In this thesis, we present a mathematical model that represents internal gravity waves and convection generated by a thermal forcing in the atmosphere. The goal is to investigate the mechanisms by which deep heating and convection in the lower atmosphere generate internal gravity waves. We consider a two-dimensional two-layer model of the atmosphere comprising an upper layer with stable stratification and an unstable convective lower layer. The governing equations are based on the equations for conservation of mass, momentum and energy for a fluid. The thermal forcing is represented by a
nonhomogeneous term in the energy conservation equation. We study different configurations depending on the vertical structure and depth of the thermal forcing.
First, we derive exact analytical solutions for the linearized equations for the case where the perturbation amplitude is independent of time. Next, we examine the case where the perturbation amplitude is time-dependent and derive exact solutions in each layer. The linear gravity wave solution approaches the steady solution in the limit of infinite time, but the linear solution for the convection grows exponentially with time. We
also carry out a weakly-nonlinear analysis of the gravity wave problem and examine the evolution of the mean flow with time due to the nonlinear interactions of the waves.
We carry out numerical simulations for each of the linear time-dependent one-layer models and then for the two-layer model. The results of the gravity wave simulations are in very good agreement with the analytical solution. In order to generate finite-amplitude convection, it is necessary to include the viscous and heat conduction terms in the equations. The time evolution of the convection depends on the relative
strength of the unstable stratification to the strength of the viscosity and heat conduction. This is in agreement with the situation that occurs in the well-known Lorenz model for atmospheric convection.
Our solutions can be used as to represent unresolved convective gravity wave drag in large-scale models of the atmosphere.