Collapses in weakly stratified no-stress boundary layers

The physical processes controlling mixing in the uppermost layer of the ocean are still poorly understood. Although this part of the water column is often weakly stratified due to direct solar heating or air bubble entrainment, the possible role of weak stratification has never been considered and was a priori generally believed to be negligible. Here, aiming at getting an insight into mixing mechanisms in the oceanic surface boundary layer we develop a weakly-nonlinear asymptotic model of nonlinear dynamics of linearly decaying three-dimensional long-wave perturbations in a generic no-stress weakly stratified boundary-layer flow. 

The perturbation evolution is shown to be described by a novel generalization of the essentially two-dimensional Benjamin-Ono equation modified by the explicit account of linear viscous decay and weak stratification. Within the framework of the new evolution equation an initial localized perturbation (`lump') of any given shape collapses, i.e., forms a point singularity in finite time, provided its initial amplitude exceeds a certain threshold specific for each particular initial shape, the Reynolds number, stratification, and the curvature of the basic flow vorticity at the boundary. 

For a broad range of Reynolds numbers where the linear decay is negligible the sufficient criterion for collapse has been found in a compact analytical form. The system has two attractors: the collapse and unperturbed flow; the initial perturbations exceeding the threshold collapse in a self-similar manner, while the perturbations with the amplitudes below the threshold - decay. The weak stratification and decay strongly affect the threshold. Although weak stratification does not affect linear stability properties of the flow, it might raise the threshold beyond the range of validity of weakly nonlinear model. We find the self-similar solution describing the collapses in the vicinity of the singularity. 

The axially symmetric solution is universal: it does not depend on the account of stratification and linear decay. Collapses are suggested as a mechanism resulting in coherent three-dimensional coherent structures and enhancement of mixing in linearly stable boundary layers.

Venue: Fulton Building, Room G20