Second Order Statistical Moments of the Power Spectrum of Ionospheric Scintillation

Paper considers some aspects of the relationship between scintillation level of scattered radiation and anisotropic plasma irregularities in the collision turbulent magnetized plasma. Analytical calculations are carried out using modify smooth perturbation method taking into account both diffraction effects and polarization coefficients of both ordinary and extraordinary waves. Correlation function and the wave structure function of the phase fluctuations are obtained for arbitrary correlation function of electron density fluctuations. Analytical and numerical calculations of the second order statistical moments are carried out for the anisotropic Gaussian correlation function taking into account both anisotropy factor and the inclination angle of the elongated plasma irregularities with respect to the external magnetic field. Scintillation level and the power spectrum of the intensity fluctuations are analyzed for different parameters characterizing anisotropic irregularities.


Introduction
Many excellent reviews and books of scintillation theory and observations [1,2] have been published. A high priority given to the ionospheric scintillation comes from its significant impact on satellite radio communications. Scintillation of satellite radio signals is a consequence of the existence of random electron density fluctuations within the ionosphere.
The irregularities have different spatial sizes and usually are elongated in the magnetic field direction. The connection between phase and amplitude fluctuations and the irregularities causing them has been studied in [3]. Peculiarities of the second order statistical moments of scattered radiation in collision magnetized plasma using the ray (-optics) approximation have been investigated in [4]. Scintillation spectra of scattered ordinary and extraordinary waves in the turbulent collisionless magnetized plasma have been considered in [5]. Analytical and numerical calculations were carried out for the anisotropic Gaussian spectrum taking into account diffraction effects. Statistical moments of the spatial power spectrum of multiple scattered ordinary and extraordinary waves in the turbulent collision magnetized plasma with aligned anisotropic electron density irregularities were investigated in [6,7] using the modify smooth perturbation method.
In this paper we consider some aspects of the relationship between scintillation and ionospheric irregularities. Analytical and numerical calculations are carried out for the anisotropic Gaussian spectral function containing important experimentally measured parameters: anisotropy factor and slope angle of elongated plasma irregularities with respect to the lines of force of geomagnetic field. Scintillation level is estimated for different anisotropy factor and the angle of inclination of elongated irregularities with respect to the external magnetic field. Numerical calculations are based on the experimental data.

Statistical Moments of Scattered Radiation in the Ionospheric Plasma
Electric field in the turbulent collision plasma satisfies the wave equation: (1) with the components of the dielectric permittivity [8]: where: where: , upper sign (index 1) corresponds to the ordinary wave; the lower sign (index 2) to the extraordinary wave.
Fluctuation of the phase of scattered electromagnetic wave caused by electron density fluctuations satisfying the boundary condition 1 ( , , 0) x y k k z ϕ = = 0 = applying the modify smooth perturbation method [6,7,9] normal to the direction of the principle plane has the following form: (4) where , , , .
Second order statistical moment of the phase fluctuations is: where: V k is the arbitrary 3D spatial spectral function of elec-tron density fluctuations, the asterisk indicates the complex conjugate, L is the mean distance between the observer and the plasma irregularities; x ρ and y ρ are the distances between observation points spaced apart in the principle and perpendicular planes, respectively. The wave structure function of the phase fluctuations is [3]: Usually in scintillation studies, the fluctuations are characterized by the scintillation index whose properties can be related to those of the ionospheric irregularities by the application of the diffraction theory. For weak scattering the scintillation level 4 S is expressed via second order moments of the phase fluctuations [10]: where: is the Fresnel wavenumber. The sinusoidal term is responsible for oscillations in the scintillation spectrum.
If the satellite and/or the ionosphere moves relative to the receiver, temporal variations of both intensity and phase are recorded. We shall use the assumption of "frozen-in" irregularities. If rigid plasma irregularities are assumed to be moving in the X direction with apparent velocity x V transverse to the line of the sight path, the power spectrum The Fresnel frequency is the Fresnel radius, the sinusoidal term is responsible for oscillations in the scintillation spectrum.

Numerical Calculations
Scintillation level and the power spectrum we investigate for two frequencies of an incident electromagnetic waves 3 MHz ). An RH-560 rocket flight was conducted from Sriharikota rocket range (SHAR) to study electron density and electric field irregularities during spread F [11]. Large scale structures, with vertical scale sizes up to a few tens of km, are seen at the altitude from 150 km to 257 km. Small-scale structures having scale sizes of the order of few hundred meters are also seen, superimposed on large-scale structure in the entire region. A patch of very intense irregularities can be seen in 210-257 km region. Irregularities of a range of scale sizes starting from a few hundred meters to a few tens of kilometers are observed in this patch. Spaces receiver measurements made at Kingston, Jamaica, show [12] that irregularities causing the scintillation of signals from earth satellites are closely aligned along the magnetic field lines in the F-region, and the dip angle of the irregularities in the ionosphere is within 0 16 . Numerical calculations are carried out for the irregularity model describing by the anisotropic Gaussian 3D correlation function of the electron density. Spectra of electron density fluctuations, scintillation measurements provide information about other irregularity parameters used in scintillation modeling: axial ratio along and across the magnetic field, orientation of plasma irregularities, slope of the in-situ spatial spectrum of the irregularities and the in-situ drift velocity of irregularities.
The anisotropic 3D Gaussian spectral function having certain mathematical advantages has the following form [ where: is the mean-square fractional deviation of electron density. This spectral function contains parameter (the ratio of longitudinal and transverse linear sizes of plasma irregularities) measuring the anisotropy of irregularities and the inclination angle of elongated irregularities with respect to the external magnetic field. Anisotropy of the shape of irregularities is connected with the difference of the diffusion coefficients in the field where: (3) and (4) for a given altitude of the ionosphere.  is associated with a significant filtering, non-fully developed diffraction pattern, whereas 1 ϒ >> is associated with a fully developed pattern. Shaded area corresponds to the transition region between these two regions. and becomes stationary at 800 ϒ = .     . Numerical calculations show that first minimum for the ordinary wave is at: 1

Conclusion
Statistical characteristics of scattered electromagnetic wave have been obtained for arbitrary correlation function of electron density fluctuations and the angle between the wave vector of an incident wave and the direction of line of force of an external magnetic field. Analytical calculations were carried out for the anisotropic Gaussian correlation function of electron density fluctuations containing the anisotropy factor of elongated plasma irregularities and the angle of inclination of the irregularities to the external magnetic field. Numerical analyses were carried out for 3 MHz incident wave for the high latitude ionosphere. Analyses of the normalized scintillation level of scattered radiation show that stationary oscillations are generated in the collision magnetized plasma (at the collision frequency 0.01 s ≈ ) with small-scale plasma irregularities having characteristic spatial scale 80 m. Scintillation spectrum reaches maximums and minimums with periodicity 3 degree substantially depending on the frequency of an incident wave. Scintillation level increases in proportion to the inclination angle of elongated irregularities with respect to the line of forces of an external magnetic field. At the frequency 40 MHz the curves of the normalized scintillation level tend to the saturation at big anisotropy factors in proportion of the slope angle, while at the frequency of an incident wave 3 MHz scintillation level less sensitive to the geometric parameters of the ionospheric plasma irregularities. Minimums of the power spectrum of the intensity fluctuations for both ordinary and extraordinary waves satisfy the condition: 1 : 2 : 3 : 4 and so on. Inflection points approximately are the same for both waves.