2.1 SGS parameterization

Common SGS parameterizations in physical space are the Smagorinsky (Reference Smagorinsky1963) and the dynamic Smagorinsky (Germano et al. Reference Germano, Piomelli, Moin and Cabot1991) models, in which the SGS momentum stress $\unicode[STIX]{x1D70F}_{ij}$ and density flux $h_{j}$ are related to the resolved rate-of-strain tensor $\bar{\unicode[STIX]{x1D634}}_{ij}$ and density gradient $\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}/\unicode[STIX]{x2202}x_{j}$ using the eddy viscosity $\unicode[STIX]{x1D708}_{r}$ and diffusivity $\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D705}}$ coefficients, respectively, given by

(2.7a-d ) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{ij}=2\unicode[STIX]{x1D708}_{r}\bar{\unicode[STIX]{x1D634}}_{ij},\quad h_{j}=\unicode[STIX]{x1D708}_{\unicode[STIX]{x1D705}}\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}x_{j}},\quad \unicode[STIX]{x1D708}_{r}=c_{s}\unicode[STIX]{x1D6E5}^{2}(2\bar{\unicode[STIX]{x1D634}}_{ij}\bar{\unicode[STIX]{x1D634}}_{ij})^{1/2},\quad \unicode[STIX]{x1D708}_{\unicode[STIX]{x1D705}}=\frac{\unicode[STIX]{x1D708}_{r}}{Pr_{t}},\end{eqnarray}$$

where $c_{s}$ , $\unicode[STIX]{x1D6E5}$ and $Pr_{t}$ are the Smagorinsky coefficient, grid spacing and turbulent Prandtl number, respectively. In the classical Smagorinsky (Reference Smagorinsky1963) model, $c_{s}=(0.17)^{2}$ , as is proposed by Lilly (Reference Lilly1967). In the dynamic Smagorinsky model, however, $c_{s}$ is not constant and is a time- and space-dependent variable, which is calculated dynamically by employing a coarse-filter scale $\tilde{\unicode[STIX]{x1D6E5}}>\unicode[STIX]{x1D6E5}$ (see e.g. Khani & Waite (Reference Khani and Waite2015), for more information about $c_{s}$ calculations). Similarly, we can propose a turbulent-viscosity model in wavenumber (Fourier) space, for which the spectral SGS kinetic and potential energy transfers are related to the kinetic and potential energy spectra at the cutoff wavenumber $k_{c}$ using spectral eddy-viscosity and diffusivity coefficients, respectively (see e.g. Kraichnan (Reference Kraichnan1976), for more information about the Kraichnan eddy-viscosity model).

4.1 Kinetic energy and SGS dissipation spectra

Figure 1 shows the time-averaged compensated horizontal wavenumber kinetic energy spectra for hyperviscosity simulations, DNS and LES, where the horizontal energy spectra are multiplied by $k_{h}^{5/3}\langle \unicode[STIX]{x1D716}\rangle ^{-2/3}$ . The horizontal axes are normalized by the buoyancy wavenumber $k_{b}$ , for which in each panel $k_{h}/k_{b}=1$ shows the location of the buoyancy wavenumber. Constant horizontal compensated spectra in the intermediate horizontal wavenumbers imply a $-5/3$ power law in the inertial subrange of the horizontal wavenumber kinetic energy spectra that is in line with the Lindborg (Reference Lindborg2006) hypothesis for stratified turbulence. For DNS, LES and hyperviscosity simulations, the $-5/3$ spectral slope is seen over the horizontal wavenumbers $0.5\lesssim k_{h}/k_{b}\lesssim 1$ when $N=0.5$ , $0.3\lesssim k_{h}/k_{b}\lesssim 0.8$ when $N=1$ , $0.2\lesssim k_{h}/k_{b}\lesssim 0.9$ when $N=2$ , and $0.1\lesssim k_{h}/k_{b}\lesssim 0.8$ when $N=4$ (see figure 1 a–d). Also, these constant compensated spectra are followed by a bump (or shallowness) around $k_{h}\sim k_{b}$ that indicates a non-local energy transfer from larger scales to the buoyancy scale $L_{b}=2\unicode[STIX]{x03C0}/k_{b}$ (consistent with hyperviscosity simulations, DNS and LES of Waite (Reference Waite2011), Khani & Waite (Reference Khani and Waite2013, Reference Khani and Waite2014, Reference Khani and Waite2015)). These bumps are seen around $k_{b}\approx 8$ , $16$ , $30$ and $60$ , when $N=0.5$ , $1$ , $2$ and $4$ (see the vertical arrows in figure 1, indicating the location of the buoyancy wavenumber $k_{b}$ ). It is clear that the bump (or shallowness) occurs at $k_{h}/k_{b}\approx 1$ in all panels of figure 1. For example, for the dynamic Smagorinsky simulation with $n=512$ and $N=2$ , the bump (or shallowness) happens around $k_{h}\approx 30$ that is very close to $k_{b}=29.9$ (see table 1 and the dash-double-dotted red line in figure 1 c). Overall, these results show that the LES runs capture the fundamental characteristics of stratified turbulence that are in line with DNS and hyperviscosity simulations.

