__Keywords:__ionosphere regularly inhomogeneous medium radio wave propagation numerical simulation methods of by-characteristics caustic singularities integration of rapidly oscillating functions

E.B. Ipatov, E.A. Palkin, D.S. Lukin, D.E. Ipatov

The paper presents a numerical algorithm to calculate the structure of wave fields in the propagation of short radio signals in regular inhomogeneous ionosphere of the Earth. For computer modeling of the characteristics of an inhomogeneous medium in which an electromagnetic wave is distributed has been chosen the standard model International Reference Ionosphere 2001. The same model also contains a model of the geomagnetic field, which was used in calculating the effective permittivity of the ionosphere plasma. Model data, specifying the spatial structure of the ionospheric plasma are smoothed by using of cubic smoothing splines with a window from 0.1 MHz to 0.25 MHz plasma frequency. Smoothing allows to provide the continuity of first derivatives of the plasma frequency in the spatial coordinates.
To describe the fine structure of the diffraction of an electromagnetic field in the focusing areas the wave field is presented in the form of one-dimensional integral of a rapidly oscillating function by the method of the Maslov canonical operator. All functions that included with the integral representation are constructed by the method of bicharacteristics on the basis of numerical integration of the system of six ordinary differential equations. For the additional control of accuracy of integration the system of differential equations the dispersion relation is used which is an invariant of the system of equations. When creating a program that performs the integration of systems of ordinary differential equations the modified Kutt Merson method is used with an automatic choice of the integration step. The paper presents the results of calculation of the field in the vicinity of the focus corresponding to singularities of type «Butterfly». The test calculations show that a complex diffraction feature of «Butterfly» is observed in the frequency range greater than 5 MHz. For this type of focusing a comparison of the obtained solution with a solution that is built into the framework of geometrical optics is performed. In the areas where the method of geometric optics is applicable, the two solutions are different by no more than one percent. We have developed numerical algorithms and software packages allow you to "bring up the number of" solving a wide class of physical problems.

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