Momentum Flux and Flux Divergence of Gravity Waves in Directional Shear Flows over Three-Dimensional Mountains

Linear mountain wave theory is used to derive the general formulas of the gravity wave momentum flux (WMF) and its vertical divergence that develop in directionally sheared flows with constant vertical shear. Height variations of the WMF and its vertical divergence are studied for a circular bell-shaped mountain. The results show that the magnitude of the WMF decreases with height owing to variable critical-level height for different wave components. This leads to continuous-rather than abrupt-absorption of surface-forced gravity waves, and the rate of absorption is largely determined by the maximum turning angle of the wind with height. For flows turning substantially with height, the wave momentum is primarily trapped in the lower atmosphere. Otherwise, it can be transported to the upper levels. The vertical divergence of WMF is oriented perpendicularly to the right (left) of the mean flow that veers (backs) with height except at the surface, where it vanishes. First, the magnitude of the WMF divergence increases with height until reaching its peak value. Then, it decreases toward zero above that height. The altitude of peak WMF divergence is proportional to the surface wind speed and inversely proportional to the vertical wind shear magnitude, increasing as the maximum wind turning angle increases. The magnitude of the peak WMF divergence also increases with the maximum wind turning angle, but it in general decreases as the ambient flow Richardson number increases. Implications of the findings for treating mountain gravity waves in numerical models are discussed.