The horizontal and vertical SGS energy transfer spectra (eddy dissipation spectra) are shown in figure 2 (see appendix A, and also Khani & Waite (Reference Khani and Waite2014, Reference Khani and Waite2015), for calculations of the SGS eddy dissipation spectra). Interestingly, eddy dissipation spectra peak at large horizontal and small vertical scales, consistent with anisotropic dissipation in stratified turbulence (in line with effective eddy-viscosity results of Khani & Waite (Reference Khani and Waite2013), Remmler & Hickel (Reference Remmler and Hickel2014), Khani & Waite (Reference Khani and Waite2016)). The horizontal wavenumber eddy dissipation spectrum for the Kraichnan LES with $n=768$ and $N=2$ has a cusp at $k_{h}\approx k_{c}$ because the Kraichnan eddy-viscosity model increases near $k_{c}$ (see e.g. Kraichnan Reference Kraichnan1976; Métais & Lesieur Reference Métais and Lesieur1992; Khani & Waite Reference Khani and Waite2013). Increased stratification or decreased resolution increases the eddy dissipation by squeezing the buoyancy-scale thickness (vertical layer thickness) towards the dissipation scale or by increasing the dissipation scale (as seen in hyperviscosity simulations and DNS of Waite & Bartello (Reference Waite and Bartello2004), Bartello & Tobias (Reference Bartello and Tobias2013)). In addition, this anisotropic picture of eddy dissipation spectra is in agreement with the effective eddy dissipation spectra when a test-filter scale $\unicode[STIX]{x1D6E5}_{test}=\unicode[STIX]{x03C0}/k_{c}$ is employed around the buoyancy scale $L_{b}$ in DNS results (see below in figure 8).

Figure 2.

The (a) horizontal and (b) vertical wavenumber SGS eddy dissipation spectra for LES. These spectra are calculated at $t=450$ , and are multiplied by wavenumber to preserve area on the log-linear axes.

4.2 Length scales

We can calculate the vertical and horizontal length scales using the corresponding averaged vertical and horizontal energy spectra in DNS, LES and hyperviscosity simulations of stratified turbulence as follows,

(4.1a,b ) $$\begin{eqnarray}{\mathcal{L}}_{v}=\frac{2\unicode[STIX]{x03C0}\displaystyle \int _{0}^{\infty }\langle E(k_{v})\rangle \,\text{d}k_{v}}{\displaystyle \int _{0}^{\infty }k_{v}\langle E(k_{v})\rangle \,\text{d}k_{v}},\quad {\mathcal{L}}_{h}=\frac{2\unicode[STIX]{x03C0}\displaystyle \int _{0}^{\infty }\langle E(k_{h})\rangle \,\text{d}k_{h}}{\displaystyle \int _{0}^{\infty }k_{h}\langle E(k_{h})\rangle \,\text{d}k_{h}},\end{eqnarray}$$

where $\langle E\rangle$ , $k_{h}$ and $k_{v}$ are the averaged directional resolved energy spectrum, and the horizontal and vertical wavenumbers, respectively. Figure 3 shows the ratio ${\mathcal{L}}_{v}/L_{b}$ versus $L_{b}/\unicode[STIX]{x1D6E5}$ , which is an indication of how sufficiently the buoyancy scale $L_{b}$ is resolved in the LES of stratified turbulence. For comparison, the results from hyperviscosity simulations and DNS are also shown. Khani & Waite (Reference Khani and Waite2014, Reference Khani and Waite2015) have shown that there is a threshold on the grid spacing $\unicode[STIX]{x1D6E5}$ , for which LES of stratified turbulence captures the fundamental characteristics of stratified turbulence. This threshold depends on $L_{b}$ and varies with different SGS models: $L_{b}/\unicode[STIX]{x1D6E5}>4.17$ for the dynamic Smagorinsky model, $L_{b}/\unicode[STIX]{x1D6E5}>5.89$ for Smagorinsky model and $L_{b}/\unicode[STIX]{x1D6E5}>2.13$ for the Kraichnan eddy-viscosity model (see Khani & Waite (Reference Khani and Waite2014, Reference Khani and Waite2015), these thresholds are also shown with vertical dashed lines in figure 3). It is clear that the ratio ${\mathcal{L}}_{v}/L_{b}$ approaches to $1$ when $L_{b}$ is well resolved in the LES, or when the buoyancy Reynolds number $Re_{b}=\langle \unicode[STIX]{x1D716}\rangle /(\unicode[STIX]{x1D708}N^{2})$ is sufficiently large in the DNS (note that $\unicode[STIX]{x1D6E5}\propto L_{d}$ in DNS, so the ratio $L_{b}/\unicode[STIX]{x1D6E5}$ is a measure of the size of the sub-buoyancy-scale inertial range; see figure 3, and also figure 6 in below). This trend is in line with an important feature of stratified turbulence, in which the vertical length scale is scaled by the buoyancy scale $L_{b}$ (i.e. ${\mathcal{L}}_{v}\sim L_{b}$ , see e.g. Waite Reference Waite2011; Khani & Waite Reference Khani and Waite2013).

