Mathematical Modeling of the Singularities of Caustic Structure of Electromagnetic Waves Formed by Traveling Ionospheric Disturbances

It is well known that when studying traveling ionospheric disturbances (TIDs) using Doppler methods, multipath regions arise, which are called S-shaped curves and are usually associated with the formation of caustics. It is well known that when studying traveling ionospheric disturbances using Doppler methods, multipath regions arise, which are called S-shaped curves and are usually associated with the formation of caustics (A
3
). In this work, a model of the ionosphere was chosen, consisting of three layers: layer F2, layer F1 and layer E. In addition, a model TID is considered. In our model, TID propagates horizontally in the direction from the radiation source to the right receiving device with given dimensions and speed. To calculate ray trajectories, we used the bicharacteristic system of Hamilton-Lukin equations. The bicharacteristics method was actively used in studying the propagation of frequency-modulated radiation of electromagnetic waves in the Earth’s ionosphere, taking into account absorption and external magnetic field, and was also used for mathematical modeling of the propagation of space-time radio signals in stationary and non-stationary plasma. In this study, nonstationarity is associated with the movement of the TID. The case is considered when the magnetic field strength vector lies in the ray exit plane. The work carried out mathematical modeling of the ray structure of radio waves that occurs during the movement of TIDs in the upper ionospheric layer. The change in the frequency of radiation along the Earth’s surface caused by the movement of the TID (Doppler frequency shift) was studied. The results of studies of the caustic structure that arises during weakly inclined sounding of the Earth’s ionosphere containing moving ionospheric disturbances are presented. It is shown that in this case a caustic structure of the "cusp" type (catastrophe A
3
) arises, which is reflected in Doppler curves in the form of S-shaped lines and in curves describing the dependence of delays on distance in the form of time loops. The Doppler frequency shift along the rays caused by TID has been studied.