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ISSMI'98 | ISSMI'98 |
DEPENDENCE OF ORDINARY WAVE ANOMALOUS ABSORPTION ON THE SOUNDING PROBLEM PARAMETERS A.G. Bronin (RSU, Rostov-on-Don, Russia) S.M. Grach (NIRFI, Nizhni Novgorod, Russia) N.A. Zabotin (RSU, Rostov-on-Don, Russia) |
Poster paper full text
Introduction. Transformation of electromagnetic wave into plasma waves plays an important role in different processes taking place in ionospheric plasma. In particular it causes significant attenuation of ordinary wave in the region of upper-hybrid resonance. According to modern point of view transformation of ordinary wave into plasma waves plays an important role in small-scale random irregularities generation when ionospheric plasma is heated by power radio wave [1,2].
Determination of anomalous attenuation of ordinary wave was earlier carried out by different methods. Among them the averaging of radio wave intensity transfer equation over the fast oscillations of electron density in the framework of dynamic theory [1,3], random phases approximation [2,4,5], or calculation of effective dielectric constant tensor for resonance region of plasma with random irregularities [6 9].
It is seemed that the most natural and consequent approach to calculate the value of anomalous attenuation is the one that based on determination of transformation cross-section of ordinary wave on the ground of the general method of scattering or transformation cross-section calculation for waves in plasma [10]. As far as the authors know such approach was not described in literature yet. Transformation cross-section was used in [11] where it was derived from transfer equation for energy spectral density. However, consideration in [11] was limited to the case of cold plasma (thermal motions of particles were neglected) and quasi-longitudinal propagation.
Section I of present paper contain the general equations used to determine the anomalous attenuation. The total energy of scattered field is determined from these equations in Section II. General expressions for transformation cross-section and anomalous attenuation, which are valid for the random media with spatial and frequency dispersion, are obtained in Section III. These expressions are simplified for the case of cold collisionless plasma in Section IV. The results of numeric calculations of anomalous attenuation are discussed in Section V.
1.Initial equations. Let us assume the relative deviation of electron density to be small:
(1)
where is the deviation of electron density from the
average value
,
where
. The electric field
of the wave in the plasma is described by the wave equation:
(2)
where the nucleus of dielectric permeability integral operator is of the following structure:
(3)
unit diagonal tensor,
,
dielectric constant tensor of homogeneous plasma.
By the averaging of equation (2) over the ensemble of realizations of random irregularities and consequent subtraction of averaged equation from the initial one can get the equations for the mean field and scattered field of the wave:
(4)
(5)
where
(6)
The changes in spatial and angular distribution of irregular component of the wave field due to multiple scattering do not affect the rate of energy of incident wave scattered into irregular component. Therefore we may use single-scattering approximation and simplify equation (5):
(7)
Assuming that the incident wave is plane wave and applying Fourier
transform to (4) - (7), we obtain:
(8)
where
,
,
dispersion
tensor,
matrix of algebraic adjuncts of dispersion tensor.
2. Energy of scattered field. One may interpret equation (7) as an equation for
electric field ,
created by the current with density
. The total energy of the field in this
case is
(9)
By substitution of expression (8) for spectral
density of current and electric field into (9) and averaging the
quantity over
the ensemble of realizations of random irregularities, we get:
(11)
where is unit polarization vector of incident wave,
spatial and frequency spectrum of
random irregularities, V Х T the volume of plasma and the
time of its interaction with wave filed. Assuming that the
infinitely small attenuation is present we can use the limiting
transition [10]:
and finally obtain the following expression for the mean total energy of the scattered:
(12)
3. Anomalous attenuation. Total scattering (transformation) cross-section is
defined as ratio of mean power, scattered from unit volume to the
modulus of energy flux of incident wave total [10,11]:
(13)
where is refractive index of incident wave,
is the angle between
the group velocity vector and the wave vector of incident wave.
