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.

Full-wave solutionfor a monochromatic VLF wave propagating through the ionosphere R = re f l \ r e f line where Sref and Si„c are time-averaged energy fluxes at the upper boundary of height in reflected and incident wave correspondingly. The outputs of calculations are shown in Fig. 2. In these figures, the values o f R corresponding to relatively large perpendicular wave vectors, thus, large angles of incidence, for which waves do not propagate in the atmosphere, are marked by points; values of R for the cases when waves reach the ground are marked by circles. Graphs related to six angles of incidence are plotted. Upper graphs that correspond to small angles of incidence and, thus, propagation in the earth-ionosphere cavity, show a quasiperiodic behaviour o f the reflection coefficient as a function of wave frequency. With increasing angle of incidence, waves in the atmosphere become evanescent, and the quasiperiodic feature of R ceases. This suggests that this feature is due to resonance properties of the earth- ionosphere cavity. The quasiperiodic dependence of the reflection coefficient on frequency has earlier been pointed out by Hayakawa and Ohtsu (1972). To understand the quasiperiodic dependence o f the reflection coefficient on wave frequency, we first remember that at the ground, the total energy flux is equal to zero, so that the reflected energy flux is equal (in magnitude) to the incident one. Thus, a deviation of the reflection coefficient from unity at h = 600 km is only due to energy absorption in the ionosphere. The peculiarity o f this absorption consists in that maximum absorption takes place in a small region ~30km at the height Habs about 90 km. Although at these heights the wave field structure is quite complicated and differs significantly from sinusoidal one typical of atmospheric region, the magnitude of the electric field that determines the wave field absorption still has a quasiperiodic structure with a characteristic length equal to (in dimensional units) / _ n_ л с k ~ (0 ' The wave field absorption will then have a maximum and, consequently, the reflection coefficient will have a minimum if Habs = (n+ 1 / 2 ) 1, where n = 0 , 1 ,2 ,..., thus, f 1 4 n + - CO к с H a b s This gives a period of variation of the reflection coefficient over frequency: A ^D КС / ~ 10 ra d / 51 , or AF ~ 1.1kHz. To verify this interpretation, we calculated the reflection coefficients with the ground level artificially shifted towards the absorption region, which indeed resulted in the corresponding increase of the period AF (see Fig. 3). In this work, we have presented a new approach to full-wave description of VLF wave penetration through the ionosphere. In this approach, the problem of numerical swamping, which constitutes the main difficulty of previous considerations, is resolved analytically, making numerical calculations plain and straightforward. The developed approach is based on the method of successive approximations which, in its present form, is applicable only to the case of wave incidence on the ionosphere at small angles. Investigation of the problem for arbitrary angles of incidence will be the subject of further work. The results of the present study imply a frequency modulation in wide­ band VLF measurements of whistler mode waves reflected from the lower ionosphere. The mechanism of this modulation explained in the present work also hints at a possible quasiperiodic frequency dependence of VLF spectral intensity observed on the ground. C o llis io n fr e q u e n c y p ro file fo r th e n ig h t-tim e io n o s p h e re Fig. 1 Electron density and collision frequency profiles. 75