A note on nonlinear corrections to the Ekman layer pumping velocity

High order corrections to the Ekman pumping formula are derived analytically by a regular perturbation expansion in the external flow Rossby number. General shear lines V(x) and general circular vortices V(r) are considered. The results extend previous calculations based on the assumption of constant vorticity in the exterior fields. At each order in the expansion, higher derivatives of the basic external flow are introduced, thus rendering the pumping more and more nonlocal as additional orders in Rossby number are included. For general profiles of exterior velocity, the constant vorticity similarity and perturbation solutions may be corrected using the formulas presented here.