The momentum flux profiles produced by trapped lee waves

Orographic gravity waves (also known as mountain waves) cause the atmosphere to exert a drag force on mountains. By Newton’s 3^rd law, the mountains exert an equal and opposite force on the atmosphere. It is clear from linear wave theory how to develop a framework for representing this reaction force in parametrizations for vertically propagating waves in climate and weather prediction models: the waves break and dissipate either due to critical levels (where the wind speed is perpendicular to the horizontal wavenumber vector, or zero), or due to the progressive decrease of density with height. But the situation is more complicated for trapped lee waves, which propagate horizontally near the surface, and where the wave energy is alternately reflected at the ground and at an elevated layer where the waves become evanescent. It is clear that boundary layer friction should be responsible for most of the dissipation of trapped lee waves, but it is not clear, even in the inviscid approximation, what form the wave momentum flux profiles that force the large-scale mean flow will take. This is due to the complications associated with the fact that trapped lee waves have both horizontal and vertical momentum (and pseudo-momentum) fluxes, which oscillate indefinitely with the wave phase downstream of the orography. No mechanism equivalent to critical levels, or density decay with height, acting on vertically propagating mountain waves, is available for trapped lee waves. In this study, this limitation is overcome by accounting for the effects of weak friction. While for an inviscid trapped lee wave train, the horizontally integrated momentum flux is ill-defined (except at the surface), in a dissipative problem where friction exists, no matter how small, the wave train necessarily decays downstream, and so is spatially bounded. This allows the areally integrated effect of the trapped lee wave to be expressed in terms of the divergence of the vertical flux of horizontal wave momentum (as for vertically-propagating waves). On the other hand, the form of the momentum flux profile (which defines this divergence) is different from any form that could be inferred from inviscid theory, although it is independent of the magnitude of friction, as long as this is small. These results from linear theory are compared with high-resolution numerical simulations of trapped lee waves for the two-layer atmosphere of Scorer, which confirm the form of the momentum flux profiles, and suggest that these may be independent of the adopted form of friction, at least to some extent. The results therefore facilitate the formulation of parametrizations for trapped lee waves with a much more solid physical  basis, and are likely to be generalizable to other atmospheric profiles.