Figure 3.

The ratio ${\mathcal{L}}_{v}/L_{b}$ versus $L_{b}/\unicode[STIX]{x1D6E5}$ for DNS, LES and hyperviscosity simulations in stratified turbulence. The dynamic Smagorinsky, regular Smagorinsky and Kraichnan SGS models are considered in the LES runs. Vertical dashed lines (from left to right for the Kraichnan, dynamic Smagorinsky and Smagorinsky parameterizations, respectively) represent the threshold on $L_{b}/\unicode[STIX]{x1D6E5}$ , above which LES could capture fundamental characteristics of stratified turbulence (see e.g. Khani & Waite Reference Khani and Waite2014, Reference Khani and Waite2015). In DNS runs, $\unicode[STIX]{x1D6E5}\propto L_{d}$ , so the ratio $L_{b}/\unicode[STIX]{x1D6E5}$ is a measure of the size of the sub-buoyancy-scale inertial range.

Therefore, we can simplify the equation for vertical Froude number $Fr_{v}=u_{rms}/(N{\mathcal{L}}_{v})$ as follows

(4.2) $$\begin{eqnarray}Fr_{v}\sim \frac{u_{rms}}{NL_{b}}\sim \frac{L_{b}}{L_{b}}\sim 1,\end{eqnarray}$$

when $L_{b}/\unicode[STIX]{x1D6E5}\gg 1$ in the LES of stratified turbulence. On the other hand, when $L_{b}/\unicode[STIX]{x1D6E5}\lesssim 1$ , ${\mathcal{L}}_{v}\gg L_{b}$ , and so

(4.3) $$\begin{eqnarray}Fr_{v}=\frac{u_{rms}}{N{\mathcal{L}}_{v}}\sim \frac{L_{b}}{{\mathcal{L}}_{v}}\ll 1,\end{eqnarray}$$

in the LES of stratified turbulence, as seen in figure 3 (consistent with the behaviour of viscosity-affected stratified turbulence when $Re_{b}\lesssim 1$ , as seen in DNS run with $n=256$ and $N=2$ ; also see Brethouwer et al. Reference Brethouwer, Billant, Lindborg and Chomaz2007). In the next section, we will show that these two different vertical Froude number regimes in (4.2) and (4.3) have a critical influence on the scale analysis of the SGS mixing efficiency in LES of stratified turbulence.

4.3 Mixing efficiency

According to equations (2.4)–(2.7a-d ), we can write the irreversible SGS mixing efficiency in LES as follows

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{i}=\frac{\langle (\unicode[STIX]{x1D6FD}^{2}/Pr_{t})(\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}/\unicode[STIX]{x2202}x_{j})(\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}/\unicode[STIX]{x2202}x_{j})\rangle }{\langle 2\bar{\unicode[STIX]{x1D634}}_{ij}\bar{\unicode[STIX]{x1D634}}_{ij}\rangle +\langle (\unicode[STIX]{x1D6FD}^{2}/Pr_{t})(\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}/\unicode[STIX]{x2202}x_{j})(\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}/\unicode[STIX]{x2202}x_{j})\rangle }.\end{eqnarray}$$

Next, we are interested in performing scale analysis on $\unicode[STIX]{x1D6FE}_{i}$ by using the scale of resolved variables in the right-hand side of equation (4.4). If we assume ${\mathcal{L}}_{v}\ll {\mathcal{L}}_{h}$ , which is a reasonable assumption in stratified turbulence (see table 1), we can write (following to Khani & Porté-Agel Reference Khani and Porté-Agel2017)

