Long-time evolution of density layers and interfaces in forced stably-stratified flows

TL;DR

The study investigates the long-time evolution of density layers in forced, non-rotating stably stratified flows using direct numerical simulations of the Boussinesq equations with hyperdiffusion and large-scale friction. By exploring three stratifications ($N\in\{3,6,18\}$) and extending simulations to about $O(10^4)$ turnover times, the authors observe the emergence of density staircases for intermediate and strong stratification, with two coarsening pathways: decay of weaker interfaces and merging of adjacent interfaces. The buoyancy-flux versus buoyancy-gradient relation follows a non-monotonic Phillips-Posmentier-like curve, supporting an anti-diffusive mechanism that sustains layering, while intermittency is localized near layer boundaries and linked to interactions between turbulent mixing and layer formation. These results clarify the late-time dynamics of stratified turbulence and provide benchmarks for understanding mixing, energy distribution, and the role of large-scale damping in layered flows.

Abstract

Stably stratified fluids subject to sustained forcing are known to develop step-like density "staircases", where nearly homogeneous layers alternate with thin interfaces of strong stratification. However, long-time numerical investigations of this phenomenon have been limited by the intrinsically slow evolution of large-scale modes and the sensitivity of stratified turbulence to physical parameters. We present direct numerical simulations of forced Boussinesq flows for three stratification strengths (Fr = 0.42, 0.22, 0.076) and of unprecedented time extensions - up to O(10000) turnover times - with the purpose of reproducing and studying the very slow coarsening of the layered state. A large-scale friction term is introduced to arrest shear-mode growth and mimic finite-domain constraints. Staircase formation is observed for both medium and strong stratified cases, following two different coarsening dynamics: interfaces decaying or merging. While kinetic energy remains quasi-stationary during interface decay, it exhibits sharp bursts during merging events. The emergence and persistence of density steps can be explained by the non-monotonic relation between buoyancy flux and buoyancy gradient. Intermittency in vertical velocity and density fluctuations is confined to the vicinity of layer-interface boundaries, indicating that strong events arise from the interaction between turbulent mixing and layer formation rather than from regions of large density gradients alone.

Figures (7)

• Figure 1: Evolution of kinetic and potential energies. (a) Log-log plots of the time evolution of potential (main panel) and kinetic (inset) energies, for three values of the Brunt-Vaisälä frequency. Time is rescaled with the large-eddy turnover time $\tau_{NL}$. (b-d) Vertical $yz$ slices of the velocity magnitude field, for increasing values of $N$ at $t/\tau_{NL} \simeq 2800$. (e-g) Vertical profiles of the parallel and perpendicular velocities for increasing $N$, computed at the same instant as (b-d).
• Figure 2: Visualizations of the buoyancy field $b(\boldsymbol{r},t)$ at different stages of the evolution for the $N=6$ case. (a) Evolution of potential-to-kinetic-energy ratio. Along the curve four instants are chosen in which snapshots of the buoyancy $b$ are visualized. (b) Buoyancy field at (comparatively) small times, displaying a moderately-perturbed linear variation along the vertical. (c) First energy plateau, with buoyancy (read density) arranged as homogeneous layers separated by three density interfaces. (d) Decay of an interface when the next plateau is reached. (e) Final stationary state, with only one interface left. (f) Vertical profiles of the buoyancy at the four instants of (b-e). (g) Horizontal slice of (c) taken at the height of the 'middle' interface. (h) Vertical slice on the $yz$ plane.
• Figure 3: Kinetic and potential energy spectra. (a) Main: isotropic kinetic spectra, for all stratification strengths, time-averaged at the last stationary state reached. Inset: instantaneous, parallel kinetic spectra computed for $N=18$ at three salient times: first kinetic excursion with corresponding potential energy increase (dash-dotted); following stationary state (dashed); second, minor kinetic surge with simultaneous potential energy growth (solid). (b) Same as (a), but for potential energy spectra. Green solid lines indicate the spectral slope $k^{-5/3}$, the magenta dashed line corresponds to $k^{-2}$. Time averages in main panels are taken in an interval of length $t/\tau_{NL} \simeq 650$. A short time-average (5 consecutive samples) is also performed in the inset plots, to decrease fluctuations.
• Figure 4: Vertical buoyancy flux as a function of (minus) the vertical buoyancy gradient, with plane- and time-averages performed on both quantities, for (a) $N=6$ and (b) $N=18$. In the former the data are from the final, one-interface state, in the latter from the three-interfaces stationary state. Points $A$, $B$ and $C$ in both panels indicate points of interest along $z$ (respectively: center of layer, interface-layer border, center of interface) as reported in figs. \ref{['fig:vertical_profiles']}(a-b). In panel (b) points measured from different interfaces -- and their adjacent half-layers -- are shown with different colors (see fig. \ref{['fig:vertical_profiles']}(b)).
• Figure 5: Evolution of the vertical gradient of the plane-averaged density fluctuation $\langle \phi \rangle$, made non-dimensional rescaling with the Brunt-Vaisälä frequency. Large negative values (in blue) correspond to density interfaces, slightly-positive values (in red) to layers. The run with $N=6$ in (a) shows decays of interfaces, while in panel (b) we observe merging of interfaces when $N=18$. The thin strips delimited by dash-dotted lines indicate where the time-averaging for fig. \ref{['fig:vertical_profiles']} is computed.