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. 1–6. 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.

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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