Intense Downwelling and Diffuse Upwelling in a Nonlinear Ekman Layer

It has long been appreciated that Ekman transport and pumping velocities are modified through interactions with underlying geostrophic currents. Nonlinearity involving interaction of the Ekman flow with itself is, however, typically neglected. This nonlinearity occurs when the Rossby number based on the Ekman velocity and horizontal length scale approaches order one values. Such values are common, for example, in the ice-ocean stress field across sharp gradients such as leads in the sea ice cover. Recent work has shown strong asymmetry in the pumping velocities, with cyclonic forcing producing diffuse upwelling and anticyclonic forcing producing sharp downwelling fronts. To better understand this dynamics, we consider the steady response to a simple specified prescription of the stress. In the (x-z) plane perpendicular to the stress, dynamics are described by the 2-D Navier-Stokes equation, with a forcing term dependent on vertical shear of velocity in the y-hat direction, specified by a pressureless momentum equation. An expansion in an Ekman-velocity based Rossby number is used to solve the system and to better understand the asymmetry. Interactions with stratification and underlying geostrophic currents are also considered, and examples of where these effects might be important are given.