Various approximations of mathematical models of internal gravity waves in the stratified atmosphere

Abstract

Two new mathematical models describing the propagation of internal gravity waves (IGWs) in a stratified atmosphere are proposed in this paper. The first model is called the non-Boussinesq gas approximation. It differs from the well-known incompressible fluid model in that the dynamics of the buoyancy force is not described by the continuity equation but by the heat conduction equation. The obtained dispersion relation differs from the dispersion relations in the incompressible fluid and anelastic gas approximations. In the second proposed model, called the general model, we abandon the Boussinesq approximation, and the system of equations includes both the continuity equation with density disturbance and the heat conduction equation. The analysis of the proposed model has shown that in the general case the maximum frequency of IGW's oscillations is equal to the Brunt-Väisälä buoyancy frequency, i.e., the results coincide with the results of the compressible fluid approximation (adiabatic approximation). However, the fundamental difference of our model is that the amplitude of acoustic waves in the adiabatic approximation decays with height, whereas in the general case it grows with height.