Statistical Moments and Scintillation Level of Scattered Electromagnetic Waves in the Magnetized Plasma

Statistical characteristics of scattered ordinary and extraordinary electromagnetic waves in the magnetized plasma are considered using the smooth perturbation method. Diffraction effects and polarization coefficients are taken into account. Second order statistical moments of scattered radiation are obtained for arbitrary correlation function of electron density fluctuations. The expressions of the broadening of the spatial power spectrum and displacement of its maximum are obtained. Wave structure functions and the angle of arrivals are calculated. Scintillation level of scattered radiation is analyzed for different parameters characterizing anisotropic plasma irregularities for the ionospheric F-region. Numerical calculations of the statistical characteristics are carried out for the three-dimensional spectral function containing anisotropic Gaussian and power-law spectral functions using the experimental data.


Introduction
At the present time peculiarities of electromagnetic waves propagation in randomly inhomogeneous media have been rather well studied [1,2].The analysis of the statistical properties of small-amplitude electromagnetic waves that have passed through a plane turbulent plasma slab is very important in many practical applications associated with both natural and laboratory plasmas Many excellent reviews and books of scintillation theory and observations in the ionosphere have been published [3][4][5] whereas statistical characteristics of scattered radiation in the turbulent magnetized plasma are less studied.In most of the papers isotropic irregularities have been considered however the presence of geomagnetic field leads to the birefringence and anisotropy.
The fluctuations in amplitude and phase (scintillation) of radio waves propagating through the ionosphere are caused by plasma irregularities in the electron density.The irregularities have different spatial scales and usually are elongated in the magnetic field direction.Ionospheric scintillation models contain the worldwide climatology of the ionospheric plasma density irregularities that cause scintillation, coupled to a model for the effects of these irregularities on radio signals.These irregularities distort the original wave front, giving rise to a randomly phasemodulated wave.A high priority given to the ionospheric scintillation study comes from its significant impact on satellite radio communications.For instance, the signal distortion caused by scintillation can degrade the performance of navigation system and generate errors in received messages.
Peculiarities of the second order statistical moments of scattered radiation in collision magnetized plasma using the ray (-optics) approximation have been investigated in [6].Statistical characteristics and scintillation level of scattered ordinary and extraordinary waves in the collision magnetized plasma normal to the external magnetic field have been considered in [7][8][9][10] using modify smooth perturbation method.Analytical and numerical calculations were carried out for the anisotropic Gaussian and power-law spectral functions taking into account diffraction effects.Scintillation index was calculated for small-scale irregularities using the "frozen-in" assumption and taking into account movement of rigid irregularities.Minimums of the power spectrum of the intensity fluctuations of scattered ordinary and extraordinary waves satisfy the "standard relationship".It was shown that the normalized scintillation level growth in both non-fully-developed diffraction pattern and in transition zone increasing anisotropy factor.Rising orientation angle scintillation level decreases and splashes arises in fully developed scintillation region.
In section 2 of this paper, stochastic differential equation of the phase fluctuation has been obtained in the principle plane containing wave vector of an incident wave and the external magnetic field.Polarization coefficients and diffraction effects are taken into account.In section 3, second order statistical moments of scattered radiation for arbitrary correlation function of electron density fluctuations.Numerical calculations are carried out for new spectral function combining anisotropic Gaussian and power-law spectral functions in section 4 using experimental data applied to the F-region of ionosphere.Normalized scintillation level is calculated for different anisotropy factor and orientation angle of elongated plasma irregularities with respect to the external magnetic field.Conclusion is presented in section 5.

