The influence of boundary layer pumping on an externally forced, synoptic-scale flow is examined. The results follow earlier theories of stratified incompressible Boussinesq flow theories in that the spin-down time scale and the penetration depth of the influence of boundary layer pumping are inversely proportional to the stratification and proportional to the horizontal length scale of the flow. The present development is performed in isentropic coordinates to construct estimates applicable to the atmosphere. This analysis indicates that boundary layer pumping could be synoptically important in the lower troposphere under conditions of significant surface stress and tropospheric stratification.
The estimate of the stratified penetration depth scale is used to construct a simple homogeneous model to examine order-of-Rossby-number corrections to the quasi-geostrophic vorticity dynamics. Such corrections result from the influence of accelerations in both the free flow and boundary layer. It is found, for example, that vorticity adjustments due to various interactions between boundary layer pumping and accelerations are less important than predicted by scale analyses. Results are interpreted for the case of topographically forced flow.