Using the expression for the energy of scattered field (12),
obtained in previous section we get the following expression for
cross-section
(14)
Attenuation of incident wave cause by transformation into plasma waves may be found by integration of total transformation cross-section along the ray path [12]:
Using the relation , where h is vertical coordinate,
we obtain the following expression:
(15)
Expression (15) is rather general as it
describes the attenuation of men field both due to scattering in
the same mode and transformation into different mode. The
concrete mechanism of attenuation is selected by the proper
choose of the root of dispersion equation when removing the
integration with the help of delta-function.
To make the further analysis possible we must
use the concrete model of irregularities spectrum . It is well-known
that both natural and artificial ionospheric irregularities are
strongly stretched along the direction of lines of force of
geomagnetic field and the ratio of spectrum characteristic scales
in collinear and orthogonal to geomagnetic field direction
exceeds the order of magnitude. Therefore we may use
approximation of infinitely stretched irregularities and
represent their spectrum in the following form:
(16)
where orthogonal and collinear in relation to
direction of geomagnetic filed vector components of vector
. It is also taken
into account in (16) that characteristic time of changes in
ionospheric irregular structure usually much greater then the
period of the radio wave and so we may consider the
irregularities to be stationary. Substitution of (16) into
expression (15) gives:
(17)
where ,
is
plasma frequency, value
is the root of equation
corresponding to plasma waves. Quantity
is determined from
the analysis of dispersion curves for plasma wave in the resonant
region. The wave vector of plasma wave takes its smallest value
near the reflection point of ordinary wave, so the value of
may be found from
equation
.
4. Anomalous attenuation in the cold plasma approximation. Expression (17) may significantly simplified when the particle thermal motions are neglected e.g. in the approximation of cold collisionless plasma. In this case the following representation is valid
(18)
where are refractive indexes of ordinary and extraordinary
waves,
,
angle between
the wave vector of scattered wave and geomagnetic field vector,
is electron
cyclotron frequency. The last term in (18) correspond to
generation of plasma waves transformation takes place near
the upper-hybrid resonance where condition
is hold. This
condition gives us
.
In the region where plasma is transparency for electromagnetic
waves the relation
, takes place, where the unit polarization vector is
parallel to wave vector of plasma waves
, so we can obtain the following
expression for anomalous attenuation in cold plasma:
(19)
It must be outlined that when the particle
thermal motions are neglected propagation of plasma waves
strictly speaking is not possible and condition corresponds to
generation of electrostatic plasma oscillations. Hence we must
approximately take into account corrections on thermal motions,
mainly in determination of
. The analysis of exact dispersion equation for
plasma waves with corrections on thermal motions gives the
following estimation
. It means that only the irregularities with
characteristic scales less then the length of wave of incident
radiation contributes into anomalous attenuation.
Let us estimate the value of different terms
under the integral sign in equation (19). In the resonant region , and because
ordinary wave is transversal
. Using the approximate expression
, where
is the group velocity
of ordinary wave and taking into account the double pass of the
way we get the expression
(20)
Expression for anomalous attenuation (20) is equivalent to the expression obtained earlier in paper [2].
4. Results of numeric calculations. Determination
of anomalous attenuation magnitude from the exact formulae
(17) or (19) may be carried out only with the help of numeric
methods. In numeric calculation we may include corrections on
thermal motions of particles using the general expression (17).
It is more convenient to use integration over rather then
(because the interval
of integration is finite in this case). Omitting complicated
intermediate calculations we present here only the final result:
, (21)
where is dielectric constant tensor with account of
thermal corrections,
- dielectric constant tensor of cold collisionless
plasma, components of
are given in [10],
is the angle between the wave vector and
geomagnetic vector,
satisfies the equation
,
,
,
Х
- wave vector and
unit polarization vector of ordinary and plasma waves
respectively,
refractive index of ordinary wave,
refractive index of plasma wave, where
is the thermal velocity of electrons, other variables were
determined in the previous section.
It is interesting to compare results of numeric calculations of exact expression (21) with the estimations of anomalous attenuation calculated with the help of approximate expression (20), which was used in some earlier publications. Not only the magnitude of anomalous attenuation may be the object of comparison but also its dependence upon different parameters of ionospheric plasma and spectrum of irregularities.