(4.5a,b ) $$\begin{eqnarray}\frac{g}{\unicode[STIX]{x1D70C}_{0}}\frac{\unicode[STIX]{x2202}\bar{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x2202}x_{j}}\sim N^{2},\quad \bar{\unicode[STIX]{x1D634}}_{ij}\sim \frac{u_{rms}}{{\mathcal{L}}_{v}},\end{eqnarray}$$

and so,

(4.6) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{i}\sim \frac{N^{2}/Pr_{t}}{2(u_{rms}^{2}/{\mathcal{L}}_{v}^{2})+N^{2}/Pr_{t}}\sim \frac{1}{2Pr_{t}Fr_{v}^{2}+1}.\end{eqnarray}$$

Moreover, in the stratified turbulence regime $Fr_{v}\sim 1$ (as is given by (4.2), also see e.g. Billant & Chomaz Reference Billant and Chomaz2001; Waite & Bartello Reference Waite and Bartello2004; Lindborg Reference Lindborg2006; Waite Reference Waite2011; Khani & Waite Reference Khani and Waite2013; Khani Reference Khani2015), which results in

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{i}\sim \frac{1}{2Pr_{t}+1}.\end{eqnarray}$$

This scaling theory is supported by simulations at different turbulent Prandtl numbers $Pr_{t}$ , which are discussed below in § 4.6. We can write an equality relation for the scaling theory (4.7) as follows

(4.8) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{i}=\frac{c}{2Pr_{t}+1},\end{eqnarray}$$

where $c$ is a proportionality constant, and we may estimate it using LES data. We have employed the least-squares method to measure $c$ using our LES runs, where the turbulent Prandtl number $Pr_{t}$ varies from $0.2$ to $2$ . It is shown that the estimated value for the coefficient $c$ depends on the SGS scheme and grid spacing, but is fairly close to $1$ (see appendix B). Therefore, we set $c\approx 1$ in the equality relation (4.8). Assuming $Pr_{t}\approx 1$ (see below in figure 9 and also in Khani & Waite Reference Khani and Waite2016), we can obtain a theoretical estimate for the irreversible SGS mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ in the stratified turbulence regime as follows

(4.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{i}\approx {\textstyle \frac{1}{3}},\end{eqnarray}$$

which is slightly larger than the value that is observed in recent DNS studies (see e.g. Venayagamoorthy & Stretch Reference Venayagamoorthy and Stretch2010; Maffioli et al. Reference Maffioli, Brethouwer and Lindborg2016). It is important to mention that all parameters and variables are scaled except the turbulent Prandtl number, which is approximated by $1$ , to obtain equation (4.9); therefore, this equation gives an estimate for the scale of the irreversible mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ . In the rest of this section, we will show that the LES results are in good agreement with those from DNS and hyperviscosity simulations and the scale analysis in equation (4.9).

Figure 4.

The irreversible mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ versus (a) the vertical Froude number $Fr_{v}$ and (b) the horizontal Froude number $Fr_{h}$ , for DNS, LES and hyperviscosity simulations. We have considered $2\unicode[STIX]{x03C0}$ factors in definitions of $Fr_{v}$ and $Fr_{h}$ to remove the effects of $2\unicode[STIX]{x03C0}$ factors, which are used in definitions of ${\mathcal{L}}_{v}$ and ${\mathcal{L}}_{h}$ , respectively, as seen in (4.1). The violet cross sign in panel (a) shows the theoretical scale analysis of $\unicode[STIX]{x1D6FE}_{i}$ for LES approach as given in (4.9).

