4. Analysis of the Influence of Periodical IGW on Propagation
of Radio Waves of the VLF Range
[18] We find the expression for the Hertz potential
P for the case
of reflection of the VLF radio waves from a
horizontally irregular ionosphere.
[19] We assume that the modulation of the impedance
h(x) is caused
by IGW propagating in the horizontal direction and having a
wavelength
lg. It is known that the presence of IGW may
lead not only to spatial, but to time variations in the medium
parameters as well with the frequency
W which is
determined from the dispersion equation
 | (15) |
where
wg = (g/H)1/2,
kg = 2p/lg,
g is the gravity acceleration, and
H is the ionospheric scale height. For the accepted in the
estimations typical values
lg = 500 km,
g = 9.8 m s
-2, and
H = 10 km, from relation (15) we find
W
7.2 
10-3 Hz. That corresponds to the
period
T = 2 p/W 14 min. Since the frequency
W is very small as compared to the frequencies of
electromagnetic waves of the VLF range, one may with a high degree
of accuracy neglect in the performed calculations the time
variations of
h while determining the amplitude and the phase
of the radio waves reflected from the lower ionosphere.
[20] In the considered case, the dielectric permeability of the lower
ionosphere region is real and may be written in the form
e (x) = e(0) + e1(x), where
e(0) = const is the regular component,
e1(x) = a cos (kg x - f0) is the disturbance
caused by IGW, and the phase
f0 is independent of
x.
[21] The perturbations of impedance can be presented in the form
h (x)=h0 + h1 (x), where
h0 = const is a
regular part and
h1 (x)= a1 cos (kgx-f0 ) is a
perturbation caused by the IGW. The phase
f0 does not depend
on
x. According to the proposed method of the problem solution,
we perform a conformal transformation
x(u) (its form is
determined by the profile of the impedance
h(x) = 1/(e (x))1/2 under fixed time
t ). Then with the accuracy
to the terms
(a2/2) 
1, we obtain
 | (16) |
where
u = x + i z and the coefficients
B1,2 = ((1 + a)1/2 
(1 - a)1/2)/2 (here
a = - a/e0 ) are such that at
z = h the relation
b(x,h) = |x'(u)| = (1 + a cos (kg x - f0))1/2 is true.
Therefore the obtained conformal transformation
x (u) corresponds to the impedance
h (x) = h0(1+a cos (kgx-f0))1/2 which under condition
a2/8 1 can be approximately presented in the form
h (x) h0 (1+(a/2) cos (kg x-f0). From
here it follows that
a1 =a/2. Finally, we find the
field of the wave reflected from the ionosphere in the form of a
product of the undisturbed solution
P0 and the multiplier
P(x)/P0(x) which takes into account the influence of the
disturbance
e1(x). Here
P0= (J0/e0 i w ) H0(2)(k0(x2 + (2h)2)1/2) is the Hertz
potential corresponding to the field of the reflected wave at the
absence of the boundary disturbance ( e1 =0 ).
Calculating
N(x,z) in formulae (10) and (13) we assume that in
the lower medium in the
0
Formula (13) corresponds to the impedance depending on only the
horizontal coordinate
x. Actually, the derived relations may be
used also in the case when the impedance modulation occurs also in
time, if the corresponding frequency of the changes is much less
than the frequency of the electromagnetic wave. To do that, one
should substitute the constant phase
f0 to
f0 = Wt where
W is the IGW frequency.
|
|
Figure 3
|
[22] On the basis of the obtained results (see (13) and (14)),
numerical calculations were performed for the relation
| P/P0 | of the module of the Hertz potentials in the
disturbed and undisturbed cases. Figure 3 shows the dependencies
of
|P/P0| on time at various values
h = 70, 80, and
90 km (curves 1, 2, and 3, respectively) for the fixed distance
x=950 km from the source to the observational point. Here the
normalized amplitude of the disturbance in the dielectric
permeability and the IGW wavelength are
a = 0.3 and
lg = 400 km, respectively. It follows from the data
analysis that the time period of the disturbance in the
observational point coincides with the IGW period. The amplitude
of the variations of
|P/P0| increases with an increase of
the vertical distance
h from the observational point to the inhomogeneous boundary.
|
|
Figure 4
|
[23] Figure 4 shows similar curves under the following values of the
parameters:
lg = 400 km and
h=90 km for various values
of the normalized amplitude of the disturbance in the dielectric
permeability
a :
a =0.1,
a =0.2, 0.3 (curves 1, 2, and 3, respectively). These values of the
a parameter correspond to changes in the value of the relative
deviation of the electron concentration
DN/ N0 under the
action of IGW within the limits
d 0.1 
0.25. The
disturbed field variations appear to be proportional to the
disturbance amplitude, whereas the values of
| P/P0| decrease with an increase of
a .
|
|
Figure 5
|
[24] Figure 5 shows the dependencies of
|P/P0| on time for
various IGW wavelengths under
a =0.3 and
h=90 km.
Curves 1, 2, and 3 correspond to
lg = 300 km,
lg = 400 km, and
lg = 800 km, respectively. The time period
of the
| P/P0| function decreases with a decrease of the
wavelength, this fact being confirmed also by formula
(15)
under the taken values of the parameters.
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