## 1. Introduction

[2] Turbulent flows under conditions of stable density stratification occur ubiquitously in lakes, oceans and the atmospheres of earth and other planets. Accurate predictions of turbulent transport in a stably stratified fluid column are central to the analyses of heat transport, or buoyancy flux in general circulation models (GCMs). Such predictions are made difficult by the interplay between internal waves and turbulence which occur in a stably stratified fluid column. Complications arise in the partition when the value of the integral time scale of the turbulent flow approaches the buoyancy time scale of the fluid column because of the anisotropization of turbulence and generation of internal waves [*Sukoriansky and Galperin*, 2005; *Baumert and Peters*, 2005].

[3] Key to numerical predictions are turbulent diffusivities for scalar and momentum transport respectively defined as

where is the buoyancy flux, is the Reynolds shear stress. The ratio *K*_{m}/*K*_{ρ} is termed the turbulent Schmidt number Sc_{t} (rather than turbulent Prandtl number) as salinity gradients in water are used for stratification in the experiments. Prescription of a functional dependence of Sc_{t} on Ri = N^{2}/S^{2} is a turbulence closure scheme [*Kantha and Clayson*, 2000]. Here the buoyancy frequency *N* is defined by the gradient of density *ρ* as *N*^{2} = (−*g*/*ρ*_{o})(d*ρ*/d*z*), and g is the gravitational acceleration. *S* = d*U*/d*z* is the velocity shear. Analyses of field, lab and numerical (RANS, DNS and LES) data sets have led to the consensus that Sc_{t} increases with Ri [see *Esau and Grachev*, 2007, and references therein]. Accordingly, various forms have been suggested for Sc_{t} = f (Ri), for example, *Zilitinkevich et al.* [2007] gives:

where Sc_{o} is the value of the turbulent Schmidt number in the absence of density stratification (Sc_{o} ≈ 1) and C = 0.3. Other forms have been suggested [e.g., *Pacanowski and Philander*, 1981; *Mellor and Yamada*, 1982]. *Cane* [1993] noted that comparison of the closure scheme of *Pacanowski and Philander* [1981] with oceanic thermocline data gave insufficient mixing at low values of Ri and too much at high values of Ri. The lack of agreement arises because Sc_{t }is difficult to resolve when the magnitudes of the vertical momentum flux or density gradients are small [*Esau and Grachev*, 2007]. These limitations arise for both conditions of strong stability (Ri → ∞) and weak stability (Ri → 0). Difficulties are compounded as there are few studies with direct measurements of the buoyancy flux and the diffusivity *K*_{ρ} is typically inferred from gradient profiles of the mean density.

[4] There is renewed interest in the precise algebraic form of this dependence following the demonstration by *Noh et al.* [2005] that numerical simulations in their GCM of the equatorial mixed layer were more realistic with the inclusion of a Sc_{t }dependence on stratification (i.e., Ri). Recent analyses of field data of the stable atmospheric boundary layer from polar regions [*Yague et al.*, 2001; *Esau and Grachev*, 2007] also showed an additional feature namely that Pr_{t}(=Sc_{t}) values varied by more than an order magnitude at any given value of Ri. Similar spread of Sc_{t} values is evident in oceanic data [e.g., *Peters et al.*, 1988]. The reason for this spread or scatter is not currently known, and a unique functional dependence between Sc_{t} and Ri remains elusive. An emerging problem in the interpretation of data of Sc_{t} and Ri is the recognition of shared variables (density and velocity gradients) in plots of Sc_{t} versus Ri as this leads to self-correlation [*Klipp and Mahrt*, 2004; *Grachev et al.*, 2007]. The purpose of this paper is not to evaluate the effects of self-correlation but rather to emphasize other influences on Sc_{t}. In particular, we examine the role of advection on Sc_{t}. We undertook lab experiments to examine the form of the potential functional dependence as well as to investigate the large scatter in the values of Sc_{t}. The analysis described below reveals that Sc_{t} is not just a function of Ri but also dependent on an additional non-dimensional parameter T*, i.e.,

where T* is the ratio of advective time scale to the eddy turnover time scale with explicit definitions below.