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at
z
D
0. The second strings in the matrices consist of zeros because ıb
y
D
0
everywhere under the ionosphere including at
z
D
d. This means that the magnetic
field due to Alfvén mode cannot penetrate through the conducting ionosphere to
generate the magnetic perturbations in the atmosphere on the ground surface. So, in
this model only the TE mode which includes the components ıb
x
, ıb
z
, and ıe
y
can
contribute to the magnetic variation in the atmosphere.
As is seen from these two expressions for
M
.
w
/
, the propagation of the ULF per-
turbations through the conducting ionosphere and the neutral atmosphere towards
the ground is accompanied by the wavefield damping with the exponential factor
exp .
kd/. It should be noted that in the low-frequency limit the transfer matrix is
not a function of frequency. One may suppose that these conclusions hold not only
for the cases examined above but also for an arbitrary angle and wave vector
k
.
6.4.4
Correlation Matrix of Random Fields
In this section the MHD waves propagating from the magnetosphere towards the
ionosphere and the ionospheric wind-driven currents are treated as the random
functions of coordinate and time. Let ı
B
.
r
;t/ and ı
E
.
r
;t/ be the random elec-
tromagnetic variations at the point
r
D
.x;y;
z
/ produced by the fluctuations of
the random electromagnetic fields in the ionosphere. In practice, the mean value of
the random magnetic variations is close to zero. Therefore, we will be interested
in the correlation matrix/product moment, which has the form
nm
r
;t;
r
0
;t
0
D
ǝ
ıB
n
.
r
;t/ıB
m
r
0
;t
0
Ǜ
;
‰
.
B
/
(6.79)
where the brackets
hi
denotes the averaging over all available realizations of the
random process, the symbol
denotes a complex-conjugate value and the inferior
indexes n and m are taken on the values x, y, and
z
. This correlation matrix
describes the spatial and temporal correlation of the field components ıB
n
.
r
;t/ and
ıB
m
.
r
0
;t
0
/ taken at different points
r
and
r
0
, and at different time t and t
0
.
In a similar fashion we may introduce the correlation matrix, ‰
.E/
nm
, of the electric
field fluctuations. Notice that Eq. (
6.79
) satisfies both real and complex random
fields. In a similar fashion we may introduce the correlation matrix of the forcing
function fluctuations, ‰
.m/
nm
and ‰
.
w
/
nm
.
It is clear that the spectral amplitudes of the forcing functions, ı
b
.m/
.!;
k
/
and ı
I
.
w
/
.!;
k
/, and of the magnetic, ı
b
.
k
;!;
z
/, and electric, ı
e
.
k
;!;
z
/, field
fluctuations are random functions as well. By contrast, the transfer matrices are con-
sidered to be deterministic/given functions. The spectra of random electromagnetic
fluctuations on the ground surface are related to the spectral amplitudes, ı
b
.m/
.!;
k
/
and ı
I
.
w
/
.!;
k
/, through the linear equation (
6.75
).
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