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.

“Physics o fAuroral Phenomena”, Proc. XXXIIIAnnual Seminar, Apatity, pp. 73 - 76, 2011 / ^ \ Polar © Kola Science Centre, Russian Academy of Science, 2011 V vM i Geophysical W У Institute FULL -W AVE SOLUT ION FOR A MONOCHROMAT IC VLF W AV E PROPAGAT ING THROUGH THE IONOSPHERE I.V. Kuzichev, D.R. Shklyar (Space Research Institute o fRAS, Moscow) Among the many problems in whistler study, wave propagation through the ionosphere is one o f the most important, and the most difficult at the same time. Characteristics of propagation, such as reflection and transmission coefficients, are needed for interpretation of satellite and ground-based observations of VLF waves. Consequently wave penetration through the ionosphere has been in the focus o f research since the beginning of whistler study: Budden (1985) (general full-wave analysis, including the problem of numerical swamping); Pitteway (1965) (analysis o f differential equations governing the wave fields inside a horizontally stratified ionosphere in the case of oblique wave incidence from below; indication and discussion o f the problem o f numerical swamping); Helliwell (1965) (detailed consideration in the framework o f ray theory revealing the most essential features o f the phenomenon); Pitteway and Jespersen (1966) (a comprehensive numerical study o f wave propagation through the ionosphere, including calculations of reflection and transmission coefficients); Hayakawa and Ohtsu (1972) (theoretical treatment of transmission and reflection of downgoing whistlers in the case o f longitudinal propagation, using model plasma density and collision frequency profiles). The difficulty in considering VLF wave passage through the ionosphere is, after all, due to fast variation o f the lower ionosphere parameters as compared to typical VLF wave number. This makes irrelevant the consideration in the framework of geometrical optics, which, along with a smooth variation of parameters, is always based on a particular dispersion relation. Although the full-wave analysis in the framework of cold plasma approximation does not require slow variations of plasma parameters, and does not assume any particular wave mode, the fact that the wave of a given frequency belongs to different modes in various regions makes numerical solution of the field equations not simple. As is well known (see, e.g. Ginzburg and Rukhadze, 1972), in cold magnetized plasma, there are, in general, two wave modes related to a given frequency. Both modes, however, do not necessarily correspond to propagating waves. In particular, in the frequency range related to whistler waves, the other mode is evanescent, i.e. it has a negative value of N 2 (the refractive index squared). It means that one o f solutions of the relevant differential equations is exponentially growing, which makes a straightforward numerical approach to these equations despairing. This well known difficulty in the problem under discussion is usually identified as numerical swamping (see, e.g. Budden, 1985; Nagano et al., 1975, and earlier works mentioned above). Resolving the problem of numerical swamping becomes, in fact, a key point in numerical study of wave passage through the ionosphere. As it is typical of work based on numerical simulations, its essential part remains virtually hidden. Then, every researcher, in order to get quantitative characteristics of the process, such as transmission and reflection coefficients, needs to go through the whole problem. That is why the number o f publications dealing with VLF wave transmission through the ionosphere does not run short. In this work we develop a new approach to the problem, such that its intrinsic difficulty is resolved analytically, while numerical calculations are reduced to stable equations solvable with the help o f a routine programme. Another goal of the work is to present all equations and related formulae in an undisguised form in order that the problem may be solved in a straightforward way once the ionospheric plasma parameters are given. We consider the problem of wave propagation in the ionosphere in case of wave incidence on the ionosphere from above when the angle o f incidence is small enough. We used the model o f plane medium in which all ionospheric plasma parameters depend only on height h and do not depend on horizontal coordinates £ and у (у axis is directed westward, £ axis completes the right-hand system { y , A}) . The wave field was assumed to be monochromatic, harmonic in £ and independent o f у (all variables are dimensionless): <e = R e j £ ( A ) e x p ( j * r £ - / Q f ) j , and the similar formula is for the magnetic field. Horizontal component o f the wave vector к for a given wave frequency is defined by angle incidence of the wave. Having assumed such field structure one can easily obtain from general Maxwell’s equations following equation for the field amplitudes E ( Л ) , В ( h ) : d E d E . -------- — = - i n В . ; ------------ = i d B „ + i K E h , к E v = fi В h ; d h 4 d h y y d В у d h d В ( d h n { e ] E s + i e 2 c o s a E y + £ ] l E h '^-, ( l ) i fi e i E y + i s j с о s a E ^ - i f 2 s i n a £ л ; /' e 2 s i n a E y + s 3 E h . = i = i к В h + 73