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

Abstract

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.