A bound on Ekman pumping

Abstract

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 nontrivial 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.Plain Language Summary Turbulence acts as a friction to the wind. This drives a net flow of air from high to low pressure. The resulting accumulation of mass drives an upward vertical motion in a low-pressure system and vice versa in a high-pressure system. This study shows that the vertical velocity caused by turbulent friction has a maximum value.