About one property of the dispersion equation for latitudinal acoustic-gravitational waves

Heading: 
Space and Atmospheric Physics
1Kryshtal, AN, 1Voitsekhovska, AD, 2Cheremnykh, OK, 2Cheremnykh, SO
1Main Astronomical Observatory of the NAS of Ukraine, Kyiv, Ukraine
2Space Research Institute of the National Academy of Sciences of Ukraine and the State Space Agency of Ukraine, Kyiv, Ukraine
Space Sci. & Technol. 2023, 29 ;(4):067-077
https://doi.org/10.15407/knit2023.04.067
Publication Language: Ukrainian
Abstract: 
Acoustic-gravity waves are an example of processes that largely determine the dynamics of the Earth's atmosphere. This is due to the fact that the sources of these waves are located throughout the height of the atmosphere, from the very “bottom”, where earthquakes, volcanic emissions, tsunamis, tornadoes, etc., occur, and to the very “top”, where perturbations of the solar wind, magnetic storms, and precipitation of particles in high latitudes are active. All these phenomena lead to the active energy exchange between all layers of the Earth's atmosphere and the interaction of wave disturbances of significantly different scales ¾ from several thousand kilometers to several hundred meters, and this ¾ to the appearance and development of processes of convection and turbulence in the environment. It seems that only nonlinear processes should dominate under such conditions. To a large extent, it is true, but at the same time, observations indicate that in many cases in the process of propagation of acoustic-gravity waves (AGW), the effects can be comprehensively described within the framework of the linear approximation of perturbation theory and well-developed theory of oscillations. At the same time, when creating models of the process, it turned out to be appropriate to use sufficiently justified physical approximations, such as isothermality of the atmosphere, its unlimitedness in the horizontal direction and compressibility in the vertical direction.
         Taking into account the real scales of the AGW, it is possible to neglect the curvature of the Earth's surface and consider it locally flat at any point of the surface and use the Cartesian coordinate system X, Y, Z in the calculations. To describe the environment, it makes sense to use non-dissipative hydrodynamics and in an equilibrium state - the hydrostatic equilibrium equation and barometric equation. The above-mentioned approximations and the mathematical apparatus of the theory of oscillations and the theory of differential equations allow when studying the initial system of equations describing the dynamics of AGW, to obtain a dispersion equation in the form of a polynomial of the fourth degree relative to the angular frequency of rotation ω as a function of the normalized wave vector of disturbance  (AGW). AGW spectrum is a spectrum of the atmosphere's own oscillations in the form , and its obtaining can be considered as the final solution to the initial problem if we ignore the obvious influence on the AGW spectrum of the angular frequency of rotation of the atmosphere Ω, which must necessarily be present in the dispersion equation due to the influence of the Coriolis force. The formal reason for the absence of the components of the vector Ω in the dispersion equation (DE) is the fact that the  is a minimum of two orders of magnitude smaller than the characteristic rotation frequency of the atmosphere , which is equal to the acoustic cutoff frequency. At the same time, the improvement of modern atmospheric observation equipment places increases the requirements for the accuracy of DE model solutions. In this sense, the resolution of DE in the work [9] чому тут 9 див укр варіант, …(замість цього ТРЕБА ПОСТАВИТИ Перший автор та ін., рік) can be considered as a “zero-order” solution with a small parameter . In addition, according to the method of obtaining, this solution is approximate. By definition, the solution obtained in the work [10] (чому тут 10 див ур варіант, замість цього ТРЕБА ПОСТАВИТИ Перший автор та ін., рік) by taking into account terms  in the modified DE is more accurate. But it is also approximate, although more accurate.
             In this work, we study in detail the dispersion equation for latitudinal AGW. The need for such consideration, as will be shown, is a consequence of the structure of this equation, namely the presence of a linear frequency term in it. Preliminary analysis showed that existing mathematical methods do not provide an unambiguous solution to this equation. This suggests the need to study possible solutions of the equation in terms of their coincidence with previously obtained ones for some partial cases. Such research allows us to choose the right decision. In the proposed study, we have shown that the Euler-Lagrange method allows, under certain additional conditions, to obtain an exact solution of the modified equation for AGW in closed analytical form.
Keywords: acoustic-gravity waves, dispersion equation, Earth's atmosphere, Euler-Lagrange method
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