A weakly nonlinear analysis is applied 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 baroclinic configuration in a three-dimensional region where 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. By making the quasi-geostrophic approximation and considering a special case where the background flow velocity depends on a linear combination of the meridional (south–north) variable and the vertical variable, perturbation equations describing these wave interactions are derived. Nonlinear interactions between the waves give rise to a divergence of momentum and energy flux, which affects the mean flow. Asymptotic expressions are obtained for the wave-induced mean flow velocity and temperature that result from the combined effects of the Rossby waves and gravity waves. The situation where both types of waves are present is compared with the case with Rossby waves only. While Rossby waves alone cause a westward acceleration of the mean zonal velocity in the vicinity of the critical line, the addition of gravity waves can cause an eastward acceleration, as well as mean temperature changes. The configuration studied is consistent with observations of enhanced temperatures in the mesospheric inversion layer that occurs in the middle atmosphere in situations where both planetary waves and internal gravity waves are present.
Al-Zahrani et al. (2025) studied this question.