An approximate formula for the vertical flux of horizontal momentum (momentum flux for brevity) generated by topography is derived. A steady non-rotating hydrostatic 3-dimensional linear problem is considered. The height of topography is a function of both horizontal coordinates, and its environmental flow has vertical shears of both magnitude and direction. Because of the directional shear, critical levels are present continuously in the vertical direction. In the presence of the critical levels, the momentum flux vector is azimuthally filtered continuously in the vertical direction, and becomes a function of the vertical coordinate. The formula shows that, except for the azimuthal filtering, the momentum flux vector is approximately the same as that for a uniform environmental flow, whose velocity is equal to the original environmental flow velocity at the ground.
Vertical velocity induced by a meso-scale straining flow in the free atmosphere in the presence of a viscous boundary layer (i.e., Ekman pumping) is analytically investigated. Meso-scale in this note means that the Rossby number Ro is less (but not extremely less) than unity. Assuming that the horizontal velocity in the free atmosphere is nondivergent at the lower boundary, a meso-scale formula of Ekman pumping is derived. In this formula, the induced vertical velocity is expressed in terms only of the horizontal velocity and its horizontal derivatives at the lower boundary of the free atmosphere, and other terms (e.g., geostrophic velocity) do not enter the formula. In order to derive the formula, an inviscidly balanced approximation is made. This approximation means a replacement of the horizontal velocity in momentum equation, which is materially differentiated, by the velocity of the free atmosphere. In this approximation, the momentum equation in the free atmosphere remains primitive. The derived formula shows, e.g., the following. For a horizontally uniform straining flow, the induced vertical velocity is less than that in Ekman's formula. For a purely deformational flow, the induced vertical velocity becomes negative.