Theory of second-order cyclotron resonance as related to the origin of discrete VLF emissions in the magnetosphere

Recent analytical and numerical results concerning the role of the second-order cyclotron resonance effects in formation of discrete emissions in the magnetosphere are reviewed. Peculiarities of whistler cyclotron interactions with energetic particles having sharp (step-like or beam-like) distribution functions evolving in space and time are studied. Formation of such distributions is considered, and an analytical self-consistent theory of the second-order cyclotron resonance effects is developed. In particular, characteristics of electron beams produced by the interaction of a VLF wave packet from a ground-based transmitter are studied. It is shown that spatial and temporal gradients of the parallel velocity of the beams formed can be opposite to the case of a pure adiabatic motion of a single particle. Such a behavior can be significant for the generation of secondary emissions. It is proven that the optimal conditions for the instability occur for a nonstationary quasi-monochromatic wavelets whose frequency changes in time. The theory developed permits one to estimate the wave amplification and spatio-temporal characteristics of these wavelets. Numerical results on beam formation and generation of secondary emissions are presented. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 42, No. 8, pp. 713–727, August 1999.