An averaging method for nonlinear laminar Ekman layers

We study steady laminar Ekman boundary layers in rotating systems using an averaging method similar to the technique of von Kármán and Pohlhausen. The method allows us to explore nonlinear corrections to the standard Ekman theory even at large Rossby numbers. We consider both the standard self-similar ansatz for the velocity profile, which assumes that a single length scale describes the boundary layer structure, and a new non-self-similar ansatz in which the decay and the oscillations of the boundary layer are described by two different length scales. For both profiles we calculate the up-flow in a vortex core in solid-body rotation analytically. We compare the quantitative predictions of the model with the family of exact similarity solutions due to von Kármán and find that the results for the non-self-similar profile are in almost perfect quantitative agreement with the exact solutions and that it performs markedly better than the self-similar profile.