Statistical Characteristics of New Type Internal Wave in the Ionospheric F Region

Second order statistical moments of new internal MHD wave in the ionospheric F region are investigated analytically by geometrical optics approximation. Degree of a curvature of a constant phase surface and the variance of an instant frequency measuring by experiment has been obtained for arbitrary correlation function of electron density fluctuations. Energy exchange between the internal wave and turbulent plasma flow is considered calculating the mean energy flux density in the first and second approximations. Numerical calculations are carrying out for both anisotropic Gaussian correlation function and power law-spectrum characterizing elongated plasma irregularities using experimental data of satellites and incoherent scatter radar observations.


Introduction
At present time it was established that statistical characteristics of wave depend on as the correlation features of the variable parameters of a chaotically random medium, as well as on the dispersion and the type of waves. The features of transverse electromagnetic waves and low-frequency waves in different turbulent media were investigated quite well [1][2][3]. However fluctuation features of the longitudinal waves and different MHD waves are not investigated in detail. Propagation of MHD waves in the turbulent plasma flow is of great interest in both cosmic and laboratory plasma.
The Earth magnetic field in the ionosphere generates small and medium-scale MHD waves: Alfven and magneto-acoustic waves. The last one are fast (propagation velocity more that 1 km/s) and short-period (of the order of 5-20 min) waves generating by elasticity of the geomagnetic lines of force. Alfven waves are generated due to tension of the geomagnetic lines of force and its velocity depends on the orientation of the wave vector with respect to the geomagnetic field 0 H . They can be very slow (10-50 m/s) and long period (1-2 days) when the wave vector k is almost transversal to 0 H , and fast, when vectors k and 0 H are parallel. Some peculiarities of the second order statistical moments of scattered Alfven and magneto-acoustic waves in the turbulent plasma flow were considered in [4][5][6].
We suppose that the external magnetic field In the case of low-pressure plasma neglecting thermal velocities and ions motion under the action of the wave field, if electron density fluctuations are smooth and plasma nonstationarity is caused by turbulent motion, applying geometrical optics method [7] the following conditions are fulfilled: where l is characteristic linear scale of electron density irregularities. In this case we will neglect transformation of different waves and consider one of the most interesting case when collision frequency between plasma particles is small with respect to the frequency of an incident wave.
In the absence of temperature stratification, dissipation and compression only internal waves are generated in the ionospheric F-region under the action of geomagnetic field [8,9]. New Alfven-type short-wave 3 ( 10 λ ≤ km) and low-frequency inertial wave generating due to the Ampere electromagnetic force has been discovered in the ionospheric F region [10,11].
In this paper statistical characteristics of scattered new inertial MHD wave in the ionospheric F-region are considered for the first time. Several specific features are arising at MHD wave propagation in the ionospheric plasma with spatial-temporal chaotically varying parameters, particularly energy exchange between the wave and nonstationary medium.

Formulation of the Problem
In the F region where the Hall effect is absent and 0 H = const, the linearized system of Helmholtz-Freedman equations for internal waves has the following form [10,11]: The condition 0 div = V means that considered below perturbations are transverse waves; V is perturbation of the neutral particles velocity, u is vector potential having velocity dimension. Characteristic frequencies of the internal waves caused by geomagnetic field are: where: is the iconicity of ionospheric medium varying in 9 10 − up to 3 10 − in a height interval 80-600 km, M is the mass of an ion or molecule, n N is the concentration of neutral particles, e and N are charge and concentration of electrons. Since the dynamic equations of system (2) are linear and sources and dissipation of energy are not taken into account, the energy of inertial waves cannot change in time and hence the wave frequency 0 ω should be real.
One new branch of the Alfven-type low-frequency MHD wave has been revealed in the ionospheric F-region [10,11]: where: a a k V ω = The group velocity of this wave (velocity of propagation) is normal to wave vector k ; its absolute value and projection in direction 0 H would be written as: From equations (5) and (6) follow, that when the group velocity has a component directed upward, the phase velocity has a component directed downward and vice versa [10,11]. Short inertial waves in the ionosphere belong to the class of very low-frequency (VLF) electromagnetic waves.
From equation (5) also follows that the information transfer to the Earth by the slow MHD waves by means of geomagnetic lines of force 0 H = const would not be complete, because a part of the information is lost owing to the second term in (5). From equation (4) follows that numerical values of the frequency of internal waves depend strongly on angle θ . They reach the maximum at 0 θ = and disappear at 0 90 θ = . and is random function of the spatial coordinates and time. Taking into account geometry of the task we obtain stochastic differential equation for the phase fluctuation:

