The Surface Drag and the Vertical Momentum Fluxes Produced by Mountain Waves in Flows with Directional Shear

In gravity wave drag (GWD) parametrization schemes that are necessary in numerical models of the atmosphere, both the surface GWD and the distribution of the wave momentum flux with height must be specified. The present study addresses these two aspects. Firstly, the impact of a recently-developed approach to the evaluation of surface GWD is assessed. This approach uses linear theory, but incorporates the effects of wind profile shear and curvature, by means of a second-order WKB approximation. While the theory predicts the possibility of either drag enhancement or reduction, depending on the wind profile, results obtained with the ERA-40 reanalysis data clearly indicate the predominance of local drag enhancement. However, the global impact of shear on the atmospheric axial GWD torque comes mostly from regions with predominantly easterly flow, contributing to a slight reduction of the bias found in different studies of the global angular momentum budget. The relative correction due to shear on linear GWD is found not to depend too strongly on the levels chosen for the computation of the low-level wind derivatives. Secondly, the wave momentum flux is investigated within the framework of linear theory for flow with directional wind shear over a circular mountain. The variation of the momentum flux with height is calculated for relatively large shears, extending previous calculations of the surface GWD by the authors. A WKB approximation is used to address flow with generic, but relatively slowly-varying wind profiles. The WKB approximation must be extended to third order to obtain momentum flux expressions that are accurate to second order. Inviscid, steady, non-rotating, hydrostatic flow is assumed. Since the momentum flux only varies vertically due to wave filtering by critical levels, the application of contour integration techniques enables it to be expressed in terms of simple 1D integrals. On the other hand, the momentum flux divergence (which corresponds to the force on the atmosphere that must be represented in GWD parameterizations) is given in closed analytical form. The momentum flux expressions are tested for idealized wind profiles, where they become a function of the Richardson number (Ri). These expressions tend, for high Ri, to those derived by previous authors (where wind profile effects on the surface drag were neglected and critical levels acted as perfect absorbers). The linear results are compared with linear and nonlinear numerical simulations, showing a considerable improvement upon previous models, developed for very high Ri.