Formulation of the problem
Let a plane electromagnetic wave with frequency w be incident from vacuum on a semi-infinite slab of turbulent collision magnetized plasma.We choose a Cartesian coordinate system such that XY plane is the vacuum-plasma boundary, Z axis is directed in the plasma slab coinciding with the wave vector k of the refracted wave, YZ plane (principle plane) is generated by the external magnetic field vector 0 H . Components of the second rank permittivity tensor of the collisionless magnetized plasma are [11]: is the angular gyrofrequency for the magnetic field, () N r is the electron density, c is the speed of light in vacuum, e and m are the charge and the mass of an electron, respectively; weakly vary on a distances of wavelength.We also will consider distances of waves propagation satisfying the condition 2 /1 L kl << ( l is the characteristic spatial scale of irregularities).In particular case of plane-layered medium, when components of the permittivity tensor ij e are depend on one coordinate, the obtained results are valid for arbitrary distances.
Assuming ik e to be time independent, electric field E in the turbulent magnetized plasma satisfies the differential equation: where: D is the Laplacian, ij In the first approximation fluctuation of the phase satisfies differential equation: where: (1)  11 () upper sign and index 1 j = correspond to the extraordinary wave, lower sign and index 2 j = to the ordinary wave.These waves in magnetized plasma generally are elliptically polarized.Geomagnetic field leads to the birefringence and anisotropy.
Fluctuation of the phase of scattered electromagnetic wave caused by electron density fluctuations satisfies the boundary condition 1 (, , 0 )0 Applying the modify perturbation method [12,14] the solution of equation ( 5) is: where: L is a propagation distance by electromagnetic waves in the ionospheric plasma.For simplicity index j will be withdrawn in the polarization coefficients.

Second order statistical moments of scattered radiation
Knowledge of the spectral function of the phase fluctuation allows to calculate second order statistical moments of scattered electromagnetic waves.Variance of the phase fluctuations is: where: arbitrary correlation function of electron density fluctuations, angular brackets means ensemble average.Correlation function between two observation points spaced apart at small y r and x r distances in the principle and perpendicular planes, respectively, can be written in the form: where: The phase wave structure function and the angles of arrival of scattered electromagnetic waves in the principle and perpendicular planes can be easily calculated using equations ( 8) and ( 9) [1,15]: ) r r rr rr , 2 1 2 0 ( , 0 ,) lim Correlation function of the complex field in the collisionless magnetized plasma could be written as [16,17] In the most interesting case of multiple scattering, when the phase fluctuations are strong 11 1 jj * < > >> , we can assume that they obey a normal distribution [1,2].Correlation function decreases sharply as x r and y r increase, the argument of the second exponential term can be expanded in a series [16,17]: where the phase correlation function V j i s g i v e n b y equation (9).The derivatives of the phase correlation function are taken at the point 0 xy rr == .
The 2D spatial power spectrum (SPS) of scattered radiation which is of great practical importance can be obtained by Fourier transformation from the correlation function (12) [1,2]: This characteristic is equivalent to the ray intensity (brightness), which usually enters the radiation transport equation [1,2].In the most interesting case of strong fluctuation of the phase k <> taking into account the fact that the observation points are spaced at small distance apart in the YZ plane near the point 0 y = .As a result, we have obtained: The derivatives of the correlation function of the complex phase fluctuations are taken at the point 0.  f kL pl = is the Fresnel wavenumber, l is the wavelength of an incident wave, L is a mean distance between the observer and plasma irregularities, is the Fresnel radius.The sinusoidal term is responsible for oscillations in the scintillation spectrum.The spatial autocorrelation function of the diffraction pattern could be measured with a suitable twodimensional array of sensors.
Satellite and/or the F-region plasma irregularities moving relative to the receiver, temporal variations of intensity and phase are recorded.We assume that irregularities drift across the beam of the radio signals without changing their shapes (the assumption of "frozen-in" irregularities) along the Xaxis with apparent velocity x V transverse to the line of the sight path.The power spectrum (,)

