3.1 How turbulence quantities depend on the computation of $N$

Since $Gi$ is itself a function of the buoyancy frequency $N$ , it is first instructive to see how it varies with the method of computing $N$ . A direct comparison may be made using a two-dimensional histogram of $f(Gi^{\ast },\hat{Gi}^{\ast })$ (figure 2 a), which shows that $\hat{Gi}^{\ast }$ is generally greater than $Gi^{\ast }$ , especially for turbulent regions where $Gi>1$ . Because the calculation of $\unicode[STIX]{x1D716}_{k}^{t}$ is independent of the method used to compute $N$ , this indicates that $\hat{N}^{\ast }$ is generally less than $N^{\ast }$ . The two-dimensional histogram of $f(\unicode[STIX]{x1D716}_{p}^{t\ast },\hat{\unicode[STIX]{x1D716}}_{p}^{t\ast })$ (figure 2 b) provides further evidence that $\hat{N}^{\ast }$ underestimates $N^{\ast }$ . The term $\hat{\unicode[STIX]{x1D716}}_{p}^{t\ast }$ is generally larger than $\unicode[STIX]{x1D716}_{p}^{t\ast }$ , especially for larger mixing values ( $\unicode[STIX]{x1D716}_{p}^{t}$ greater than roughly $1\times 10^{-7}~\text{m}^{2}~\text{s}^{-3}$ in figure 2 b). The observed differences in $\unicode[STIX]{x1D716}_{p}^{t}$ lead to differences in the flux Richardson number $R_{f}$ (figure 2 c): $\hat{R}_{f}^{\ast }$ is, overall, slightly larger than $R_{f}^{\ast }$ , as expected from the trend in $\unicode[STIX]{x1D716}_{p}^{t}$ . Most notably, for $R_{f}^{\ast }$ less than roughly 0.1, a region of increased $\hat{R}_{f}^{\ast }$ extends up to roughly 0.6, indicating large overestimates when lateral averaging and vertical sorting are used. It should be noted that $R_{f}$ quantities in figure 2 are limited to turbulent regions where $Gi>1$ .

If the $N$ value calculated from the three-dimensionally sorted density field is taken to be the true value, then the error caused by calculating $N$ using lateral averaging and vertical sorting can be defined as

(3.1) $$\begin{eqnarray}E(\unicode[STIX]{x1D719})=\frac{|\hat{\unicode[STIX]{x1D719}}^{\ast }-\unicode[STIX]{x1D719}^{\ast }|}{\unicode[STIX]{x1D719}^{\ast }},\end{eqnarray}$$

where $\unicode[STIX]{x1D719}$ is a turbulence quantity. The histogram in figure 2(d) shows that $E(N)$ is often $O(10^{-1})$ , and can be as large as $O(1)$ . It should be noted that, as discussed above, $\hat{N}^{\ast }$ is generally less than $N^{\ast }$ ; thus, generally, $\hat{N}^{\ast }-N^{\ast }<0$ . However, the absolute value in (3.1) guarantees that $E>0$ , allowing it to be plotted on a log scale in figure 2(d). The error in $N$ translates to errors in $\unicode[STIX]{x1D716}_{p}^{t}$ , $Gi$ and $R_{f}$ as large as $O(10)$ . The terms $E(\unicode[STIX]{x1D716}_{p}^{t})$ and $E(Gi)$ are nearly identical, as they are both proportional to $1/N^{2}$ .

For the breaking wave scenarios considered here, lateral averaging and vertical sorting of density profiles unphysically smooth out variability in $N$ , leading to mixing estimates that are larger than the true mixing values. The variability in $N$ is better captured by full three-dimensional sorting, resulting in a more accurate measure of mixing. The additional (artificial) mixing that results from the use of $\hat{N}^{\ast }$ instead of $N^{\ast }$ can be quantified using an effective mixing coefficient $\unicode[STIX]{x1D705}_{eff}$ , defined with

(3.2) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D716}}_{p}^{t\ast }=\unicode[STIX]{x1D705}\frac{\overline{|\unicode[STIX]{x1D735}b^{\prime }|^{2}}}{(\hat{N}^{\ast })^{2}}=\unicode[STIX]{x1D705}_{eff}\frac{\overline{|\unicode[STIX]{x1D735}b^{\prime }|^{2}}}{(N^{\ast })^{2}},\end{eqnarray}$$

which implies

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{eff}=\unicode[STIX]{x1D705}\left(\frac{N^{\ast }}{\hat{N}^{\ast }}\right)^{2}.\end{eqnarray}$$

The effective mixing rate $\unicode[STIX]{x1D705}_{eff}$ is clearly skewed towards values larger than the true mixing rate $\unicode[STIX]{x1D705}$ , and is, on average, 3.6 times larger than $\unicode[STIX]{x1D705}$ (figure 3).

