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the ionosphere, that is via Eqs. ( 6.69 ), ( 6.71 ) and via the continuity condition for
ı b and for the horizontal components of ı e . In this notation the wavefield of the
incident Alfvén and FMS waves in the magnetosphere is assumed to be given
functions whereas the waves reflected from the ionosphere should be found to fit
the solutions in the magnetosphere and the atmosphere. Considering the large-scale
perturbations with the scale size 2=k 10 3 km, we restrict our analysis on an
extreme case of ! kV A 30 Hz (f D !=2 5 Hz) and even on the case of
stronger inequality ! k x = 0 g 0:03 Hz (f 0:005 Hz). In such a case
the matrix M . w / for the ground surface z D d can be simplified to
0
1
ik x f =k ik x f C =k 0
ik y f =k ik y f C =k 0
f
i 0 exp . kd/
2g 3
M . w /
@
A ;
(6.76)
f C
0
where g 3 D 1 C Ǜ P . Here we made use of the following abbreviations:
k y Ǜ H C k x g 3
k
k x Ǜ H k y g 3
k
f C D
;f D
;:
(6.77)
The third column in the matrix consists of zeros because the wind velocity
component parallel to the vertical magnetic field B 0 cannot excite the magnetic
perturbations. The similar expression for matrix M .m/ can be found in the paper
by Surkov and Hayakawa ( 2008 ).
Furthermore, the study is simplified if the components of the horizontal wave
vector satisfy the requirement k x k y . This implies that the azimuthal scale
size of the perturbations is much greater than that of the meridional perturbations.
In fact, we assume a 1D distribution of the height-integrated ionospheric current,
ı I . w / .x;t/, as a source of 2D random electromagnetic fields in the surroundings.
The east-west neutral winds at the altitude range of the E layer can excite the
ring current in the ionosphere thereby producing this type of perturbations. For one
more example, it is worth mentioning that the Pc5 pulsations have a rather localized
wavefield of extension 100-200 km in north-south direction.
On the basis of this simplifying assumption, the analytic form of the transfer
matrices can be simplified. Considering the large-scale perturbations the matrix
M . w / at z D d is given by
0
1
H sin i 2 H cos
0 0 0
H sin =k x kg 2 =k x H cos =k x
i 0 exp . kd/
2g 2
M . w /
@
A ;
(6.78)
where g 2 D Ǜ P C sin . The TM mode in the atmosphere contains the components
ıb y , ıe x , and ıe z that are identical with those of the shear Alfvén wave in the
magnetosphere. Both these modes ares coupled by virtue of boundary conditions
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