# Second Order Statistical Moments in the Collision Conductive Magnetized Plasma

## Giorgi Jandieriα, Akira Ishimaruσ, Jaromir Pistoraρ & Nino MchedlishviliѠ

## ____________________________________________

ABSTRACT

Stochastic differential equation of the phase fluctuations is derived for the collision conductive magnetized plasma in the polar ionosphere applying the complex geometrical optics approximation. Calculating second order statistical moments it was shown that the contribution of the longitudinal conductivity substantially exceeds both Pedersen and Hall’s conductivities. Phase structure function allows to calculate the angle of arrivals in the main (location of an external magnetic field) and perpendicular planes for different orientation of the observation points. Broadening of the temporal spectrum containing the drift velocity of elongated ionospheric irregularities is investigated. Numerical calculations are carried out for the experimentally observing power-law spectrum of electron density fluctuations containing anisotropy factor of elongated ionospheric irregularities using the experimental data.

Authorα: International Space Agency Georgian Society, Tbilisi, Georgia.

σ: Department of Electrical Engineering, University of Washington, Seattle, USA.

ρ: Nanotechnology Centre, VSB-Technical University of Ostrava, Ostrava-Poruba, Czech Republic.

Ѡ: Department of Informatics and System Management, Georgian Technical University, Tbilisi, Georgia.

- INTRODUCTION

Radiation of electromagnetic waves in the ionospheric plasma is a problem of great interest from both a theoretical and practical point of iew. The geomagnetic field plays a key role in the dynamics of plasma in the ionosphere and irregularities having different spatial scales usually elongated in the direction of an external magnetic field. Statistical methods have been proposed to treat radiation in randomly inhomogeneous media [1,2].

Phase structure functions and the angle-of-arrival (AOA) of scattered electromagnetic waves in the turublent magnetized plasma have been considered in [3,4] applying the stochastic eikonal equation. Investigation of the statistical moments in the turbulent conductive ionospheric plasma is of great practical importance. Collision between plasma particles leads to the absorption of scattered electromagnetic waves. Components of the conductivity tensor in homogeneous medium have been obtained in [5] account being taken both declination and inclination of the geomagnetic field. Statistical characteristics of the spatial power spectrum of scattered radiation, broadening and shift of its maximum in the conductive collision magnetized plasma were calculated analytically and numerically in [6].

In the present work the dispersion equation is derived calculating attenuation of electromagnetic waves propagating in the conductive collision homogeneous magnetized plasma. In section 2 the dispersion equation of an attenuation of oblique incident plane wave penetrating in a conductive homogeneous magnetized plasma is obtained. In section 3 complex phase and stochastic differential equation of the phase fluctuations are derived account being taken both dielectric permittivity and conductivity fluctuations of the collision magnetized turbulent plasma. The solution satisfies the boundary conditions. Second order statistical moment – phase correlation function of scattered radiation is obtained for arbitrary correlation function of electron density fluctuations. Different statistical characteristics of scattered electromagnetic waves are investigated analytically in the complex geometrical optics approximation in the conductive collision ionospheric plasma with randomly varying magnetoionic parameters. Numerical calculations are carried out in Section 4 for experimentally observing power-law correlation function of electron density fluctuations including anisotropy factor of electron density fluctuations using the experimental data. Conclusion is given in Section 5.

# FORMULATION OF THE PROBLEM

Vector of the electric field satisfies the wave equation:

, (1)

where: is the wavenumber of an incident wave having frequency ; is the Laplacian, is the Kronecker symbol, , are second rank permittivity and conductivity tensors of the turbulent conductive collision magnetized plasma which are random functions of the spatial coordinates.

The ambient external magnetic field is directed vertically upwards along the Z-axis (polar ionosphere), wave vector of a refractive plane electromagnetic wave in the absorptive random medium is located in the YOZ plane (main plane) of the Cartesian coordinate system. We suppose that Components of the second rank permittivity tensor and conductivity tensors of the magnetized plasma are [7,8]:

,, ,, (2)

where: , , , , , ,

,

,

,

, , and are magneto-ionic parameters of the ionospheric plasma,

is the electron plasma frequency, is the electron density which is a random function of the spatial coordinates, e and are the charge and mass of an electron, c is the speed of light in vacuum, is the collision frequency between plasma particles; , and are the longitudinal, transverse (Pedersen) and Hall’s conductivities, respectively, is the electron or ion collision frequency with the neutral molecules, and are the angular gyrofrequencies of an electron and ion, respectively; is the mass of ion. At high frequencies the influence of ions can be neglected.

