On the onset of trapped mountain waves in stably stratified turbulent boundary layers

Atmospheric boundary layers (ABLs) with weak stratification are relatively well-described by similarity theory. With common strong stratification, similarity theory becomes unreliable and interactions with the mean flow assume a variety of scenarios. In this work, we focus on trapped mountain waves developing in a turbulent ABL, using the Monin-Obukhov similarity theory to define a mixing length model. Solutions are obtained analytically in the outer region, where the inviscid approximation holds, and by solving numerically a sixth order differential equation in the inner region, close to the surface. An expression for the reflection coefficient is derived using matched asymptotic expansions to describe the vertical structure of solutions. It is shown that the response of the ABL to a downward gravity wave strongly depends on the stratification and favors the onset of trapped lee waves for small Richardson numbers. The influence of the roughness is interpreted using a pseudo critical level below the surface. A spatial linear stability analysis is used to demonstrate that coexisting preferential modes dominate downstream the mountain. The vertical structure of these modes depends in a nontrivial way on the horizontal wavenumber. Finally, the implications of these results are discussed for a family of mountain ridges.