In this thesis, we use analytical and numerical methods to study a mathematical problem describing two types of waves in an atmospheric fluid flow, namely planetary Rossby waves resulting from the rotation of the Earth and internal gravity waves resulting from the gravitational and buoyancy forces. We consider a three-dimensional or baroclinic configuration and make the quasi-geostrophic approximation in the special case where the background flow velocity depends on a linear combination of the meridional (south-north) variable and the vertical variable. Both types of waves are forced at the same location and propagate upwards and southwards to a critical line where they interact with and modify the background mean flow and background mean temperature. We carry out a weakly nonlinear analysis to derive equations describing the rate of change of the wave-induced mean velocity and temperature with time due to the nonlinear interactions in the vicinity of the critical line. We also carry out numerical simulations using finite difference approximations and spectral methods. We compare the results obtained with combined gravity waves and Rossby waves with those obtained with Rossby waves only. We find that with the additional effects of the gravity waves, the background mean flow becomes more eastward, and the background mean temperature is higher around the critical line. In addition, in the presence of gravity waves, the amplitude of the Rossby wave zonal velocity is reduced, and the Rossby wave temperature is increased. These conclusions are consistent with observations of planetary wave dissipation and enhanced temperatures in the mesospheric inversion layer that occurs in the middle atmosphere.