Similarity theory based on the Dougherty–Ozmidov length scale

Abstract

Local similarity theory is suggested based on the Brunt-Vaisala frequency and the dissipation rate of turbulent kinetic energy instead the turbulent fluxes used in the traditional Monin-Obukhov similarity theory. Based on dimensional analysis (Pi theorem), it is shown that any properly scaled statistics of the small-scale turbulence are universal functions of a stability parameter defined as the ratio of a reference height z and the Dougherty-Ozmidov length scale which in the limit of z-less stratification is linearly proportional to the Obukhov length scale. Measurements of atmospheric turbulence made at five levels on a 20-m tower over the Arctic pack ice during the Surface Heat Budget of the Arctic Ocean experiment (SHEBA) are used to examine the behaviour of different similarity functions in the stable boundary layer. It is found that in the framework of this approach the non-dimensional turbulent viscosity is equal to the gradient Richardson number whereas 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 shear and viscous dissipation. The turbulence framework based on the Brunt-Vaisala frequency and the dissipation rate of turbulent kinetic energy may have practical advantages for estimating turbulence when the fluxes are not directly available.Comment: Manuscript submitted to Quarterly Journal of the Royal Meteorological Society (04 April 2014

A Model for Turbulence Spectra in the Equilibrium Range of the Stable Atmospheric Boundary Layer

“…Because of this stratification (Bolgiano, ; Dougherty, ; Lumley, ; Phillips, ; Weinstock, ), the spectra deviate from the scaling of isotropic turbulence. It is worth noting that the Dougherty‐Ozmidov scale (Dougherty, ; Ozmidov, ) has been suggested (Gargett et al, ; Grachev et al, ; Lang & Waite, ; Li et al, ; Waite, ) to characterize the largest scale of isotropic turbulence under stratification, below which the isotropic scaling is still expected.…”

Aviation Turbulence

“…The fact that the Monin-Obukhov similarity scales are based on fluxes prevents applications of the theory in higher regions of the atmosphere, even within the local formulation of Nieuwstadt (1984). To overcome these difficulties, an alternate conceptual approach of the gradientbased similarity was introduced by Sorbjan (2008a, b), Sorbjan (2010), and subsequently elaborated by Sorbjan and Grachev (2010), Sorbjan (2012), Sorbjan and Czerwinska (2013), Grachev et al (2015), Sorbjan (2014), andSorbjan (2016). The approach was primarily developed for the shear-dominated, stable atmospheric boundary layer, but could also be applied in other stably-stratified turbulent flows.…”

“…One might notice a scatter of observational points within each bin, related to multi-level observations. Details of the turbulent and profile measurements, accuracy, calibration, data processing, and data-quality criteria for the SHEBA dataset can be found in the papers of Grachev et al (2005Grachev et al ( , 2013Grachev et al ( , 2015 and references therein.…”

“…The following three scales can be proposed within the ε-based scaling (Sorbjan and Balsley 2009;Grachev et al 2015;Sorbjan 2016):…”

“…21.4a, b. The data points in the figures were obtained by Grachev et al (2015), based on the observations made during the SHEBA experiment. The authors imposed several methodological restrictions of observations, based on the values of the Richardson numbers Ri and Rf.…”