Plane Internal Gravity Waves with Arbitrary Amplitude

Abstract

Nonlinear equations called the Stenflo equations are usually used for the analytical description of the propagation of internal gravity waves in the Earth’s upper atmosphere. Solutions in the form of dipole vortices, tripole vortices, and vortex chains are previously obtained by these equations. The Stenflo equations also describe rogue waves, breathers, and dark solitons. If disturbances cease to be small, then their profiles are usually deformed, and, presumably, they cannot be considered plane waves. This study shows that this is not always the case for internal gravity waves and that these waves can propagate as plane waves even with large amplitudes. An exact solution of the system of nonlinear Stenflo equations for internal gravity waves that contain nonlinear terms in the form of Poisson brackets is given. The solution is obtained in the form of plane waves with arbitrary amplitude. To find a solution, the original system of equations is transformed. It is split into equations for the stream and vorticity functions as well as equations for the perturbed density. To solve the obtained equations, the procedure of the successive zeroing of Poisson brackets is applied. As a result, linear equations that allow one to find the accurate analytical solutions for internal gravity waves in the form of plane waves with arbitrary amplitude are obtained. By solving these linear equations in two different ways, we have analytically found expressions for the perturbed quantities and the dispersion equation. The nonlinear equations obtained for the current, vorticity, and perturbed density functions can be used to find other nonlinear solutions. The given solutions in the form of plane waves with arbitrary amplitude may be of interest for the analysis of the propagation of internal gravity waves in the Earth’s atmosphere and the interpretation of experimental data.