## 1 Introduction

[2] The viscous dissipation rate of turbulence kinetic energy, *ε*, is an important property of turbulent flows. It physically represents the conversion of kinetic energy into thermal energy due to viscous forces. The dissipation rate of turbulence kinetic energy is frequently used to characterize the dynamics of turbulence, e.g., in connection with energy transfer across the inertial subrange and for length and time scale characterization of the flow.

[3] Atmospheric flows are usually affected by stratification due to temperature variations, which significantly alters the intensity and structure of the flow field. In stably stratified flows, the buoyancy force tends to reduce turbulence intensities and the associated mixing, whereas in unstable flows, buoyancy acts to enhance turbulence by increasing the vertical momentum exchange.

[4] Because of the directional preference of buoyancy, it is clear that the energetic scales of motion in a stratified, turbulent flow are anisotropic. Ever since *Kolmogorov* [1941], it has been argued that, at least for high Reynolds numbers, directional information is lost as energy propagates from the large to the small scales of turbulence across the inertial subrange. Hence, anisotropy on the large scales will not result in small-scale anisotropy. However, for the last decades, evidence against this belief has grown steadily, as discussed in more detail by, e.g., *Wyngaard* [2010, p. 319].

[5] In the context of stratified shear flow, many authors [e.g., *Yamazaki and Osborn*, 1990; *Thoroddsen and Van Atta*, 1992; *Smyth and Moum*, 2000] currently hold the view that there exists a certain buoyancy Reynolds number, defined as Re_{b}=*ε*/*ν**N*^{2}, where N is the Brunt-Väisälä frequency, below which some degree of anisotropy prevails even at the dissipative scales. *Reif and Andreassen* [2003] have theoretically shown that the concept of local isotropy is formally inconsistent with the Navier-Stokes equations in homogeneously sheared turbulent flows affected by stable stratification.

[6] In the laboratory setups and atmospheric field trials, measurements of fluctuating velocity gradients are limited to a few components only. Direct access to the dissipation rate is therefore only available in direct numerical simulations (DNS) of canonical flow problems. Although the DNS approach is limited to fairly low Reynolds numbers, it nevertheless has become a very valuable research tool in order to elucidate the physical characteristics of turbulent fluid motion. In order to provide reasonable estimates of the rate of viscous dissipation in real-life flows, empirical models based on measurable quantities are therefore needed.

[7] Dissipation rate models can be grouped into two different categories; algebraic and integral models. The latter kind of estimates are based on, for instance, spectra or structure functions [e.g., *Limbacher*, 2010; *Xu and Chen*, 2012]. The present study is concerned with the former class of models.

[8] One of the most commonly used algebraic formulas is based on the assumption of isotropic turbulence [*Taylor*, 1935]. This enables the dissipation rate to be computed using only one (out of nine) velocity gradient correlations. One inherent limitation of these models is, however, that they do not depend on the temperature field which becomes dynamically important in stratified turbulence.

[9] As a consequence of the anisotropic nature of stratified turbulence, *Thoroddsen and Van Atta* [1992] suggested that more refined dissipation rate estimates should utilize the Brunt-Väisälä frequency in order to account for stratification. Crude estimates of this kind had already been discussed by *Weinstock* [1981]. Another way of implicitly allowing for stratification is to adopt the assumption of local axisymmetry [*Batchelor*, 1946], which is the basis for models such as that by *George and Hussein* [1991]. Although these models are based on a sound physical basis, they require too many components of the dissipation rate, i.e., 〈*∂*_{k}*u*_{i}*∂*_{k}*u*_{i}〉, to be measured, which makes these models a less desirable choice in real-life flows.

[10] A DNS of a Kelvin-Helmholtz instability in a stably stratified flow was reported by *Werne et al.* [2005]. This statistically unsteady flow simulation provides turbulence statistics which can be used to calculate both dissipation rate estimates and the true value of the dissipation rate. In the present paper, a new model for the dissipation rate in a stably stratified environment is proposed and compared to two existing algebraic turbulence dissipation rate models, and the DNS data of *Werne et al.* [2005].