Deepening of the wind-mixed layer: A model of the vertical structure

Abstract

A new non-linear model of mixing and convection based upon a modelling by two buoyant interacting fluids is applied to the deepening of the wind-mixed layer. In view of simple algebra, the model is one-dimensional, frictionless, and neglects the turbulence production by the mean-flow shear in the thermocline. The potential-energy increase required for deepening is thus supplied by the turbulence input at the surface (turbulent erosion model). A non-similar analytical solution is found in the case of a well-mixed layer bounded below by a sharp thermocline, treated as a boundary layer. This solution is valid if the frictional Richardson number, Ri, defined as the ratio of the total mixed-layer buoyancy to the friction-velocity squared, is much greater than unity. The model predicts an entrainment rate proportional to Ri^-1, and a ratio of thermocline thickness to mixed-layer depth of the order of Ri^-3/4. The thermocline shallows as h^-1/2, as the mixed-layer depth, h, increases with time. The vertical structure throughout the mixed layer and thermocline is given by the solution, and vertical profiles of mean values and vertical fluxes are plotted. These profiles are comparable to those obtained by turbulence-closure numerical models.