Mechanism generating reverse buoyancy flux at the small scales of stably stratified turbulence

TL;DR

This work addresses how, in stably stratified turbulence, the small-scale buoyancy flux reverses sign and transfers turbulent potential energy back into turbulent kinetic energy. It derives an exact scale-decomposition of the buoyancy flux using a filtering approach, linking the local scale-local term ⟨P_B2^ℓ⟩ to ramp–cliff structures identified by Bragg & de Bruyn Kops, and shows that at small scales ⟨𝔅⟩ ≈ −ℓ^2⟨P_B2⟩, with the non-local term ⟨𝔅^{SG}⟩ contributing at larger scales. Direct numerical simulations for Pr=1 and 7 validate the theory, revealing that the sign reversal originates from the local mechanism, intensifying with Pr, and that ramp–cliff alignments govern both local and non-local buoyancy flux components via the density gradient orientation relative to gravity. These findings have implications for closure models in large-eddy simulations and deepen the understanding of energy transfers in stratified turbulence, linking small-scale reversals to ramp–cliff dynamics and scale-dependent flux partitioning.

Abstract

Previous studies have shown that at the small-scales of stably stratified turbulence, the scale-dependent buoyancy flux reverses sign, such that there is a conversion of turbulent potential energy (TPE) back into turbulent kinetic energy (TKE) at these scales. Moreover, the magnitude of the reverse flux becomes stronger with increasing Prandtl number $Pr$. Using a filtering analysis we demonstrate analytically how this flux reversal is connected to the mechanism identified in Bragg \& de Bruyn Kops (JFM 2024 Vol 991 A10) that is responsible for the surprising observation that the TKE dissipation rate increases while the TPE dissipation rate decreases with increasing $Pr$ in stratified turbulence. The mechanism identified by Bragg \& de Bruyn Kops, which is connected to the formation of ramp-cliff structures in the density field, is shown to give the scale-local contribution to the buoyancy flux. At the smallest-scales this local contribution dominates and explains the flux reversal, while at larger scales a non-local contribution is important. Direct numerical simulations (DNS) of 3D statistically stationary, stably stratified turbulence in the strongly stratified regime confirm the theoretical analysis, and indicate that while on average the local contribution only dominates the buoyancy flux at the smallest scales, it remains strongly correlated with the buoyancy flux at all scales. The results show that ramp-cliffs are not only connected to the reversal of the local buoyancy flux but also the non-local part. At the small scales (approximately below the Ozmidov scale), ramp structures contribute exclusively to reverse buoyancy flux events, whereas cliff structures contribute to both forward and reverse buoyancy flux events.

Figures (5)

• Figure 1: Lin-log plots of $\langle\mathcal{P}^\ell_{B2}\rangle/\sigma^3_{\tilde{A}}$ as a function of scale for $Pr_1=1$ (blue) and $Pr_2=7$ (orange) with $Fr_1 \approx 0.08$ (solid) and $Fr_2 \approx 0.16$ (dashed).
• Figure 2: Buoyancy flux $\langle \mathcal{B} \rangle$ as a function of scale $\ell$ for $Pr=1$ (blue) and $Pr=7$ (orange) with $Fr_1 \approx 0.08$ (solid) and $Fr_2 \approx 0.16$ (dashed). Black vertical dashed line marks the approximate Ozmidov scale $l_{O}/\eta \sim 20$, exact values in table \ref{['tab:Parameters']}.
• Figure 3: Ratio of the scale-local component to the total buoyancy flux $\langle \mathcal{B}^{{\rm l}}\rangle/\langle\mathcal{B}\rangle$ as a function of filter-scale $\ell$ for $Pr_1=1$ (blue), $Pr_2=7$ (orange) with $Fr_1 \approx 0.08$ (solid) and $Fr_2 \approx 0.16$ (dashed). Black vertical dashed line marks the approximate Ozmidov scale $l_{O}/\eta \sim 20$, exact values in table \ref{['tab:Parameters']}.
• Figure 4: Averages of the total buoyancy flux $\mathcal{B}$, and its scale-local and non-local decompositions $\mathcal{B}^{{\rm l}}$ and $\mathcal{B}^{{\rm nl}}$, respectively, conditioned on $\gamma=\hat{\bm{e}}_{\tilde{B}} \bm{\cdot} \hat{\bm{e}}_z$ and normalised by the average TPE dissipation rate $\langle\chi\rangle$. Blue (orange) lines show the values for $Pr_1=1$ ($Pr_2=7$), whereas the solid (dashed) lines correspond to $Fr_1\approx0.08$ and $Fr_2\approx 0.16$, respectively. Panels I, II, III, and IV correspond to fields filtered at scales $\ell/\eta \approx 0.7$, $\ell/\eta \approx 12$, $\ell/\eta \approx 26$, and $\ell/\eta \approx 90$, respectively.
• Figure 5: Correlation coefficient of $\mathcal{B}$ with the scale local term $\mathcal{B}^{{\rm l}}=-\ell^2 \mathcal{P}^\ell_{B2}$ (Eq.\ref{['eq:partition_PB']}) for $Pr_1=1$ (blue), $Pr_2=7$ (orange) with $Fr_1 \approx 0.08$ (solid) and $Fr_2 \approx 0.16$ (dashed). Black vertical dashed line marks the approximate Ozmidov scale $l_{O}/\eta \sim 20$, exact values in table \ref{['tab:Parameters']}.