THREE-DIMENSIONAL MAGNETOGRADIENT WAVES IN THE UPPER ATMOSPHERE

General dispersion equation has been obtained for three-dimensional electromagnetic planetary waves, from which follows, as particular case Khantadze results in one-dimension case. It was shown that partial magnetic field line freezingin as in one-dimension case lead to the excitation of both “fast” and “slow” planetary waves, in two-liquid approximation (i.e. at ion drag by neutral particles) they are represent oscillations of magnetized electrons and partially magnetized ions in E region of the ionosphere. In F region of the ionosphere using one-liquid approximation only “fast” planetary wave will be generated representing oscillation of medium as a whole. Hence, it was shown that three-dimension magnetogradient planetary waves are exist in all components of the ionosphere, and as exact solutions, with well-known slow short-wave MHD waves, are simple mathematical consequence of the MHD equations for the ionosphere. Indexing terms/


INTRODUCTION
For the first time generalization of slow Rossby-type planetary waves taking into account latitudinal gradient of the geomagnetic field in the early seventies was given by Tolstoy [1] and independently by Khantadze [2]. Emphasizing their hydrodynamic nature, these waves Tolstoy called slow hydromagnetic gradient waves. In the subsequent papers [3,[5][6][7] Khantadze for the first time shown that fast planetary waves of the electromagnetic nature should be exist in the Earth upper atmosphere, in both E and F regions of the ionosphere. These waves in [8,9] were called magnetogradient waves of Khantadze. In the above-stated papers [2][3][4][5][6][7] classification of magnetogradient planetary waves (fast and slow), hydrodynamic end electromagnetic nature of these waves and the anisotropic nature of their propagation caused by curvature of lines of force of the geomagnetic field along the Earth parallels was for the first time given. Assessment of parameters of the considered waves, and also linear and nonlinear theories of magnetogradient waves, is given in [10][11][12][13][14]. Experimentally these waves were recorded in [8,15,16]. In the listed above papers generally one-dimensional and twodimensional magnetogradient waves propagation were considered. Meanwhile numerous observations confirm that the speed of propagation of large-scale wave perturbations having the electromagnetic nature except horizontal, always has vertical component, i.e. these waves are significantly three-dimensional [8,15,16].
As is well-known [17][18][19][20], without compressibility and temperature stratification (excepting a planetary boundary layer of the troposphere) Coriolis force [ 2 ] C   FV Ω becomes the main defining force in the equations of movement of the free atmosphere. The gyroscopic Coriolis force gives to the atmosphere additional stratification. In particular, angular velocity of the Earth rotation  , which is function of latitude of the place  , ingenerates the atmospheres speed gradients, and its latitudinal gradient  -inhomogeneity in the medium.  Indeed, in the considered approximation the three-dimension Helmholtz equation for a velocity vortex rot V has the following form [17][18][19]: from which visually follows that the first term in the right sight of equation (1) really generates in the atmosphere shortwave inertial waves [19], the second -Rossby planetary waves [17,18]. In absence of rotation of Earth wave movements disappear, and nonlinear Helmholtz equation for function of current will describe only convective movement of the atmosphere.

EQUATIONS OF THE IONOSPHERE OF INCOMPRESSIBLE ELECTROCONDUCTIVE LIQUID AND FORMULATION OF THE PROBLEM
In this paper magnetogradient waves of Khantadze are generalized on three-dimensional case. In the upper atmosphere, since height of 130 km and above, the magnetic pressure of the geomagnetic field prevails over pressure of neutrals and ionospheric plasma. Therefore in the upper atmosphere in the wave processes proceeding in an ionosphere along with parameters   Taking into account that for the planetary-scale waves effects of compressibility and temperature stratification play a minor role [6,17,18], we will seek the solution of this set of equations in the form of three-dimensional internal waves for a half-space [20]: -arbitrary wave number,  -the frequency which should be definition,   ,  -longitude) [17,18] can be written in the form: where u is the vector potential of the vorticity determined by equation     [7,11].
Linearizing the above-stated set of equations (2)-(4) in long-wave approximation 34 (~10 10   km) we will receive system of the equations which are investigated further [7,11,20]: Vector of the geomagnetic field

DISPERSIAN EQUATIONS AND ESTIMATION OF THE WAVE PARAMETERS FOR THE SET OF EQUATIONS (5)-(7)
Considering that the solution of system (5)- (7) as it was noted above, we look for in the form of internal three-dimensional , neglecting for simplicity action of the Coriolis force, from (5)-(7) we will receive: Khantadze can be easily obtain from the equation (8): From here in E region of the ionosphere (where 1   ), taking into account R    [3,5] we will find the following two branches of oscillations: а) for fast magnetogradient planetary wave (high-frequency branch): 1 3sin In F region of the ionosphere ( 0)   we will have only one branch of the fast magnetogradient planetary waves propagating both in positive and negative directions: From the expressions (11)- (13) follow that the phase speed of fast planetary waves (formula (11)) does not depend on the wave number, do not experience dispersion and they propagate one-dimensional, however slow Rossby type waves in E region and fast waves in F region of the ionosphere are strongly dispersive. The considered waves have all-planetary character and they can be exited at all latitudes of Earth. As in planetary waves horizontal wavenumbers satisfy the conditions , x y z k k k  , the formulas (11)-(13) received above can be simplified: gr z c  -on the contrary, from the upper layers of the ionosphere to the lower one. Now, at least, two permanently thermal sources of waves in the upper atmosphere are well-known: one of them is at the height of 80 km, where because of strong turbulence of the environment there is an attenuation of the acoustic-gravity waves, tidal and planetary waves going from below, and the seconda high-latitude thermal source at the heights of 350-400 km, where kinetic energy of high energetic magnetosphere particles going from above turns into heat [20]. Characteristic vertical linear scale of planetary waves -an order of a scale of heights, which in the troposphere is about 8 km, and 30 km in E region and 50 km in F region of the ionosphere. Characteristic horizontal linear scales of planetary waves along parallel and along meridian are an order of 34 10 10  km. Therefore the ionosphere for such large-scale wave processes is represented in the form of a thin film and analytical consideration of waves can be carried out according to the known theory of "small water" [17,18]. For planetary waves owing to these conditions , x y z k k k  full wave vector k will have the direction, close to a vertical.
Let's note that such inclination of the line of constant phase of planetary waves very often is registered at observation in the atmosphere [18].
It is also necessary to note that the considered fast magnetogradient waves, as shown in [9], transfer ionospheric perturbations on global distances along parallels and meridians. Numerical values of phase speed of fast magnetogradient H c -waves, calculated using experimental data are provided in [8] where it is shown that parameters of the Sun. In view of that this paper has character of brief communication we refrain from further specification of the considered waves. In more detail numerical values of parameters of magnetogradient waves, their altitude profiles for different levels of activity of the Sun, time of day and seasons are carried out in [21].

CONCLUSION
Summarizing, we can claim that unlike two-dimensional planetary waves which can extend only in the horizontal direction three-dimensional slow and fast magnetogradient planetary waves should mainly propagate in the vertical direction ( , ) z x y k k k  , which well is confirmed by observations in the upper atmosphere [15,20].