Influence of a viscous boundary layer on trapped lee waves

The backward reflection of a stationary Gravity Wave (GW) propagating toward the ground is examined in the linear viscous case and for large Reynolds numbers Re. In this case, the stationary GW presents a critical level at the ground because the mean wind is null there. When the mean flow Richardson number at the surface (J) is below 0.25, the GW reflection by the viscous boundary layer is total in the inviscid limit Re~infinite. The GW is a little absorbed, when Re is finite, and the reflection decreases when both the dissipation and J increase. When J>0.25, the GW is absorbed for all values of the Reynolds number, with a general tendency for the GW reflection to decrease when J increases. As a large ground reflection favors the downstream development of trapped lee wave, the fact that it decreases when J increases explains why the more unstable boundary layers favor the onset of mountain lee waves.

The fact that the GW reflection depends strongly on the Richardson number indicates that there are some correspondences between the dynamics of trapped lee-waves and the dynamics of Kelvin-Helmholtz instabilities. Accordingly, and on one classical example, it is shown that some among the neutral modes for Kelvin-Helmholtz instabilities that exist in an unbounded flow when J<0.25, can also be stationary trapped-wave solutions in the presence of a ground and in the inviscid limit Re~infinite. When Re is finite, these solutions are affected by the dissipation in the boundary layer and decay in the downstream direction. Interestingly, their decay rate increases when both the dissipation and J increase, as does the GW absorption by the viscous boundary layer.