Approximate solution for the VLF eigenvalues near cut-off frequencies in the nocturnal inhomogeneous Earth-ionosphere waveguide

An approximate wave solution for the VLF modes near cut-off frequencies in the high-latitude Earth-ionosphere waveguide is obtained. The vertical inhomogeneity of the lower D-region of the ionosphere is included in the form of a simple exponential profile of the parallel conductivity. The E-region is represented as a magnetoactive plasma with a sharp bottom, at which the electron density increases strongly and then varies slowly at the scale of the local wavenumber. Approximate expressions are derived for the phase velocities and attenuation rates of the VLF modes near cut-off frequencies (⨍ c 1 ≈ 1.5–1.7 kHz, ⨍ cm≈ m⨍ c 1,m = 1,2,… , in the night-time ionosphere conditions). They describe a strong dispersion of the QTE modes (left-hand polarized near corresponding cut-off frequencies). Taking into account the D-region vertical inhomogeneity results in a small strengthening of the QTE modes' dispersion, but it is in principle necessary for correct description of the VLF signal amplitude (in particular, tweek-atmospherics and whistlers with additional traces in the Earth-ionosphere waveguide).