The evolution of a turbulent Ekman layer

Abstract

The evolution of shear flow at the sea surface, following the sudden application of wind stress, is modeled as a wall layer—outer layer combination of turbulent flow. The outer layer eddy viscosity is proportional to eddy length scale and hence to the penetration of the shear flow, so that it grows initially, then approaches a constant asymptote. The analytical solution for time‐dependent eddy viscosity has a structure very similar to the classical Ekman‐Fredholm solution for constant viscosity. In physical terms the spontaneous depth limitation of the shear layer at the surface of a rotating viscous fluid remains effective in turbulent flow even though the viscosity is not constant in space or time. When a sharp density interface limits the downward penetration of a surface shear layer at a ‘mixed’ layer depth h, the same rotating viscous fluid effect controls the shear across the interface and hence the entrainment rate. Therefore the equilibrium depth of a mixed layer remains proportional to u*/ƒ, as the shear layer in a homogeneous fluid (u* is the friction velocity; ƒ is the Coriolis parameter). A detailed model of the equilibrium mixed layer may be constructed by a combination of a free surface wall layer, an outer layer, and an interface wall layer. The behavior of this model is similar overall to the classical model of Rossby and Montgomery (1935) and leads to bulk formulae of identical structure.