Mountain waves developing inside and aloft stably stratified turbulent boundary layers

A linear theory of the trapped mountain waves that develop in a turbulent boundary layer is presented. The theory uses a mixing-length turbulence model based on Monin–Obukhov similarity theory. First, the backward reflection of a stationary gravity wave propagating toward the ground is examined. Three parameters are investigated systematically: the Monin–Obukhov length (Formula presented.), the roughness length (Formula presented.), and the limit value of the mixing length (Formula presented.) aloft the “inner” layer. The reflection coefficient appears to depend strongly on the Richardson number aloft the inner layer ((Formula presented.), with (Formula presented.) the von Kármán constant), with the reflection decreasing when the stability (Formula presented.) increases. The influence of the roughness and mixing lengths on the reflection is explained in terms of the depth of a “pseudo”-critical level located below the surface, with the reflection decreasing when the depth of the “pseudo”-critical level decreases. The preferential modes of oscillations occurring in the presence of mountain forcing are then analysed, with the decay rate of the trapped waves downstream increasing when the reflection decreases. At a certain point nevertheless, when the absorption is large but the boundary-layer depth deep enough, trapped modes appear that interact little with the surface.