A Comprehensive Analysis of Nonlinear Corrections to the Classical Ekman Pumping

Abstract

The role of each nonlinear term in the vorticity and divergence equations derived from the primitive equations to the steady nonlinear Ekman pumping is examined for uni-directional flows with a small Rossby number comprehensively. For uni-directional flows with a constant horizontal shear, the perturbation analysis in the first order of a small Rossby number show that the contribution from the nonlinear divergence term in the vorticity equation to the Ekman pumping is the most dominant. That from the vertical advection term in the vorticity equation is the second dominant. The nonlinear terms from the divergence equation are less important for the nonlinear corrections to the classical Ekman pumping. In addition, the effect of the nonlinear divergence term to the Ekman pumping is opposite from that of the other nonlinear terms. The latter result also can be verified by qualitative discussions. Furthermore, the analytical solutions for constant horizontal shear flows can be utilized even for uni-directional flows with general horizontal shear. In this case, it is found that the contribution from the nonlinear divergence and horizontal advection terms in the vorticity equation to the Ekman pumping is the most dominant. That from the vertical advection and tilting terms in the vorticity equation is the second dominant. Some comments on the pre-existing approximate models of the nonlinear Ekman pumping are made.