On the ionospheric propagation of VLF waves generated by currents in the lower ionosphere

The consistent description of ELF/VLF wave propagation in the ionosphere is essential for understanding the properties of the wave phenomena observed both at the ground and satellites. Due to strong inhomogeneity of the lower ionosphere coupled with collisional damping of a wave, the geometrical optics approach is not valid, while the full-wave approach suffers from the problem of numerical instability. In this paper, we implement a rather general and powerful method avoiding the numerical instability to study propagation of the wave generated in the lower ionosphere. Our calculations demonstrate the quasi-periodic behaviour of the field in the upper ionosphere, which is associated with the Earth-ionosphere waveguide. The field distribution varies significantly with frequency, from being nearly sign-definite to oscillating with comparable positive and negative values, producing the ring structure in wave-vector space. The field distribution at the ground is studied and shown to have resonant structure defined by two parameters: number of the resonance, which depends on wave frequency, and spatial scale of the Earth-ionosphere waveguide.