Asymptotic Solutions for Atmospheric Internal Gravity Waves Generated by a Thermal Forcing in an Anelastic Fluid Flow with Vertical Shear

Asymptotic solutions are derived to model the development of atmospheric internal gravity waves generated by latent heating in a two-dimensional configuration involving a vertically-sheared background flow. The mathematical model comprises nonlinear partial differential equations derived from the conservation laws of fluid dynamics under the anelastic approximation where the background density and temperature vary with altitude. The latent heating is represented by a horizontally-periodic but vertically-localized nonhomogeneous forcing term in the energy conservation equation. This generates gravity waves that are considered as perturbations to the background flow and are expressed as perturbation series, with the leading-order contributions being the solutions of linearized equations. Taking into account the nonlinear terms at the next order gives expressions for the effects of the waves on the background mean flow. Due to the vertical shear, there is a critical level where momentum and energy are transferred from the wave modes to the mean flow. The asymptotic solutions show that the wave–mean-flow interaction is nonlocal and occurs over the range of altitudes from the thermal forcing level up the critical level. This is in contrast to what occurs in the case of waves forced by an oscillatory lower boundary, where the interaction is typically localized around the critical level. It is found that the wave drag is negative above the thermal forcing level, making the mean flow velocity more negative, but it becomes positive as the waves approach the critical level, indicating wave absorption in this region. There is wave transmission through the critical level, as well as absorption, and the extent of transmission depends on the depth of the latent heating profile. The mean potential temperature is reduced above the thermal forcing level and enhanced at the critical level, a situation that could ultimately lead to the development of convective instabilities.