Momentum transport by boundary-layer turbulence causes a weak
synoptic-scale vertical motion. The classical textbook solution for the
strength of this Ekman pumping depends on the curl of the surface
momentum flux. A new solution for Ekman pumping is derived in terms of
the curl of the geostrophic wind and a term that depends in a
non-trivial way on the vertical profile of the turbulent momentum flux.
The solution is confined to a boundary-layer regime that is vertically
well mixed and horizontally homogeneous. The momentum flux is computed
from a commonly used bulk surface drag formula and a flux-jump relation
to capture the entrainment flux of momentum at the top of the boundary
layer. It is found that the strength of Ekman pumping is bounded. The
weakening of Ekman pumping for enhanced turbulent surface friction can
be explained from the fact that it will reduce the magnitude of the
horizontal wind. It is demonstrated that entrainment of momentum across
the top of the boundary layer tends to diminish the large-scale
divergence of the wind. As momentum transport is parameterized in
large-scale models, the analysis is relevant for the understanding and
interpretation of the evolution of synoptic-scale vertical motions as
predicted by such models.