Physics of auroral phenomena : proceedings of the 33rd Annual seminar, Apatity, 02 - 05 March, 2010 / [ed.: A.G. Yahnin, A. A. Mochalov]. - Апатиты : Издательство Кольского научного центра РАН, 2011. - 206 с. : ил.

There all E with indices are the components of dielectric tensor in coordinate system {%, y, h} ( a is an angle between £ axis and external magnetic field): I. V. Kuzichev, D.R. Shklyar 2 - 2 £ ! = £ j c o s a + £ 3 s i n a 2 . 2 £ 3 = £ 3 c o s a + e , s in a c o s a 1 + ( CO 1 ( 0 M eff со = 1 - ■ I — i o n s m ® ^ e ) ’ 3 ffl (ffl + / V . ) M eff We assume the Earth to be perfectly conductive, hence tangential components o f the electric field equal to zero at the surface, at the upper boundary of height (which was set equal to 600 km) the field is the superposition of incident wave with given amplitude and reflected wave. System (1) consists of four ordinary differential equations and but for numerical swamping it could be integrated numerically in a straightforward way. So we need to find a way to regularize this instability. It can be noticed that for enough high heights in ionosphere following inequalities are fulfilled: I £ 3 | » | S 2 | » | S x I , which allows us to use some kind of method of successive approximations. Now it becomes more convenient to use another coordinate system (x, y, z) where z axis is along external magnetic field. Passing on to these coordinates we can easily obtain following equations for the new components o f the electromagnetic field: d 2 A , d A 7 + 77 A , = - iA у A 2 ; ( 2 ) d h d 2 A. d h 2 ц A 2 = - iA d h dA i E . d h s e c a ‘ 2 > + r A \ ; E A h / 2 0 ; У ; A - - ( E x + i E y s e c a \)e‘ Я A/2 n , M, r = i>£2 ); Л = К t a n a . Having assumed that the angle of incidence is small enough, so that | £ - enough high heights (for h > 80-90 km) coefficients in (2) will be related as following: r\ 2 2 , >> к , we can expect that for t] 0 ц > > m a x - £ / !}■ so we are able to perform the expansion of system ( 2 ) in this small parameters. Zero order equations will be d 2 A L + Л о A i = 0 ; d h ' A 2 = 0 ; and the first order: d 2 A . TL + r/ A , = 0 ; d h A 2 = - ( 3 ) 2 П , - i A dA (o) d h г л ; ( 0 ) Zero order equations and the first equation in (3) do not need anything besides aforementioned inequalities but the second relation in (3) is correct only in assumption of slow variation o f plasma parameters. These equations are stable and can be easily integrated. For low altitudes exact system (1) is provided to be stable enough, hence one can integrate it numerically Moreover the regions, where exact system (1) and approximate system (3) are applicable, overlap, consequently it is possible to join the solutions of these systems and finally we can obtain unique solution o f the problem in the whole ranee of altitudes. 8 Using the developed method we perfonn calculations o f electromagnetic field and reflection coefficient R defined by formula below for a set of ionospheric parameters obtained from IRI model (see Fig. 1 ). 74