HIE-FDTD method for underwater VLF electromagnetic wave propagation in ionosphere-air-seawater media

The study of underwater very low frequency (VLF) electromagnetic wave propagation in ionosphere-air-seawater media is of great significance for cross-air-sea interface communication. For the fine-layered structure, the conventional finite-difference time-domain (FDTD) method was limited by the minimum grid size, making it difficult to increase the time step. In this paper, the hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) method was proposed to predict the propagation characteristics of underwater VLF electromagnetic waves in ionosphere-air-seawater media. Iteration formulas of the HIE-FDTD method for dispersive media in the perfectly matched layer (PML) absorbing boundary were derived in cylindrical coordinates. A propagation model for electromagnetic waves radiated from an underwater VLF source in the three-layer ionosphere-air-seawater medium was constructed. The influence of ionospheric variations on underwater and above-water propagation was revealed. The results show that the HIE-FDTD method achieves comparable accuracy to the conventional FDTD method, with significantly improved efficiency. Underwater VLF waves radiated from a source attenuate severely and are almost unaffected by the ionosphere, while their propagation in the above-water air is significantly modulated by the ionosphere. Strong absorption by the daytime ionospheric D-layer leads to increased signal attenuation, whereas the disappearance of the D-layer at night significantly improves propagation conditions. During the day-night transition, gradual changes in ionospheric parameters cause mode coupling and conversion, resulting in significant oscillations and non-uniform attenuation of the field strength distribution.