Numerical calculations
The incident electromagnetic wave having frequency of 3 M H z ( Irregularities of a range of scale sizes starting from a few hundred meters to a few ten of kilometers are observed in these patches.Data obtained from spaced receiver measurements made at Kingston, Jamaica (during the periods August 1967-January 1969 and June 1970-September 1970) show that the irregularities between heights of 153 and 617 km causing the scintillation of signals from the moving earth satellites (BE-B and BE-C) are closely aligned along the magnetic field lines in the F-region [21].The dip angle of the irregularities with respect to the field lines was within 0 16 .The anisotropic spectral features in the F-region are defined for Gaussian and Power-law spectra.
Observations (Tbilisi, 0 41 43N) of drift small-scale irregularities in the ionospheric F-region show [22] that they have elliptic form, the ratio of axes basically varies from 1 t o 3 .A n i s o t r o p y a x i s i s m a i n l y o r i e n t e d a l o n g t h e geomagnetic field of lines.Drift of small-scale irregularities mainly has S-W direction.The most probable values of drift velocity is in the range of 40-100 meter/sec.
Measurements of satellite's signal parameters moving in t he i o no s p he re s ho w t h a t i n F-region of the ionosphere irregularities have power-law spectrum.3D power-law spectral correlation function of electron density irregularities with a power-law index p has been proposed in [23,24].The corresponding spectral function has the form: ( ) where: << spatial spectrum could be written as [23,24]: is the gamma function [25].We will use new spectrum of electron density irregularities combining anisotropic Gaussian and powerlaw spectra [10]: = is the anisotropy factor -the ratio of longitudinal and transverse characteristic linear sizes of plasma irregularities, 0 g is the orientation angle of elongated ionospheric plasma irregularities with respect to the magnetic lines of force.The shape of electron density irregularities has a spheroidal form.Anisotropy of the shape of irregularities is connected with the difference of the diffusion coefficients in the field align and field perpendicular directions.
Experimental investigations of Doppler frequency shift of ionospheric signal and measurement by translucence of satellite signals show that index of the power-law spectrum of electron density fluctuations is in the range of 3.8 4.6 p ££ ( 4) p < >» [26].Experimental observations of backscattering signals from the artificially disturbed region of the ionosphere by the powerful HF radio emission shows that a lot of artificial ionospheric irregularities of the electron density are stretched along the geomagnetic field.Power-law spectral index was within the limits 1.4 4.8 p =f or different heating sessions using "Sura" heating facility in the frequency range of 4 .79 ¸MHz (ordinary mode) with the effective radiated power 50 70 ¸MW beamed vertically upwards [27].
Scintillation spectra are in agreement with 3D power-law irregularity spectrum with an exponent around -4 [19].This e x p o n e n t o f -4 i s i n a g r e e m e n t w i t h t h e i n s i t u measurements of the one-dimensional irregularity spectrum derived from rockets and satellites.

Conclusions
Statistical characteristics of scattered electromagnetic waves in the turbulent magnetized ionospheric plasma are calculated solving stochastic differential equation for the phase fluctuations taking into account boundary condition, diffraction effects and polarization coefficients for both ordinary and extraordinary waves.Variance and correlation function were obtained for arbitrary correlation function of electron density fluctuations.These second order statistical moments allows to estimate the broadening and shift of maximum of the SPS of scattered radiation, and also investigate scintillation effects in the F region of the ionosphere using the experimental data.Numerical calculations are carried out for 3 MHz incident wave and 3D anisotropic spectral function of electron density fluctuations characterizing anisotropic plasma irregularities containing both anisotropy factor and orientation angle of elongated plasma irregularities with respect to the external magnetic field.
It is shown that displacement and the width of the SPS for the ordinary and extraordinary waves tends to the saturation increasing anisotropy factor.Shift of maximum of the SPS strongly depends on the orientation angle of anisotropic plasma irregularities, particularly, varying angle in the interval 00 0 30 displacement of its maximum increases six times.The angle-of-arrivals in the principle plane less than in normal direction.Phase scintillation index for small-scale irregularities fast growth in proportion to the orientation angle and reaching maximum slowly decreases inversely proportion to the characteristic linear scale of plasma irregularities.Small level scintillations is associated with both positive and negative intensity fluctuations, while the large levels primarily corresponds to the positive intensity fluctuations.
Sinusoidal type oscillations are observed in the intensity spectrum and are attributed to a Fresnel filtering effect for plasma irregularities having characteristic spatial scale less than the Fresnel radius.These oscillations satisfy the "standard relationship".Scintillation level allows to calculate the spectral width (first 1S

