Analytical properties of the Green's function of the equation of linear internal gravity waves in stratified media with model distributions of buoyancy frequency

This paper presents a theoretical study of the analytical properties of the Green's function for the internal gravity wave equation for two model density distributions of a stratified, inviscid medium. Integral representations of the solutions are obtained in a linear formulation using the Fourier transform. The selection of a single-valued form for the resulting analytical solutions is discussed. The resulting analytical constructs, using integral convolution, enable the study of wave fields generated by arbitrary nonlocal and nonstationary disturbance sources in real natural stratified media. The obtained asymptotic results enable the investigation of wave disturbances that can be recorded using radar and optical systems. They provide information not only about the sources of generation but also about the characteristics of the marine environment. This is important, among other things, for studying the response of the marine environment to various hydrodynamic disturbances and improving methods for remote sensing of the sea surface. Initial and boundary conditions for specific disturbance sources should be determined from the results of direct numerical modeling of the complete system of hydrodynamic equations or from purely evaluative semi-empirical considerations, allowing for the adequate approximation of real non-local disturbance sources by a certain system of model sources. The resulting analytical solutions enable the calculation of the fundamental amplitude-phase characteristics of the excited far fields of internal gravity waves under certain generation conditions, and, furthermore, the qualitative analysis of the resulting solutions, which is important for the correct formulation of more complex mathematical models of the wave dynamics of real natural stratified media. These model solutions subsequently enable the derivement of representations of wave fields taking into account the actual variability and non-stationarity of such media.

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