An approximate analytical solution for the baroclinic and variable eddy diffusivity semi-geostrophic Ekman boundary layer

The WKB method has been used to develop an approximate solution of the semi-geostrophic Ekman boundary layer with height-dependent eddy viscosity and a baroclinic pressure field. The approximate solution retains the same simple form as the classical Ekman solution. Behaviours of the approximate solution are discussed for different eddy viscosity and the pressure systems. These features show that wind structure in the semi-geostrophic Ekman boundary layer depends on the interaction between the inertial acceleration, variable eddy viscosity and baroclinic pressure gradient. Anticyclonic shear has an acceleration effect on the air motion in the boundary layer, while cyclonic shear has a deceleration effect. Decreasing pressure gradient with height results in a super-geostrophic peak in the wind speed profile, however the increasing pressure gradient with height may remove the peak. Anticyclonic shear and decreasing the variable eddy viscosity with height has an enhanced effect on the peak. Variable eddy viscosity and inertial acceleration has an important role in the divergence and vorticity in the boundary layer and the vertical motion at the top of the boundary layer that is called Ekman pumping. Compared to the constant eddy viscosity case, the variable eddy diffusivity reduces the absolute value of Ekman pumping, especially in the case of eddy viscosity initially increasing with height. The difference in the Ekman pumping produced by different eddy diffusivity assumptions is intensified in anticyclonic flow and reduced in cyclonic flow.