Similarity Theory Based on Dougherty-Ozmidov Scaling

We investigate local similarity theory for the stably stratified boundary layer that is based on the Brunt-Väisälä frequency and the dissipation rate of turbulent kinetic energy instead of on turbulent fluxes as used in the traditional Monin-Obukhov similarity theory (MOST). A buoyancy length scale constructed from these two variables was originally suggested by Dougherty (1961) and independently by Ozmidov (1965) and herein is referred to as the Dougherty-Ozmidov length scale. In oceanography, this length scale is known as the Ozmidov length and is widely used to describe small-scale turbulence. Based on dimensional analysis (Pi-theorem), the Brunt-Väisälä frequency, the dissipation rate of turbulent kinetic energy, and the buoyancy parameter can be considered as the governing variables ("Dougherty-Ozmidov scaling system") that define other variables in the stable atmospheric boundary layer at the height z. We show that any properly scaled statistics of the turbulence are universal functions of a stability parameter defined as the ratio of height z and the Dougherty-Ozmidov length scale; in the limit of z-less stratification, the Dougherty-Ozmidov length is linearly proportional to the Obukhov length. We also find that, in the framework of this approach, the non-dimensional turbulent viscosity is equal to the gradient Richardson number and the non-dimensional turbulent thermal diffusivity is equal to the flux Richardson number. These results are a consequence of the approximate local balance between production of turbulence by the mean flow and viscous dissipation. The proposed approach is equivalent to traditional MOST and its applicability in stable conditions is limited by inequalities, when both gradient and flux Richardson numbers are below a "critical value" about 0.20–0.25. Here, we use measurements of atmospheric turbulence made at five levels on a 20-m tower over the Arctic pack ice during the year-long Surface Heat Budget of the Arctic Ocean experiment (SHEBA) to examine the behavior of various similarity functions derived in the framework of Dougherty-Ozmidov scaling.