If oblique incident plane wave penetrates into homogeneous conductive collision magnetized plasma at arbitrary refractive angle to the external magnetic field from equation (1) we obtain set of equations:

,

,

, (3)

where: , ,

, ;

is the polar angle between the projection of an incident wavevector on the XOY plane and the Y axis. Complex refractive index [9] of the collision magnetized plasma: contains the refractive coefficient of homogeneous plasma and the absorption coefficient :

, , (4)

where: ,

signs corresponds to the ordinary and extraordinary waves. Determinant set of equations (3) is:

, (5)

where: ,

,

,

,

,

.

The solution of equation (5) determines the attenuation of an incident wave propagating in the collision conductive homogeneous plasma for arbitrary angle .

# STATISTICAL MOMENTS IN THE CONDUCTIVE COLLISION MAGNETIZED PLASMA

In this Section calculate statistical characteristics of scattered electromagnetic waves assuming that the characteristic spatial scale of the elongated ionospheric irregularities exceeds the wavelength of an incident wave. This assumption enables us to utilize complex geometrical optics approximation ignoring the interaction between the normal waves account being taken that the phase fluctuations substantially exceed amplitude fluctuations. Application of this method imposes well-known restrictions on the distance traveled by the wave in the inhomogeneous medium. Wave field we introduce as [9]:

,

, (6)

here: is the phase fluctuation of a scattered wave, , here is constant value and is a random function of the spatial coordinates. Dielectric permittivity is a sum of the constant mean and fluctuating terms (, the angular brackets indicate the ensemble average). The second term contains , which can be easily reproduced from equation (1).

Substituting equation (6) into (1) fluctuating phase satisfies stochastic differential equations:

, (7)

where:

,

,

,

.

Using a double Fourier transform approach and the boundary condition , the solution of the equation (7) yields:

, (8)

here is the distance propagating by the wave in the conductive collision magnetized plasma satisfying the condition ( is the characteristic spatial scale of electron density fluctuations), coefficients: , , and are:

,

, (9)

where: , , ,, .

If waves propagate along the ambient external magnetic field () , i.e. no dumping caused due to conductivity fluctuations; at angle we obtain . Scattered electromagnetic waves dumped stronger in proportion to the angle .

Correlation function of the phase fluctuations is:

where and are components of the wavevector perpendicular to the external magnetic field, and are the distances between observation points spaced apart at a small distance in the main and perpendicular planes, respectively. The regular phase difference between two observation points are neglected. Equation (10) includes both field-aligned and transversal characteristics linear scales of anisotropic electron density irregularities. If , exponential term in (10) can be expended into a series and in this limiting case statistical characteristics of the phase fluctuations in the geometrical optics approximation are proportion to a distance L travelling by the wave in the turbulent plasma. This statement is valid beyond of its application [1,2].

In the theory of waves propagation in the turbulent ionosphere usually are interested in both amplitude and phase fluctuations, however in different type systems the registering parameter is the frequency. In general, intensity of the frequency fluctuations of scattered electromagnetic waves depends on: 1) the geometry of the task (thickness of a turbulent conductive collision magnetized plasma slab, angle between the wave vector of an incident wave and the ambient magnetic field; 2) characteristics spatial scale of elongated plasmonic structures (account being taken anisotropy factor and the inclination angle of ionospheric irregularities with respect to the external magnetic field); 3) absorption caused by the collision of electrons with other plasma particles. In this case frequency fluctuations caused due to scattering on the turbulent plasmonic structures put natural restrictions on the accuracy of measurements. Knowledge of the phase correlation function allow to calculate broadening of temporal spectrum of scattered radiation:

, (11)

here: is the distance between observation points in the plane perpendicular to the direction of wave propagation, is the angle between the vector and the drift velocity of frozen in plasmonic structures. In this case new allocated direction is appeared – the velocity of the ionospheric irregularities. From equation (11) it is possible to calculate and measure horizontal drift velocity of the plasmonic structures if other parameters are known and vice-versa.

Phase fluctuations are responsible for fluctuations of the AOA which can be measured by interferometer systems. As a part of a radar propagation effects program at the Millstone Hill radar facility [10], AOA has been measured with a single mono-pulse tracking system. Phase wave structure function allow to calculate AOAs in the main and perpendicular planes:

, (12)

where: and are nondimensional parameters.

