: Collapses in the oceanic Ekman boundary layer

Mixing in the uppermost part of the water column is crucial for modelling air-sea interaction, yet it remains poorly understood, especially the processes under strong wind conditions. The Ekman boundary layers are a salient feature of the air-sea interface. In the the Ekman boundary layers the current velocity vectors always rotates, making two components of the basic flow vorticity comparable and, thus, the boundary layer three dimensional. Linear instabilities of the homogeneous steady Ekman layers were examined and  found to occur for sufficiently large turbulent Reynolds numbers.  Here, we derive a model of  nonlinear instabilities of 3d Ekman layer  in deep ocean taking into account also a possible weak stratification of the boundary layer caused by air entrainment due to wave breaking or solar heating. The model exploits the observation that the corresponding linearized boundary value problem  always supports a “vorticity wave” mode which is often decaying. Employing an asymptotic procedure utilizing smallness of the boundary layer thickness to the characteristic wavelength of perturbations  scaled as  inverse Reynolds number squared we derive a novel nonlinear evolution equation with a pseudo-differential dispersion.  We take into  account viscosity and weak stratification  in the boundary layer. Within the framework of this equation a  wide class of initial conditions, which we a priori specify, leads to `collapses’ of localized perturbations, that is an initial perturbation becomes more and more localised and its amplitude becomes infinite in finite time forming a point singularity. We derived a self-similar solution describing these collapses. The mechanism of collapse is essentially nonlinear. A new insight into linear instabilities has been also  obtained.  The collapses are expected to result in intense mixing and even temporary destruction of the boundary layer.