Figure 3.

Histogram of the effective mixing rate $\unicode[STIX]{x1D705}_{eff}$ due to lateral averaging and vertical sorting. The vertical dotted line shows the true value of $\unicode[STIX]{x1D705}=1\times 10^{-6}~\text{m}^{2}~\text{s}^{-1}$ (Arthur et al. (Reference Arthur, Koseff and Fringer2017) use a Prandtl number $Pr=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}=1$ ). The vertical solid line shows the mean value of $\unicode[STIX]{x1D705}_{eff}/\unicode[STIX]{x1D705}=3.6$ .

3.2 The flux Richardson number $R_{f}$ as a function of $Gi$

The method of computing $N$ has a strong effect on the functional relationship between $R_{f}$ and $Gi$ , which is often used to parameterize mixing in the ocean. Here, $R_{f}$ is calculated as a weighted mean in bins of $Gi$ (figure 4 a). As in the calculation of $f$ (described at the beginning of this section), area weighting is used to account for the non-uniform grid in the simulations of Arthur et al. (Reference Arthur, Koseff and Fringer2017). Thus, larger (smaller) grid cells, which take up a larger (smaller) portion of the computational domain, contribute more (less) to the mean value of $R_{f}$ . To show the spread of the data in terms of the Gibson number, $f$ is also calculated in each $Gi$ bin (figure 4 b). Due to computational restrictions on $Gi$ associated with the DNS, many of the data have relatively low (non-turbulent) values of $Gi<1$ . Since these values imply laminar flow, they are omitted in figure 4. It should be noted that calculations of $R_{f}$ for larger values of $Gi$ are based on a small subset of the data (figure 4 b), probably explaining the wiggles in the $\hat{R}_{f}^{\ast }$ curve.

Figure 4.

(a) Comparison of mean $R_{f}^{\ast }$ as a function of $Gi^{\ast }$ with mean $\hat{R}_{f}^{\ast }$ as a function of $\hat{Gi}^{\ast }$ . The $\hat{R}_{f}^{\ast }$ calculation of Mater & Venayagamoorthy (Reference Mater and Venayagamoorthy2014b ) using the data of Shih et al. (Reference Shih, Koseff, Ivey and Ferziger2005) is included as well. (b) Histograms of $Gi^{\ast }$ and $\hat{Gi}^{\ast }$ .

The term $\hat{R}_{f}^{\ast }$ , which reaches a peak of nearly 0.6 for $\hat{Gi}^{\ast }\approx 10^{3}$ , is generally larger than $R_{f}^{\ast }$ , which has maximum values between 0.2 and 0.3 for $Gi^{\ast }<10^{2}$ . For $Gi^{\ast }>10^{2}$ , a sharp drop in the mixing efficiency occurs and $R_{f}^{\ast }$ approaches 0. A similar result was found for $R_{f}$ by Shih et al. (Reference Shih, Koseff, Ivey and Ferziger2005), Walter et al. (Reference Walter, Squibb, Woodson, Koseff and Monismith2014) and others; see figure 12 in Walter et al. (Reference Walter, Squibb, Woodson, Koseff and Monismith2014). Because $\hat{Gi}^{\ast }$ is generally larger than $Gi^{\ast }$ , the decline of $\hat{R}_{f}^{\ast }$ for large values of $\hat{Gi}^{\ast }$ occurs at a larger value of $\hat{Gi}^{\ast }$ (approximately $\hat{Gi}^{\ast }=10^{3}$ ).

The large discrepancies that arise in $R_{f}$ when lateral averaging and vertical sorting are used on unsteady inhomogeneous breaking wave data emphasize the potential effect of $N$ on mixing estimates in the ocean. If $Gi=10^{2}$ , local vertical sorting causes $R_{f}$ to be overestimated by roughly a factor of 2. This increases to a factor of roughly 10 when $Gi\geqslant 10^{3}$ . In contrast, the $\hat{R}_{f}^{\ast }(\hat{Gi}^{\ast })$ calculations of Mater & Venayagamoorthy (Reference Mater and Venayagamoorthy2014b ) using the stratified shear flow data of Shih et al. (Reference Shih, Koseff, Ivey and Ferziger2005) generally follow the present $R_{f}^{\ast }(Gi^{\ast })$ curve, but depart substantially from the present $\hat{R}_{f}^{\ast }(\hat{Gi}^{\ast })$ curve. This is probably due to the homogeneous nature of the turbulence studied by Shih et al. (Reference Shih, Koseff, Ivey and Ferziger2005), which could allow local vertical sorting to achieve a more accurate measure of the true background stratification. In the inhomogeneous flow studied here, $\hat{N}^{\ast }$ varies spatially, and may therefore be a less appropriate measure of  $N$ .