Mountain wave drag due to trapped lee waves

When air flows over a mountain barrier, gravity waves are generated in the lee of the mountain. Depending on the upstream atmospheric profile of wind and stability, and the horizontal length scale of the mountain, two generic wave trains may be forced: an upwardly propagating wave train, or an evanescent, trapped lee wave train which propagates horizontally. As a consequence of the generation of either type of wave train a cross-mountain pressure gradient exists, implying a pressure force on the mountain. To counteract this force on the mountain there exists an equal and opposite force acting to decelerate the mean atmospheric flow. According to classical gravity wave theory this deceleration occurs in regions of wave dissipation where there is a non-zero gradient of vertical flux of horizontal momentum with height.

A primary source of dissipation is wave breaking which generally occurs at upper tropospheric or lower stratospheric levels due to upwardly propagating wave trains. Such a wave train is characterized by a non-zero vertical flux of horizontal momentum which is constant with height until it reduces in magnitude over the vertical depth of the breaking region. An equally important source of dissipation comes from the `removal' downstream of trapped lee wave trains. Various studies have shown that, averaged across the mountain, such a downstream gravity wave train generates a non-zero vertical flux of horizontal momentum which also decreases in magnitude with height. The classical Eliassen-Palm (EP) theory of wave-mean flow interaction would suggest that this vertical gradient of flux acts to decelerate the mean flow. However, it is assumed that the trapped wave train is steady and not yet dissipated, indicating that the mean flow has not been modified, and the EP theory is not applicable in this case.

The horizontal wave momentum budget cannot be closed without accounting for the upstream and downstream values of the dynamic pressure gradient (pressure + density*velocity²). In this case the mean flow deceleration from dissipation of the trapped lee wave train is not solely due to the vertical divergence of the vertical flux of horizontal momentum (as is the case in upwardly propagating gravity waves). In this presentation a general theory is developed to include the dynamic pressure contribution to the momentum budget. For both trapped and upwardly propagating orographically generated waves the dynamic pressure contribution is calculated from an analysis of the Bernoulli function on a streamline. It is shown how recent results indicating non-zero, and vertically divergent, wave momentum fluxes in a trapped lee wave train are due to the variation of dynamic pressure across the wave train. An indication of how to incorporate this theory within an orographic gravity wave drag parametrization scheme is given.