fig2\]. The collapse of the data from different runs, on to seemingly universal curves, is remarkable for all the cases except for $Ri_g > 0.2$. We would like to mention that similar scaling behavior was not found if other normalization factors (e.g., $h$) are used.

Both normalized $L_{OZ}$ and $L_b$ decrease monotonically with $Ri_g$; however, the slopes are quite different. The length scales $L_C$ and $L_H$ barely exhibit any sensitivity to $Ri_g$ (except for $Ri_g > 0.1$). Even for weakly-stable condition, these length scales are less than 25% of $\mathcal{L}$.

Based on the expressions of the OLSs (i.e., Eqs. \[OLS\]) and the definition of the gradient Richardson number, we can write:

$$\frac{L_C}{L_{OZ}} = \left(\frac{N}{S}\right)^{3/2} = Ri_g^{3/4},$$

$$\frac{L_H}{L_{b}} = \left(\frac{N}{S}\right) = Ri_g^{1/2}.$$

Thus, for $Ri_g < 1$, one expects $L_C < L_{OZ}$ and $L_H < L_b$. Such relationships are fully supported by Fig. \[fig2\]. In comparison to the buoyancy effects, the shear effects are felt at smaller length scales for the entire stability range considered in the present study.

![image](LOZ_Lb.eps){width="49.00000%"} ![image](LC_LH.eps){width="49.00000%"}

Owing to their similar scaling behaviors, $L_b/\mathcal{L}$ against $L_{OZ}/\mathcal{L}$ are plotted in Fig. \[fig3\] (left panel). Once again, all the simulated data collapse nicely in a quasi-universal (nonlinear) curve. Since in a double-logarithmic representation (not shown) this curve is linear, we can write: $$\frac{L_b}{\mathcal{L}} \equiv \left(\frac{L_{OZ}}{\mathcal{L}}\right)^m,
    \label{Lb_vs_LOZ}$$ where, $m$ is an unknown power-law exponent. By using $L_b \equiv \overline{e}^{1/2}/N$ and the definitions of $L_{OZ}$ and $\mathcal{L}$, we arrive at: $$\frac{\overline{e}^{1/2}}{N} = \left(\frac{\overline{\varepsilon}}{N^3} \right)^{m/2} \left(\frac{\overline{e}^{3/2}}{\overline{\varepsilon}} \right)^{1-m}.$$ Further simplification leads to: $\overline{\varepsilon} = \overline{e} N$; please note that the exponent $m$ cancels out