# NUMERICAL CALCULATIONS

Incident electromagnetic wave has frequency 3 MHz. Magnetoionic plasma parameters at the altitude of 260 km are: , [11]. We will use the anisotropic power-law spectral correlation function of the electron density fluctuations with the spectral index:

, (13)

where: is the mean-square fractional deviation of electron density. This spectral function contains anisotropy factor (the ratio of longitudinal and transverse linear sizes of ionospheric plasma irregularities),

indicates field aligned wave number

, is the gamma function. Ellipsoidal shape of plasma irregularities is caused due to the diffusion processes in the terrestrial ionosphere. In the polar ionosphere the magnetic field lines is oriented almost vertically formatting elongated vertical plasmonic structures. Characteristic spatial scale of electron density irregularities ranges from hundreds of meters to ten kilometers. The geomagnetic field of the high-latitude ionosphere plays an important role in the process of plasmonic structures generation. The incident electromagnetic wave propagating in the conductive randomly inhomogeneous ionospheric plasma makes angle with an external magnetic field in the main plane.

The electron density irregularities were studied by measuring the intensity fluctuations at the frequencies 430 and 1400 MHz at the Arecibo Observatory [12]. It was found that large-scale electron density irregularities aligned with the geomagnetic field having dimensions longer than ~ 2 km along and several hundred meters across the geomagnetic field were formed in the F-region of the ionosphere. From the observed power spectra drift velocities of irregularities were observed. Since the irregularities are believed to be at a height of ~ 260 km the drift velocity of the irregularities ~ 46 m/sec (eastwards). Measurements of plasma drifts performed at Jicamarca show [13] that the zonal drift is westward and about 50 m/s during the day, and eastward and up to 130 m/s during the night.

One of the important problems of plasma turbulence in the upper ionosphere is the three–dimensional (3D) spatial spectra of the turbulence at various latitudinal regions. The 3D spatial spectrum of electron density fluctuations at various latitudinal regions plays an important role in the evolution of statistical characteristics of scattered radiation. Spectral shape of irregularities in F–region of the ionosphere could be presented as a product of two functions having various dependencies on the wavenumber parallel and perpendicular to the geomagnetic field (the spectra had various inner scales in these directions). The spatial anisotropy of turbulence spectra for the geomagnetic north–south (N–S) and E–W directions has been studied in [14]. Cross-field anisotropy, whose scale is varying from km to km, plays a significant role in the phase fluctuations, where the N–S component of phase fluctuation spectra reaches the saturation. Irregularities of ionospheric F-region are strongly stretched along the geomagnetic field.

The solution of the biquadratic equation (5) at gives the attenuation of electromagnetic waves propagating in the homogeneous conductive collision magnetized plasma . We have four roots:

and

. (14)

Attenuation of electromagnetic waves in the conductive homogeneous plasma substantially depends on the refractive angle of the penetrated wave vector and the external magnetic field. For our model the imaginary part of () varies from 0.41 up to 0.86 in the interval .

Figure 1 illustrates the normalized temporal spectrum versus distance between observation points in the XOZ plane for different anisotropy factors: (curve 1), (curve 2), (curve 3), (curve 4) for the power-law spectrum (13) at (i.e. km). The width of the temporal spectrum narrows in proportion to the parameter in the perpendicular plane.

Figure 2 depicts the broadening of the normalized temporal spectrum versus anisotropy factor at fixed distances between observation points , , and different characteristic spatial scales of ionospheric irregularities. Substituting equation (13) into (11) the normalized broadening of the temporal spectrum

is expressed via McDonald function. Increasing drift velocity of the ionospheric elongated plasmonic structures, temporal spectrum of scattered electromagnetic waves broadens. Curve 1 corresponds to (m), curve 2 is devoted to (km), curve 3 - km). Increasing characteristic spatial scale of ionospheric irregularities from m up to km, temporal spectrum broadens 1.4 times.

Figure 3 depicts the dependence of the normalized broadening of the temporal spectrum in the conductive turbulent polar ionosphere at (km), (or L = 32 km) and different refractive angles: -curve 1, - curve 2, - curve 3. Temporal spectrum broadens in proportion to the refractive angle for large-scale plasmonic structures. Starting from (isotropic case) temporal spectrum of scattered radiation increases, reaching maximums the curves smoothly decreases in proportion to the anisotropy factor of elongated ionospheric irregularities. The reason is that in the geometrical optics approximation, in non-absorbing medium (neglecting fluctuations) when both amplitude and phase are real quantities, vector of the energy-flux density and the vector are collinear and directed to the normal to the phase front, while in absorptive media (collision conductive magnetized plasma) the direction of wave propagation and the direction of fast dumping of the wave are not coincide [4]. On the other hand, according to the frozen-in hypothesis disregarding fluctuations of the drift velocity of ionospheric inhomogeneities, at transversal motion of inhomogeneities the width of the spectrum is of the order ofHz which is in agreement with [14].

