Beta-Effect of Internal Inertia–Gravity Waves in a Stratified Atmosphere in the Incompressible Fluid Approximation

Abstract

This paper presents a mathematical model that describes the propagation of internal inertia–gravity waves in a stratified atmosphere under the approximations of an incompressible fluid and a traditional β-plane. It demonstrates that, in the incompressible fluid approximation, the temperature field is inconsistent with the heat conduction equation. The system of equations that describes internal inertia–gravity waves is considered in the general case, taking into account the buoyancy force, and reduced to a single equation. The solution is sought in the form of traveling plane waves. A dispersion relation has been obtained in the form of a cubic equation that represents a hypersurface in wave number space, without the assumption of small vertical wavelength. Cross-sections of this surface are plotted, and an extremum study is performed. This shows that a new frequency region appears in the low-frequency spectrum (Formula presented.) that was not present in the f-plane approximation. Here, (Formula presented.) is the Coriolis parameter, and (Formula presented.) is the latitude. Furthermore, these waves only propagate in the negative direction of the x-axis, i.e., in the opposite direction of the Earth’s rotation. It is also shown that there is a region with a minimum frequency in the “high-frequency” spectrum determined by buoyancy (Formula presented.), and that waves propagate in the negative direction as well. Thus, the dispersion surface is shown to have two extremum points. The first is a minimum in the “high-frequency” spectrum (Formula presented.) and the second is a maximum in the “low-frequency” spectrum (Formula presented.).