Three‐dimensional modeling of subionospheric VLF propagation in the presence of localized D region perturbations associated with lightning

A theoretical model of single-mode, subionospheric VLF wave propagation in the presence of localized perturbations of the nighttime D region has been developed. Such perturbations could be produced, for example, by lightning-induced electron precipitation associated with a characteristic type of phase or amplitude perturbation in VLF signals known as “Trimpi” events. Our model assumes that the ionospheric perturbation is slowly varying in the horizontal plane and that mode-coupling is therefore negligible, and accounts for (1) effects of perturbations with finite extent in the dimension transverse to the great circle (GC) path between transmitter and receiver, and (2) effects of perturbations which lie off the GC path as well as on it. The formulation used for the numerical calculations depends significantly on the mode refractive index of the ambient Earth-ionosphere waveguide and the mode refractive index in the region of the perturbation. In the calculations, values for the mode refractive index are determined from the electron-density-versus-altitude profiles of both the ambient and perturbed ionospheres. Values for changes in the amplitude and phase of a received signal were obtained from the model and compared with amplitude and phase measurements of a VLF signal received at Palmer Station, Antarctica, from the NPM transmitter (23.4 kHz) in Hawaii during energetic electron precipitation events. The large distance and all-sea path between the transmitter and receiver make it possible to represent the signal using single waveguide mode theory. The results of varying the location of a perturbation along the GC path as well as off the path in the transverse dimension, varying the horizontal scale of the perturbation, and varying the vertical density profile of the perturbation were all examined. The model shows that positive phase and/or negative amplitude shifts in the received signal are produced by perturbations centered on the GC path, whereas both positive and negative amplitude (or phase) shifts in a single-mode signal can be produced by perturbations lying in regions off the GC path. Results of the model indicate that the magnitude of the signal scattered by the perturbation towards the receiver continuously decreases with distance away from the GC path, becoming insignificant beyond ∼20 λ. On or near the GC path, it was found that the magnitude of the scattered signal was proportional to the scale of the perturbation parallel to the GC path. Using realistic values for the ground and ionospheric profile parameters, values of the shift in the amplitude and phase of the signal similar to those measured on the NPM signal received at Palmer Station, Antarctica, were obtained using this model. For example, a cylindrically symmetric perturbation of 5 λ in horizontal extent due to a 0.2 second burst of precipitating electrons of ∼2×10−3 ergs cm−2 s−1 flux density can produce amplitude changes of ∼ −0.3 dB and phase changes of ∼2°. Results from the model suggest that the ratio of the shifts in signal phase and amplitude can be used to determine the distance of the perturbation from the GC path.