Figure 4 depicts the dependence of the normalized spectrum versus angle between refractive wave and the external magnetic field at different anisotropy factors: - curve 1, - curve 2 and - curve 3. In this case (km). In the polar conductive ionosphere temporal spectrum broadens inversely proportion to the anisotropy factor for plasmonic structures having linear scale 1,6 km in the interval and then fast decreases; temporal spectrum not broadens at . So, broadening of the temporal spectrum depends on the characteristic spatial scale of ionospheric irregularities, anisotropy factor and the angle between the refractive wave and the external magnetic field.

Phase structure function allow to calculate AOAs of scattered radiation in the conductive turbulent magnetized plasma for different orientation of the observation points. Numerical calculations are carrying out for experimentally observing power-law spectral function of electron density fluctuations using the experimental data. The AOA in the main and perpendicular planes are given by:

,

. (15)

From this equation follows that the AOA in the XOZ plane exceeds the AOA in the main plane. Figure 5 and Figure 6 illustrate the dependence of the AOAs versus anisotropy factor at km). Curve 1 corresponds to the angle , curve 2 - , curve 3 - . Numerical calculations show that tends to the saturation starting at (see Table 1), while has the asymmetric Gaussian form, reaching maximums the curves fast decrease.

Particularly, at the curve 1 reaches maximum at , ; curve 2 has maximum at , ; curve 3 has maximum at , . Increasing angle maximum of the AOA in the main plane decreases and shift to the left. External magnetic field has substantial influence on the AOA in the main plane. Table 1 illustrates AOA in both planes for plasmonic structures having characteristic spatial scale 1.6 km, (or L=160 km). Estimations show that conductivity fluctuations increase AOAs of scattered radiation in the polar ionosphere.

Numerical calculations of the AOAs were carried out for small-scale ionospheric irregularities having characteristic spatial scale m () and (or L = 32 km). In this case at ,

Table 1:

at , , ; at , ; at , , . The behavior of the curves is the same than in the case of large-scale ionospheric irregularities. Study of the AOA could provide useful information about the structure of the ionospheric irregularities. Upper ionosphere places the important role in satellite communication over great distances.

# V. CONCLUSIONS

The dispersion equation is obtained describing attenuation of oblique incident radio wave on a conductive collision homogeneous magnetized plasma in the polar ionosphere. Second order statistical moments of scattered electromagnetic waves are calculated in the conductive polar ionosphere applying the complex geometrical optics approximation.

Applying the phase structure function, the AOAs are calculated numerically for experimentally observing power-law spectral function of electron density fluctuations in the main and perpendicular planes using the experimental data. Numerical calculations of the AOA versus anisotropy factor were carry out for both small and large-scale ionospheric irregularities in the conductive polar ionospheric plasma. It was shown that the AOA in the main plane has the asymmetric Gaussian form; increasing refractive angle maximum of these curves shifts to the left, while the AOA in the perpendicular plane increases in proportion to the anisotropy factor and tends to the saturation. Estimations show that conductivity fluctuations increase AOAs of scattered radiation than in magnetized plasma with permittivity fluctuations. AOAs in the main plane are less than in perpendicular plane caused due to the existence of an external magnetic field. Study of the AOA could provide useful information about the structure of the ionospheric irregularities.

Frequency fluctuations of the radio waves propagating in the conductive turbulent polar ionosphere is investigated. Correlation function of the phase fluctuations allows to calculate temporal spectrum of scattered radiation taking into account drift velocity of elongated plasmonic structures. Increasing the drift velocity temporal spectrum of scattered electromagnetic waves broadens. The broadening of the temporal spectrum depends on the characteristic spatial scale of ionospheric irregularities, anisotropy factor and the angle between the refractive wave and the external magnetic field. The width of the normalized temporal spectrum broadens in proportion to: the linear scale of ionospheric irregularities (at fixed angle ) and the angle (at fixed characteristic spatial scale of plasmonic structures). The obtained results allow to solve the reverse problem restoring distance of travelling frozen in irregularities in the polar conductive ionosphere for given drift velocity and also the angle between the observation points and the direction of irregularities motion.

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