A Quasi-Normal Scale Elimination (QNSE) Theory of Turbulent Flows with Stable Stratification

A new spectral model of turbulent flows with stable stratification is presented.
The model is derived in maximum proximity to first principles using the hypothesis of quasi-normality of turbulence stirring. The model explicitly resolves the stratification-induced disparity between the transport processes in the horizontal and vertical directions and accounts for the combined effect of turbulence and waves. The theory is based upon a mapping of the actual velocity field to a quasi-Gaussian field. The parameters of the mapping are calculated using a systematic process of successive averaging over small shells of velocity and temperature modes that eliminates them from the equations of motion. This approach offers a powerful mathematical tool for dealing with previously nearly intractable aspects of anisotropic turbulence; among these aspects are the threshold criterion for generation of internal waves and the modification of their dispersion relation by turbulence. The process of successive small scales elimination results in a model describing the largest scales of a flow. Partial scale elimination yields subgrid-scale viscosities and diffusivities that can be used in large eddy simulations. The elimination of all fluctuating scales results in RANS models. The model predicts various important characteristics of stably stratified flows, such as the dependence of the vertical turbulent Prandtl number on Froude and Richardson numbers, anisotropization of the flow filed, and decay of vertical diffusivity under strong stratification, all in good agreement with computational and observational data. The theory also yields analytical expressions for various 1D and 3D kinetic and potential energy spectra that reflect the effects of waves and anisotropy. The model's results are suitable for immediate use in practical applications and have been tested versus various data sets and in numerical weather prediction systems.