Turbulent Prandtl number and characteristic length scales in stably stratified flows: steady-state analytical solutions

TL;DR

This paper develops an analytical model to quantify how the turbulent Prandtl number varies with stability in stratified flows, relating it to length scales and the Richardson number, and validates predictions against existing data.

Contribution

It introduces a novel hybrid length scale formulation and an explicit relationship between turbulent Prandtl number and Richardson number based on variance and flux budgets.

Findings

Good agreement with existing datasets and formulations

Derived explicit relationship between Pr_t and Richardson number

Theoretical predictions for turbulence variables align with observations

Abstract

In this study, the stability dependence of turbulent Prandtl number ($Pr_t$) is quantified via a novel and simple analytical approach. Based on the variance and flux budget equations, a hybrid length scale formulation is first proposed and its functional relationships to well-known length scales are established. Next, the ratios of these length scales are utilized to derive an explicit relationship between $Pr_t$ and gradient Richardson number. In addition, theoretical predictions are made for several key turbulence variables (e.g., dissipation rates, normalized fluxes). The results from our proposed approach are compared against other competing formulations as well as published datasets. Overall, the agreement between the different approaches is rather good despite their different theoretical foundations and assumptions.