Statistical Characteristics of Scattered Internal Wave in the Ionospheric F Region
where: Using the method of characteristics in the region 0 z > we obtain: where: If electron density fluctuations are statistically homogeneous and stationary for arbitrary correlation function ( , ) N W τ ρ the variance of the phase fluctuation is: here: L is a distance traversed by wave in the ionospheric plasma along the external magnetic field. Knowledge of this function allows to estimate dumping of the mean field of scattered wave using the well-known formula [12][13][14]: One of the most important statistical characteristic in nonstationary media is the variance of instant frequency 2 1 ω < > characterizing broadening of the temporal spectrum easily measuring by experiment: The degree of curvature of a constant phase surface in the turbulent plasma is characterized by fluctuations of a unite vector of the wave normal s : In the geometrical optics approximation neglecting dissipation processes, transport equation for the wave amplitude E or the logarithmic relative amplitude can be obtained from the differential equation [7]: where: is the coefficient connecting energy density Substituting in equation (13) the expressions: we obtain stochastic differential equation of the logarithmic relative amplitude 1 χ the solution of which is: where: fluctuations of the field are small and the forward scattering approximation is valid, the usually observed scintillation The next important problem of wave propagation in nonstationary medium is the energy exchange between the inertial wave and turbulent ionospheric plasma. The mean energy flux density (EFDE) of internal wave along the z direction is the sum of two terms in zero and second approximations: Growth of the energy flux along the z-axis means the energy transfer from the medium to the wave and energy decrease vice versa -from wave to the medium. These formulae are valid for near

Numerical Calculations
Experimental observations using orbital satellite beacons and the EISCAT incoherent scatter radar show [15] [10]. Other experimental data of the slow MHD waves measured by ground-based radar systems are presented in [16]. It was shown that slow MHD waves perturbations can be propagate over distances ~ 1000 km at 2 10 ω − ≤ sec 1 − (period 10 T ≥ minute), while on global distances at 3 10 ω − ≤ sec 1 − (period 100 T ≥ minute). Phase velocity of these waves are in the interval 6 10 ÷ km/sec and less. The irregularity model is described by 3D correlation function of electron density fluctuations. Numerical calculations are carried out for both Gaussian and power-law spectra for elongated ionospheric irregularities. The spectral density function which best describes the irregularities in a randomly inhomogeneous magnetized plasma depends on the particular case. Most widely used the Gaussian spectral function in the Gaussian, which has certain mathematical advantages. However, power-law spectral densities have been receiving more attention as being physically more realistic. Knowledge of the power spectrum of ionospheric refractive index fluctuations can lead to an understanding of the physical processes that characterize the region of the ionosphere under study.
Observations by "Sura" heating facility experiment show [17] that artificial ionospheric irregularities are stretched along the geomagnetic field. It was established that the transversal scale l ⊥ of elongated irregularities varies in the range of 100 -500 m; the magnitudes of the drift velocity were within the limits 65-270 m/s (the typical velocities of ionospheric motions 0 V ≈ 60-100 m/s). Velocity 100 m/s caused by the steady drifting with the horizontal wind of scattering irregularities embedded in the ionosphere will be used in numerical calculations.
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 responsible for fluctuations of MHD waves parameters and causing the scintillation of signals from the moving earth satellites (BE-B and BE-C) are between heights of 153 and 617 km closely aligned along the magnetic field lines in the F-region [18]. Orientation of the irregularities in the ionosphere has been measured with respect to the geographic north observing a diffraction pattern of the satellite signals (41 MHz) on the ground. The dip angle of the irregularities with respect to the field lines was within 0 16 . In the geometrical optics approximation forward scattering assumption is valid: σ is the variance. If the single scattering condition is also fulfilled 2 2 0 1 N k l L σ << a medium is characterized by the Gaussian irregularity spectrum [15].
Numerical calculations are carrying out for the anisotropic Gaussian correlation function of electron density fluctuations [19]: where: ( where: ( ) x Γ is the gamma function. Substituting (17) into (9) we obtain: where: For spectral function (17) variance of an instant frequency (11) and the degree of a curvature of a constant phase surface (12) along the direction of geomagnetic field of lines (zdirection) yield: Energy exchange between inertial wave propagating in the ionospheric F region and the regular turbulent plasma flow depends on the following parameters: angles θ and 1 θ ; characteristic spatial scale of plasma irregularities || should be fulfilled as in the geometrical optics approximation diffraction effects are neglect [12]- [14], however despite of this the obtain results are good approach to the reality (especially phase)); frequencies 0 ω and 0 a ω ; wavenumber 0 k .
Substituting (17) in equation (16) we obtain: where: 0 If the second term in brace is negative this means that turbulent plasma transfer its energy to the inertial wave increasing its amplitude and intensity propagating along the geomagnetic lines of force; if second term is negative internal wave losses energy transferring it to the plasma flow and its intensity decreases.
Applying the spectral function of the spatial-temporal power spectrum of electron density fluctuation in the "frozen" turbulence ( , ) ( ) ( ) [12][13][14] from equation (9) for the variance of the phase fluctuations in the principle plane we obtain: In this case the mean energy flux density along the z-axis is: where:   in the ionospheric F region with large scale electron density irregularities near the turbulent plasma flow pumps energy growing its energy flux; increasing anisotropy factor of elongated plasma irregularities maximum of EFDE is displaced to the direction of lines of force of geomagnetic field; decreasing own frequency (1/ ) T of the turbulent ionospheric plasma irregularities interval of slop angle of elongated plasma irregularities having substantial influence on the energy flux density substantially narrows. Figure 2 illustrates normalized mean EFDE as a function of anisotropy factor for different inclination angle of elongated electron density irregularities with respect to the external magnetic field. Numerical analyses show that the mean EFDE of the internal wave is increased inversely proportion to the angle 0 γ due to energy exchange with the plasma flow. MHD wave travelling distance 1000 km near to the lines of force of geomagnetic field pumps energy from the turbulent plasma flow increasing its intensity along and normal directions to the external magnetic field. Varying slop angle in small interval