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

Abstract

In this study, the stability dependence of turbulent Prandtl number ($$Pr_t$$ P r 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 $$Pr_t$$ P r 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.

Revisiting and revising Tatarskii's formulation for the temperature structure parameter ($C_T^2$) in atmospheric flows

“…The following text has been taken verbatim from Section 14 (page 73) of [30]: "A gradient of mean temperatures will result in a systematic temperature difference between any two points at different heights. This temperature difference ∆θ is approximately given by ∆θ ≈ Γ∆z, and its square is (∆θ) 2 ≈ Γ 2 (∆z) 2 . There is also a random temperature difference between these points, whose mean square value is C 2 T (∆z) 2/3 .…”

“…In a recent study [2], we have derived closed-form solutions for a characteristic length scale (denoted as L X ) and turbulent Prandtl number (P r t ) based on the conventional budget equations. For brevity, we only report the key findings from [2] which are directly relevant for the present paper. In [2], we used Eqs.…”

“…In [2], the dissipation rates of TKE (e) and variance of temperature (σ 2 θ ) are derived to be respectively as follows:…”

“…However, there is ample evidence in the fluid dynamics and atmospheric turbulence literature that P r t monotonically increases with increasing values of Ri g . In this context, a physically-based formulation was derived by [2].…”

Revisiting and revising Tatarskii’s formulation for the temperature structure parameter ($$C_T^2$$) in atmospheric flows

“…According to Eq. ( 3), if L 0 and can be reliably measured, estimated or prescribed, then C 2 T values can be computed directly. Since it is relatively straightforward to measure or estimate in the atmosphere, over the past decades, several studies focused on the estimation of L 0 .…”