Figure 4 shows the irreversible SGS mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ versus the vertical Froude number $Fr_{v}$ in panel (a) and the horizontal Froude number $Fr_{h}$ in panel (b) for the dynamic Smagorinsky, regular Smagorinsky and Kraichnan SGS models. For comparison, the results from hyperviscosity simulations and DNS are also shown. Here, we have considered a $2\unicode[STIX]{x03C0}$ factor in definitions of $Fr_{v}$ and $Fr_{h}$ to remove the effects of the $2\unicode[STIX]{x03C0}$ factor in the definitions of ${\mathcal{L}}_{v}$ and ${\mathcal{L}}_{h}$ , respectively, as are given in (4.1). Interestingly, the curves for the irreversible SGS mixing efficiency in LES are fairly close to those of the irreversible mixing efficiency in DNS. The mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ is small when $Fr_{v}$ is small, and it goes to the value $1/3$ as $Fr_{v}$ increases towards $1$ (figure 4 a). The violet cross sign in figure 4(a) shows the theoretical value of $\unicode[STIX]{x1D6FE}_{i}$ (as given by (4.9)). Importantly, equation (4.9) is only valid in LES approach when $Fr_{v}\sim 1$ because this is the situation that ${\mathcal{L}}_{v}\sim L_{b}$ (see (4.2)). As $Fr_{v}\rightarrow 1$ , LES results approach the theoretical value $\unicode[STIX]{x1D6FE}_{i}\approx 1/3$ (see figure 4 a). It is interesting that DNS and the dynamic Smagorinsky LES perform similarly (see black and blue lines in figure 4). The maximum irreversible mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ is given by $0.33$ in DNS run with $n=768$ and $N=0.5$ , and it is given by $0.32$ ( $0.31$ ) in the dynamic Smagorinsky simulation with $n=512$ ( $256$ ) and $N=0.5$ . This behaviour shows that the dynamic Smagorinsky model performs reasonably well even at lower resolution simulations in comparison with high resolution DNS runs. The performance of the regular Smagorinsky model is less efficient in comparison with the Kraichnan model and the maximum of $\unicode[STIX]{x1D6FE}_{i}$ is $0.29$ and $0.32$ , respectively, in the Smagorinsky and Kraichnan simulations with $n=256$ and $N=0.5$ . As $Fr_{v}$ (or $Fr_{h}$ ) decreases, the irreversible SGS mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ reduces (figure 4). Also, we should mention that simulations are mostly in the range of small horizontal Froude number, and $Fr_{h}$ increases as $N$ decreases (moving from the strongly stratified to weakly stratified regime, see § 4.4). The hyperviscosity simulations show slightly higher $\unicode[STIX]{x1D6FE}_{i}$ even at $Fr_{v}\approx 0.7$ (figure 4 a). This trend could be due to the effects of hyperviscosity that constrains the dissipation range to much smaller scales in comparison with regular viscosity, and so slightly decreases the averaged kinetic dissipation rates (see table 2).

We have suggested different regimes for $Fr_{v}$ in LES of stratified turbulence based on the resolution of the buoyancy scale $L_{b}$ (see (4.2) and (4.3)). Hence, it will be important to investigate if $\unicode[STIX]{x1D6FE}_{i}$ does depend on the resolution of $L_{b}$ or not. We will do a similar investigation to study the dependence of $\unicode[STIX]{x1D6FE}_{i}$ on the resolution of $L_{o}$ in the next section. Figure 5 shows $\unicode[STIX]{x1D6FE}_{i}$ versus $L_{b}/\unicode[STIX]{x1D6E5}$ for different SGS parameterizations in LES, and also for hyperviscosity simulations and DNS (vertical dashed lines in figure 5 represent the thresholds on $L_{b}/\unicode[STIX]{x1D6E5}$ , respectively from left to right, for the Kraichnan, dynamic Smagorinsky and Smagorinsky models). It is clear that for all SGS parameterizations, as the ratio $L_{b}/\unicode[STIX]{x1D6E5}$ increases, the irreversible SGS mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ rises and approaches the theoretical value $1/3$ (see (4.9) and figure 5). This trend suggests that if the buoyancy scale $L_{b}$ is well resolved in LES, the mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ is in line with the results of the theoretical scale analysis in LES of stratified turbulence, and if the threshold $L_{b}/\unicode[STIX]{x1D6E5}$ is violated, LES of stratified turbulence underestimates the irreversible SGS mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ (figure 5). This behaviour is in agreement with DNS runs, in which the irreversible mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ increases monotonically with $L_{b}/\unicode[STIX]{x1D6E5}$ (see black line in figure 5; note that $\unicode[STIX]{x1D6E5}\propto L_{d}$ in DNS). Figure 6 shows the irreversible mixing efficiency versus the buoyancy Reynolds number $Re_{b}$ in DNS runs. Clearly, $\unicode[STIX]{x1D6FE}_{i}$ increases respectively from $0.18$ to $0.33$ when the buoyancy Reynolds number increases from $Re_{b}\approx 0.3$ , where the layered stratified flows are mainly affected by viscous sublayers, to $Re_{b}\approx 17$ , where overturning and Kelvin–Helmholtz instabilities are ubiquitous in stratified turbulence (in line with DNS results of Shih et al. Reference Shih, Koseff, Ivey and Ferziger2005).

Figure 5.

The irreversible mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ versus the ratio $L_{b}/\unicode[STIX]{x1D6E5}$ for DNS, LES and hyperviscosity simulations. See the caption in figure 3 for more information about the vertical dashed lines.

Figure 6.

The irreversible mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ versus the buoyancy Reynolds number $Re_{b}=\langle \unicode[STIX]{x1D716}\rangle /(\unicode[STIX]{x1D708}N^{2})$ for DNS runs.

