Nonlinear Atmospheric Gravity Waves in the Earth’s Isothermal Atmosphere

Modern experimental and theoretical studies of atmospheric gravity waves (AGW) indicate the need for a nonlinear consideration of these processes. First of all, this is due to the exponential growth of the amplitudes of gravity waves with height in the atmosphere, which significantly limits the possibility of applying the linear theory. In the work, analytical solutions of the system of nonlinear equations describing the propagation of atmospheric gravity waves in the isothermal atmosphere were obtained. To find the solutions, nonlinear equations obtained earlier in the model of two-dimensional motion of an ideal atmospheric gas in the Boussinesq approximation were used. The nonlinear components in these equations have the form of Poisson brackets. We found the solutions of the nonlinear equations in the form of plane waves. For this type of solution, the Poisson brackets are converted to zero. This approach allowed us to obtain analytical solutions that describe various types of nonlinear gravity waves in an isothermal atmosphere. In the linear theory of AGW, solutions in the form of plane waves are in the assumption of small amplitudes of perturbations. Unlike the linear consideration, the solutions of the nonlinear equations that were obtained do not have restrictions on the amplitude. Within the framework of the specified simplifying assumptions, solutions were obtained from the system of nonlinear equations for: (1) freely propagating internal gravity waves, (2) horizontal (evanescent) atmospheric gravity waves, and (3) important special cases of evanescent wave modes. The energy conditions for the realization of the obtained types of wave perturbations in an isothermal atmosphere were analyzed. The specified nonlinear solutions (1)–(3) are nondivergent since a system of nonlinear equations was used when obtaining them, written in the assumption of zero velocity divergence. At the same time, in the linear theory, the assumption of zero velocity divergence singles out only one f-mode from the entire AGW spectrum. That is, the application of nonlinear theory when considering gravity waves, even with significant simplifications in the original system of nonlinear equations, significantly expands the class of wave solutions in comparison with the linear theory.