Self-consistent amplitude profile of ducted VLF transmitter signal due to resonant interaction with energetic electrons in the magnetosphere

Of the various sources of whistler waves in the magnetosphere, the signals from terrestrial VLF transmitters are apparently the simplest for theoretical analysis. The role of these signals in the acceleration and pitch-angle scattering of energetic electrons in the Earth’s radiation belts has not been fully studied quantitatively, despite many decades of relevant research. The main problem in studying this issue is that to calculate the acceleration and pitch-angle scattering of particles, it is necessary to know the spatial profile of the signal amplitude. This is especially critical for phase-trapped particles, for which the energy exchange with the wave is the most significant. At the same time, the wave amplitude profile itself is determined by the interaction of the wave with resonant particles. Thus, we are dealing with a self-consistent problem, which is described by a nonlinear system of equations - a kinetic equation for the distribution function of resonant particles with a self-consistent field, and an equation for the wave field, in which the nonlinear and non-local growth rate is determined by the distribution function of resonant particles. In this paper, we develop an approach to solving this system of equations and the corresponding physical problem using the method of successive approximations.