Overall, depending on the resolution of the buoyancy scale $L_{b}$ , LES of stratified turbulence will underestimate $\unicode[STIX]{x1D6FE}_{i}$ if $L_{b}$ in not adequately resolved (similar to DNS runs with $Re_{b}\lesssim 1$ ), or LES will correctly resemble $\unicode[STIX]{x1D6FE}_{i}$ if $L_{b}$ is well resolved, i.e. $L_{b}/\unicode[STIX]{x1D6E5}\gtrsim 10$ as seen in figures 3 and 5 (similar to DNS runs with $Re_{b}\gtrsim 10$ ). In addition, the regular Smagorinsky parameterization still underestimates $\unicode[STIX]{x1D6FE}_{i}$ when $N$ is large even for the highest resolution simulations, but both the Kraichnan and dynamic Smagorinsky models represents reasonable results when $L_{b}$ is more likely well resolved (see figures 4 and 5). This behaviour is due to the highly dissipative nature of the regular Smagorinsky model (as is also reported in Khani & Waite (Reference Khani and Waite2014, Reference Khani and Waite2015)). These results suggest that reproducing correct mixing efficiency in the LES approach requires much better resolution of $L_{b}$ in comparison with, for example, reproducing correct energy spectra in LES. In the next section, we study the effects of resolving $L_{o}$ in mixing efficiency for LES of stratified turbulence.

4.4 Ozmidov-scale effects

In stratified turbulence, the ratio of the Ozmidov scale $L_{o}$ to the buoyancy scale $L_{b}$ can be scaled as

(4.10) $$\begin{eqnarray}\frac{L_{o}}{L_{b}}\sim Fr_{h}^{1/2},\end{eqnarray}$$

where we have used Taylor’s hypothesis ${\mathcal{L}}_{h}\sim u_{rms}^{3}/\unicode[STIX]{x1D716}$ . When $Fr_{h}\ll 1$ , which is the case in strongly stratified turbulent flows, $L_{o}\ll L_{b}$ (see (4.10)). Therefore, if $L_{o}$ is resolved (i.e. $L_{o}\gtrsim \unicode[STIX]{x1D6E5}$ ), the resolution of $L_{b}$ is ensured (i.e. $L_{b}\gg \unicode[STIX]{x1D6E5}$ ). In this situation ${\mathcal{L}}_{v}\sim u_{rms}/N$ (as in (4.2)), and we can scale $Fr_{h}$ as follows

(4.11) $$\begin{eqnarray}Fr_{h}=\frac{u_{rms}}{N{\mathcal{L}}_{h}}\sim \frac{{\mathcal{L}}_{v}}{{\mathcal{L}}_{h}},\end{eqnarray}$$

where ${\mathcal{L}}_{v}\ll {\mathcal{L}}_{h}$ (anisotropic motions). Using (4.10) and (4.11), we can conclude that

(4.12) $$\begin{eqnarray}\frac{L_{o}}{L_{b}}\sim \left(\frac{{\mathcal{L}}_{v}}{{\mathcal{L}}_{h}}\right)^{1/2},\end{eqnarray}$$

in the strongly stratified turbulence regime. If the turbulence regime moves from strongly stratified to weakly stratified (and so forth to the isotropic Kolmogorov turbulence), ${\mathcal{L}}_{v}\rightarrow {\mathcal{L}}_{h}$ (isotropic motions), and thus $L_{o}\rightarrow L_{b}$ (see (4.12)). As a result, weakly stratified means $Fr_{h}\sim \mathit{O}(1)$ , which is in agreement with (4.10).

Figure 7 shows the ratio ${\mathcal{L}}_{v}/L_{b}$ and $\unicode[STIX]{x1D6FE}_{i}$ versus $L_{o}/\unicode[STIX]{x1D6E5}$ in panels (a) and (b), respectively, for DNS, LES and hyperviscosity simulations. It is clear that ${\mathcal{L}}_{v}\approx L_{b}$ when $L_{o}/\unicode[STIX]{x1D6E5}\gtrsim 1$ (figure 7 a). This trend is expected because if $L_{o}$ is more likely resolved in LES of stratified turbulence where $Fr_{h}\ll 1$ , it is guaranteed that $L_{b}$ is well resolved (see (4.10)). In addition, the irreversible mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ moves towards the theoretical value $1/3$ if $L_{o}$ is resolved in LES of stratified turbulence (figure 7 b). These results show that if $\unicode[STIX]{x1D6E5}\lesssim L_{o}$ in LES of stratified turbulence, the mixing efficiency is correctly reproduced.

Figure 7.