The model of ionospheric layer with linear
profile of electron density was used for calculations the
following characteristics of the layer were chosen: thickness
100 km, cyclotron frequency 1.35 MHz. The spectrum
of irregularities was normalized to the scale 50 m, what is
corresponding to parameters of artificial ionospheric small-scale
irregularity spectra. The frequency of sounding signal was 6 MHz.
For parameter
the value of
was
chosen (corresponds to F region of ionosphere) [13]. Multiple
integrals were calculated using
nodes of Korobov quadrature formula [14].
It is well known that the main contribution into transformation
comes from the narrow region near the point v = 1-u (upper hybrid
resonance region). Unlike the case of cold plasma, in general
case plasma waves may propagate below the resonance region, but
because their refractive index in this area is large the
contribution into transformation coming from this region is
small. The range of heights giving the main contribution into
transformation was determined directly in calculations. Numeric
simulation shows that it is possible to restrict the height
interval to
.
This region was split into three parts:
;
and
,
calculations of contribution into anomalous attenuation were
performed separately for each part.
The results of calculations are presented at
Figs. 16. At Fig.1 the dependency of anomalous attenuation
magnitude on the index of power-type spectrum of irregularities,
calculated using approximate expression (20) for the different
values of angle between the geomagnetic field vector and vertical
direction, is shown. It is clearly seen that there is no
dependency upon the latitude of observation point in this case.
The same calculations using the exact expression (Fig.2) display
the expressed latitude dependency with significant decrease of
anomalous attenuation at low latitudes. At Fig. 3 the ratio of
anomalous attenuation values calculated with expressions (21) and
(20) as a function of index of power-type spectrum for different
latitudes is shown. The same ratio as a function of frequency
calculated for different values of the index
and latitude of point
N. Novgorod is shown at Fig.4. Figs.5 and 6 presents the
magnitude of anomalous attenuation as a function of index
and structural
function
in the
given scale (50 m) (this function is presented in the form of
surface at Fig.5 and at Fig.6 it is presented as a contour map of
lines of equal attenuation). It is easy to see that the magnitude
of anomalous attenuation reaches its maximal value when the index
of power-type spectrum of irregularity is approximately equal to
2.5.
Conclusion. The approach applied in this paper allows one to determine clearly the essence of approximations which are usually used in calculations of anomalous attenuation caused by transformation of ordinary wave into plasma wave on random irregularities in the resonant region of ionospheric plasma. Among this approximations are the assumption of statistic homogeneity of the media, single-scattering approximation for scattered field energy calculations, stationarity of ionospheric irregularities, and application of infinitely stretched irregularities spectrum model as well as the approximation of cold collisionless plasma with partial correction on thermal motions of particles. All these approximations are used in derivation of formula (20). Besides this the more general expressions (15) and (17) with relatively wide area of application are obtained in the paper. They allow one, in principle, to study the anomalous attenuation properties with strict account of spatial dispersion (i.e. thermal motions of particles of plasma) and expression (15) gives the possibility to include in consideration the non-stationarity of ionospheric irregularities. Numeric calculations with the help of formula (21) derived from exact expression (17) provides additional information (such as expressed latitude dependence of anomalous attenuation) which can not be obtained with the use of (20) due to its approximate character. The results of this paper may be generalized in natural way to the case of random media with smoothly varying mean characteristics and weak collision absorption.
REFERENCES.
Fig.1
Anomalous attenuation calculated with the use of simplified formula.
Fig.2
Results of anomalous attenuation numeric calculations when the thermal motions of electrons are included
Fig.3
Ratio of anomalous attenuation values calculated with the help of exact formula (21) and approximate formula (20).
Fig.4
Frequency dependence of Ratio of anomalous attenuation values calculated with the help of exact formula (21) and approximate formula (20) at the latitude of point N.Novgorod.
Fig.5
Dependence of anomalous attenuation upon structural function and the index of power-type spectrum of irregularities
Fig.6
Lines of equal attenuation as a function of tructural function and the index of power-type spectrum of irregularities