A simple description of the deepening and structure of a stably stratified flow driven by a surface stress

Abstract

The deepening and structure of a stably stratified flow driven by a surface stress are examined by means of a simple conceptual model, in which local mixing does not occur if the gradient Richardson number Ri is greater than the critical value Ric ≃ ¼ indicating onset of shear instability, and local mixing occurs instantaneously, through a simple exchange of fluid volumes, if Ri is less than Ric. The model leads to a mathematical formulation in which turbulent mixing is a gradient transport process just strong enough to maintain Ri at the critical value Ric throughout the boundary layer. Although unrealistic just beneath the stressed surface, self‐similar solutions based on this formulation plausibly describe the outer part of the boundary layer and the rate at which the layer deepens, provided that appropriately defined bulk Richardson and Reynolds numbers are sufficiently large. A comparison of model computations with existing laboratory measurements and turbulence closure simulations supports the conceptual model as an approximate representation of the processes controlling the deepening of the boundary layer and the qualitative structure in the outer part of the flow. Estimates are obtained of the region within the boundary layer and in parameter space where the formulation is realistic.