(a) The ratio ${\mathcal{L}}_{v}/L_{b}$ and (b) irreversible mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ versus the ratio $L_{o}/\unicode[STIX]{x1D6E5}$ for DNS, LES and hyperviscosity simulations. In DNS, $\unicode[STIX]{x1D6E5}\propto L_{d}$ , and therefore $L_{o}/\unicode[STIX]{x1D6E5}$ is a measure of $Re_{b}$ (i.e. large $L_{o}/\unicode[STIX]{x1D6E5}$ in DNS refers to stratified turbulence regime with $Re_{b}\gg 1$ ).

Overall, these results are in line with those from figures 3 and 5. Consequently, if $\unicode[STIX]{x1D6E5}\lesssim L_{o}$ , irreversible mixing efficiency goes to the theoretical value $1/3$ (see figures 5 and 7). In addition, the fundamental characteristics of stratified turbulence, such as anisotropic features, are more likely captured because the buoyancy scale $L_{b}$ is well resolved (see e.g. Khani & Waite Reference Khani and Waite2014, Reference Khani and Waite2015; Khani & Porté-Agel Reference Khani and Porté-Agel2017) when $\unicode[STIX]{x1D6E5}\lesssim L_{o}$ .

4.5 An a priori study using DNS results

In this section, we calculate the spectra of effective SGS energy transfer (eddy dissipation) and turbulent Prandtl number by applying a test cutoff wavenumber $k_{c}$ , which is around the buoyancy wavenumber $k_{b}$ , to DNS data (see appendix A, and also Khani & Waite Reference Khani and Waite2016, for calculations of the effective eddy dissipation spectra). Figure 8 shows the horizontal and vertical wavenumber spectra for effective eddy dissipation. These plots should be compared with the SGS eddy dissipation spectra that are shown in figure 2. We consider $k_{c}=20$ for DNS cases with $N=0.5$ , $0.75$ and $1$ , and $k_{c}=40$ for the DNS case with $N=2$ (these values are around the buoyancy wavenumber $k_{b}=2\unicode[STIX]{x03C0}/L_{b}$ ; see table 2). The effective eddy dissipation spectra also peak at large horizontal and small vertical scales (in line with the anisotropic dissipation picture, and are consistent with the corresponding SGS eddy dissipation spectra that are shown in figure 2). These trends strongly suggest that the dissipation mechanism in stratified turbulence includes anisotropic features around the buoyancy scale $L_{b}$ , and that is different than the dissipation mechanism in isotropic turbulence. When the test-filter wavenumber $k_{c}$ is fixed, increased buoyancy frequency $N$ results in increasing the non-local horizontal (i.e. $k_{h}\ll k_{c}$ ) and local vertical (i.e. $k_{v}\sim k_{c}$ ) effective eddy dissipation in simulations with $N=0.5$ , $0.75$ and $1$ , because the buoyancy scale $L_{b}$ becomes smaller than the filter width $\unicode[STIX]{x1D6E5}_{test}=\unicode[STIX]{x03C0}/k_{c}$ , and thus the anisotropic motions are mainly filtered (figure 8 and table 2). Overall, resolving the buoyancy scale in the LES approach is a critical threshold in order to capture the fundamental characteristics of stratified turbulence (in line with LES and DNS results of Khani & Waite Reference Khani and Waite2014, Reference Khani and Waite2015, Reference Khani and Waite2016; Khani & Porté-Agel Reference Khani and Porté-Agel2017).

Figure 8.

The (a) horizontal and (b) vertical wavenumber effective eddy dissipation spectra in DNS runs. The spectra are calculated at $t=450$ , and are multiplied by wavenumber to preserve area on the log-linear axes. We employ the test cutoff wavenumber $k_{c}=40$ in the DNS run with $N=2$ , and $k_{c}=20$ in other DNS runs. These plots should be compared to the corresponding SGS eddy dissipation spectra that are shown in figure 2.

The spectra of effective turbulent Prandtl number are shown in figure 9. We define $Pr_{t}(k|k_{c})$ as the ratio of the effective eddy viscosity $\unicode[STIX]{x1D708}_{e}(k|k_{c})$ over the effective eddy diffusivity $D_{e}(k|k_{c})$ (as are given in appendix A, and also in Khani & Waite Reference Khani and Waite2016). The turbulent Prandtl number spectra show approximately constant trends when $k/k_{c}\gtrsim 0.2$ (consistent with the results of Khani & Waite Reference Khani and Waite2016, and this also demonstrates that the assumption of $Pr_{t}\approx 1$ in this current work is a reasonable approximation).

Figure 9.

