Dynamics and energetics underlying mixing efficiency in homogeneous stably stratified turbulence

We studied homogeneous stably stratified turbulence in a triply periodic domain over a wide range of stratification strengths. We evaluated the statistically stationary, volume-averaged budgets of Reynolds stresses, turbulent potential energy, and turbulent vertical density flux. By separately studying the three components of the turbulent kinetic energy (TKE), we examined the role of pressure-strain correlations and observed connections between changes in the energetics to regime shifts of the mixing coefficient $\left(\mathrm{\Gamma }\right)$ as a function of the turbulent Froude number $\left({\text{Fr}}_k\right)$. As we increase stratification, we find that pressure-strain correlations become more important in producing the vertical component of TKE $\left(k_w\right)$. At the stratification strength where direct production and pressure-strain correlations equally generate $k_w$, we observe the maximum value of $\mathrm{\Gamma }$, and it remains constant as stratification is increased further. However, when we greatly increase stratification from this point, the pressure-strain correlations become the dominant source of $k_w$ with direct production becoming negligible, and this change is accompanied by the mixing coefficient decreasing from its maximum value. Finally, we find that this final transition for the mixing coefficient coincides with a sign change of the pressure scrambling term in the vertical density flux budget.

Figure 1

(a) Temporal trajectories of volume-averaged values of ${\text{Re}}_L$ and ${\text{Fr}}_k$ for four simulations with different temporal forcing strategies: constant $A$ (red squares), constant $k$ (orange diamonds), constant ${\epsilon }_k$ (blue stars), constant $k_w$ (black triangles). The four simulations only correspond to the C2, K3, D12, and V6 simulations in Table 2. (b) Volume- and time-averaged values of ${\text{Re}}_L$ and ${\text{Fr}}_k$ for all simulations in Table 2.

Figure 2

Forcing input parameter ($A/N$) plotted versus the right-hand side of Eq. (12) for the steady-state, volume- and time-averaged energy balance for our simplified system. Given that the points for all four forcing strategies lie on the “one-to-one” line, it appears that all four strategies are well-described by the steady-state balance described in Eq. (12).

Figure 3

(a) Normalized mean-squared horizontal velocity fluctuations (circles) and mean-squared vertical velocity fluctuations (triangles) versus turbulent Froude number. (b) Lumley triangle visualization of simulations colored by turbulent Froude number. In panel (a), the horizontal dashed line corresponds to the value of $1/3$, which is expected of isotropic turbulence, and the horizontal dotted line corresponds to the value of $1/2$, which is expected of two-component, disklike turbulence. In panel (b), stronger stratification (smaller turbulent Froude number) corresponds to increasingly anisotropic turbulence, corresponding to the movement from the isotropic corner of the Lumley triangle ($\xi =0$, $\eta =0$) to the two-component, disklike turbulence limit in the upper left.

Figure 4

Steady-state, volume- and time-averaged budgets of (a) TKE and (b) TPE as a function of the turbulent Froude number. The two panels correspond to Eqs. (5) and (6), respectively.

Figure 5

Steady-state, volume- and time-averaged budgets of (a) $k_H$, (b) $k_w$, and (c) $\overline{w\rho }$ as a function of the turbulent Froude number. The three panels correspond to Eqs. (9), (10), and (11), respectively.

Figure 6

(b) Volume- and time-averaged values of $\mathrm{\Gamma }$ versus ${\text{Fr}}_k$. Figures 5 and 5 are reproduced in panels (a) and (c) for convenience. The simulations exhibit four mixing regimes with the power-law slopes of $\mathrm{\Gamma }$ changing at ${\text{Fr}}_k\approx 0.1$, 0.7, and 2.1. The low Froude number slope of $\mathrm{\Gamma }\sim {\text{Fr}}_k^{0.4}$ is an empirical fit and remains an open question in the literature.

Figure 7

(a) $R_w/P_k$ and (b) $b_{33}$ plotted as a function of ${\text{Fr}}_k$ with the magnitude of Re shown in color and the least-squares fits to the DNS values shown by thin blue lines. (c) $c_3={\epsilon }_w/{\epsilon }_k$ plotted as a function of Re with the magnitude of ${\text{Fr}}_k$ shown in color and the dashed horizontal line marking the value of $c_3=1/3$, which is expected under local isotropy of dissipative scales. (d) Contour plot of ${\text{Ri}}_f$ as a function of ${\text{Fr}}_k$ and $c_3$ where (15) is evaluated for each ${\text{Fr}}_k$ by using the least-squares fits of $R_w/P_k$ and $b_{33}$ shown in panels (a) and (b) and varying $c_3$. Colored circles represent the actual values of ${\text{Ri}}_f$ from DNS.

Figure 8

Time series of volume-averaged (a) $({\epsilon }_k+{\epsilon }_p)/P_k$ and (b) $\mathrm{\Gamma }$ from simulations C2 (red), K3 (orange), and D14 (blue). Temporal averaging windows are marked by horizontal lines. Note the marked decay and growth of $({\epsilon }_k+{\epsilon }_p)/P_k$ for the constant-$A$ simulation (red) in contrast to the behavior of the constant-$k$ and constant-${\epsilon }_k$ simulations (orange and blue, respectively).