, a is the angle between k and 0 H
vectors.In reality components of the tensor ij e are parameters.The double integral in the wave number space depends only on the shape of the fluctuation spectrum but not on the strength of the fluctuations.Phase fluctuations at different observation points are not independent and they correlate, the asterisk represents the complex conjugate.
which expressions(14) and(15) correctly describe the SPS of scattered radiation is determined by the following inequalities.These conditions are not in contravention to the assumption of strong phase fluctuations because in a smoothly inhomogeneous medium mean spatial scale of plasma irregularities of the phase correlation function substantially exceeds the wavelength of scattered electromagnetic waves, l l >> ( l is characteristic spatial scale of plasma irregularities caused by electron density fluctuations)[2] and the angle of the normal to the will consider only electron density fluctuations in F region of the ionosphere.Ionospheric phase scintillation fluctuations are characterized by the scintillation index.For weak scattering of electromagnetic waves the scintillation level 4 S and the 2D phase spectral function describing 2D diffraction pattern at the ground are connected by the relationship[18]:

1 m 5 N
-) propagates along the Z-axis.Plasma parameters at the altitude of 300 km are: 0 Fresnel radius and the Fresnel wavenumber are equal to 5.5 km and 0.64 1 km -, respectively.An RH-560 rocket flight observations from Sriharikota rocket range (SHAR), India ( 0 ; apogee was 348 km) show[20] that the intermediate range irregularities (100 m -2 km) were observed in abundance in altitude regions 220-250 km and 290-320 km.

for 3
equation (3) has two complex conjugate and two real roots having opposite signs.MHz incident wave diffraction parameter is in the interval 0 0.08 ¸.Figure1depicts the 3D normalized correlation function of the phase fluctuations when two observation points are located in mutually perpendicular planes at distances x h and

Figure 2
illustrates cross-section of 3D phase correlation function(22) of field aligned plasma irregularities in the XZ plane for both the ordinary and extraordinary waves.The widths of the curves approximately are the same in the principle YZ plane as the external magnetic field has similar influence on both waves.In the XZ plane behavior of these waves strongly differ.Broadening of the 2D correlation function for the ordinary wave 2.6 time exceeds the extraordinary one.Minimums for the ordinary and extraordinary wave are at 160 x k D of the SPS in non-absorbing medium (1) is equal zero because the dependence on x k is even.Plots of the parameter y k D as a function of anisotropy factor c for

Figure 1 :
Figure 1: Normalized 3D correlation function of the phase fluctuations at .

Figure 2 :
Figure 2: Normalized correlation function versus for the extraordinary (curve 1) and the ordinary (curve 2) electromagnetic waves.

F i g u r e 6 :
Scintillation index versus anisotropy factorfor different characteristic spatial scale of plasma irregularities.

Figure 3 :
Figure 3: Plots of the displacement of the SPS as a function of anisotropy factor .

Figure 4 :
Figure 4: The width of the SPS in the XZ plane versus parameter for the extraordinary wave.

Figure 5 :
Figure 5: Plots of the phase structure function versus distance between observation points in the XZ plane.

Figure 10 Figure 7 :
Figure 7: Scintillation index versus anisotropy factor for different orientation angle of anisotropic plasma inhomogeneities.

Figure 9 :
Figure 9: The power spectrum of the phase fluctuations versus parameter for the ordinary and extraordinary waves.

n
and second 2 S n moments) computing the power spectrum and scintillation period.If "frozen-in" elongated irregularities drift along the X-axis with the velocity 100 m/sec, spectral width 1 ~10 S n mHz, period is 100 sec and 12 104 SS nn <= mHz.If elongated plasma irregularities are moving along the Y-axis, 1 ~80 S T sec and 12 118 ss nn <= mHz.Knowledge data of these oscillations allows to calculate the velocity of plasma irregularities in the principal and perpendicular planes, to estimate characteristic spatial scales and the r.m.s.electron density fluctuations for plasma irregularities smaller than the Fresnel radius.

Figure 10 :
Figure 10: The spectral width (1 st moment) of the power spectrum for different orientation angle of elongated plasma irregularities.