
currents flowing through the upper and lower portions of these sectors are oppositely
directed. The total current of each sector is proportional to the integral of the gas
velocity along the corresponding ray. For N-wave this integral is equal to zero, so
that the corresponding extrinsic current vanishes as well. As is seen from Fig. 11.15 ,
only the last shaded cells on the right and on the left contain the unbalanced currents.
Certainly, this approach is valid for the short N-waves. Usually the longitudinal size
of the acoustic waves b 1 3 km while the width of the E-layer is about 20-30 km
so that the condition b l is true.
Thus, the uncompensated extrinsic current arises only at the lower boundary
of the ionosphere. The cross section of this ring current is on the order of the
longitudinal wave size. This implies that the extrinsic current density is nonzero
only inside the narrow layer b l in the vicinity of the ionosphere bottom, that
is at z D 0. To simplify the problem, we formally assume that the velocity altitude
distribution is described by a delta-function, that is V r D bı. z /V r .r;t/. Performing