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scale. As is seen from this expression, the extrinsic current flows in the circle with
radius a and it vanishes outside this area. In this case the integrals in Eqs. ( 11.52 )
and ( 11.53 ) can be easily calculated (Surkov 1996 )
0 H
2
1=2
2B 0 V bT
a
r
2 @ z B z i ;
B z i D
G z . z ;t/; B ri D
(11.57)
B 0 V bT z r
a.2/ 1=2 . 0 H / 3=2 G ' . z ;t/:
B 'i D
(11.58)
where the functions G z and G ' are determined by Eqs. ( 11.54 ) and ( 11.55 ).
We recall that Eqs. ( 11.57 ) and ( 11.58 ) are valid in the interval t< d ,
which can be applied to the front of electromagnetic perturbations. Formally this
solution describes the case of infinite gyrotropic conductive half-space bordering
the atmosphere.
The factor exp ǚ 0 P z 2 =.4t/ is indicative of the diffusion character of the
GMP propagation across the ionospheric E -layer. Danilov and Dovzhenko ( 1987 )
have noted that this factor determines the length of an electromagnetic precursor for
acoustic wave. This effect is similar to the electromagnetic forerunner of seismic
wave that we have examined in more detail in Chap. 7 . Substituting p for in
Eq. ( 7.20 ) we obtain the estimate of the precursor length . 0 P C a / 1 , where
C a is the acoustic wave velocity.
The damping factor in Eqs. ( 11.54 ) and ( 11.55 ) is analogous to the skin effect
in conductive media. However, the oscillating factors in these equations lead to a
new property of this effect because the diffusion perturbations propagate in a form
of damped oscillation. The phase of the oscillations 0 H z 2 =.4t/ depends merely
on the Hall conductivity, which means that the effect essentially depends on the
presence of magnetized electrons in the ionospheric plasma of the E layer. The
oscillation period increases in time and the oscillations cease at t> 0 H z 2 =.4/.
By analogy with the above line of reasoning, one can estimate the “oscillatory”
length of the electromagnetic precursor as o . 0 H C a / 1 .
The same regime of diffusion has been demonstrated to be excited in the
ionosphere for the case of horizontal geomagnetic field (Surkov 1990a , b ). In
the Hall medium, the analogous type of micropulsations propagating along the
geomagnetic field has been termed the Schrödinger mode (Greifinger and Greifinger
1965 ). Another way to explain the oscillatory structure of the electromagnetic
forerunner in the magnetoactive plasma is to take into account the radiations of
helicon waves which are known as whistler mode in the geophysical practice. As
the electrons are magnetized whereas the ions are not yet, the dispersion relation for
the field-aligned helicons reads (e.g., Lifshitz and Pitaevskii 1981 )
k 2 V A
H
! D
(11.59)
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