Linear theory of momentum fluxes in 3-D flows with turning of the mean wind

Current orographically forced gravity wave-drag parametrization schemes in numerical weather-prediction models are based on two-dimensional linear momentum-flux theory. The vertical flux of horizontal-momentum vector is considered parallel to the surface-stress vector and independent of height unless the Richardson number falls below a quarter, or a critical level exists where the mean wind becomes zero in the direction of the flux vector. Non-zero gravity-wave drag is then taken to be parallel and opposite to the Bur vector. Consideration of the full three-dimensional linear problem reveals a number of fundamental differences: (i) the drag force is perpendicular to the mean wind where there is no wavebreaking, and (ii) the variation of the stress vector with height is dependent on the wave-number decomposition of the underlying orography. For a general orographic profile having a continuous Fourier spectrum spanning an azimuthal angle of 180 degrees: (a) critical levels exist at every height in the atmosphere for any wind profile, and (b) the momentum-flux vector decreases in magnitude and turns with height.