The effective turbulent Prandtl number spectra for DNS runs. The spectra are calculated at $t=450$ . We employ the test cutoff wavenumber $k_{c}=40$ in the DNS run with $N=2$ , and $k_{c}=20$ in other DNS runs. The solid and dashed magenta lines indicate values of zero and $1$ , respectively.

Table 3.

List of numerical simulations with LES when $Pr_{t}=0.2$ .

4.6 Turbulent Prandtl number effects

In stratified turbulence, $Pr_{t}\approx 1$ (as is shown in figure 9; and also see Salehipour & Peltier Reference Salehipour and Peltier2015; Khani & Waite Reference Khani and Waite2016), which leads to a scale of around $1/3$ for the irreversible mixing efficiency in LES of stratified turbulence (see (4.9)) that is in line with DNS results. Nevertheless, it is still very interesting to test the scaling theory (4.7) with different turbulent Prandtl numbers. We have performed extra LES runs when $Pr_{t}$ varies from $0.2$ to $2$ . Tables 3, 4 and 5 show simulation parameters and averaged variables for these LES runs. Using these new simulations, we can evaluate the dependency of the irreversible mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ on the turbulent Prandtl number $Pr_{t}$ as is given by scaling theory (4.7), and also we can estimate the proportionality coefficient $c$ in (4.8), as shown in appendix B. Figure 10 shows $\unicode[STIX]{x1D6FE}_{i}$ versus $Pr_{t}$ for the dynamic Smagorinsky, regular Smagorinsky and Kraichnan SGS models. For comparison, the scaling theory (4.7) is also shown with a black dashed line. It is clear that $\unicode[STIX]{x1D6FE}_{i}$ increases (decreases) when $Pr_{t}$ decreases (increases), and that LES results are in line with the scaling theory (figure 10). When $Pr_{t}\gtrsim 1$ , the scaling theory (4.7) agrees very well with the LES results, however, it slightly overestimates $\unicode[STIX]{x1D6FE}_{i}$ when $Pr_{t}<1$ . For example, the dynamic Smagorinsky model with $n=384$ , $N=1$ and $Pr_{t}=2$ gives $\unicode[STIX]{x1D6FE}_{i}=0.20$ , which equals to the scaling theory value in (4.7) when $Pr_{t}=2$ . Similar results are given by the Smagorinsky model with $n=512$ and $N=1$ , and the Kraichnan model with $n=384$ and $N=1$ . When $Pr_{t}=0.5$ , the dynamic Smagorinsky model with $n=384$ and $N=0.5$ gives $\unicode[STIX]{x1D6FE}_{i}=0.38$ , which is below the given value by the theoretical scale analysis curve with $\unicode[STIX]{x1D6FE}_{i}\approx 0.5$ . Similar trends are seen in the Smagorinsky and Kraichnan models. Also when $Pr_{t}=0.2$ , the dynamic Smagorinsky model with $n=384$ and $N=1$ gives $\unicode[STIX]{x1D6FE}_{i}=0.52$ , which is below the given estimate with scaling theory $\unicode[STIX]{x1D6FE}_{i}\approx 0.71$ . It is worthwhile mentioning that in stratified turbulence regime in which $Pr_{t}\approx 1$ (see e.g. Khani & Waite Reference Khani and Waite2016), LES results agree very well with the theoretical scale analysis value with $\unicode[STIX]{x1D6FE}_{i}\approx 0.33$ .

Table 4.

List of numerical simulations with LES when $Pr_{t}=0.5$ .

Figure 10.

The irreversible mixing efficiency $\unicode[STIX]{x1D6FE}_{i}$ versus the turbulent Prandtl number $Pr_{t}$ in LES. The black dashed line shows the scaling theory (4.7).

Table 5.

List of numerical simulations with LES when $Pr_{t}=2$ .

Overall, these results suggest that the scaling theory (4.7) agrees very well with LES results when $Pr_{t}\gtrsim 1$ . When $Pr_{t}<1$ , LES results and theoretical scale analysis still show similar behaviour (i.e. decreased $Pr_{t}$ leads to an increase in $\unicode[STIX]{x1D6FE}_{i}$ in LES runs and scaling theory). Nevertheless, $\unicode[STIX]{x1D6FE}_{i}$ is overestimated when $Pr_{t}<1$ , and this trend might be due to the dynamics of turbulent flows with $Pr_{t}\ll 1$ compared to that with $Pr_{t}\gtrsim 1$ , including stratified turbulence, and that we might be required to use different SGS parameterizations for eddy diffusivity rather than those used in equation (2.7a-d ) when $Pr_{t}\ll 1$ . This topic is an interesting research plan for future work, and is out of the scope of this current paper about stratified turbulence.