Anisotropy of emergent large-scale dynamics in forced stratified shear flows

TL;DR

This work investigates forced, stably stratified shear flows using three-dimensional direct numerical simulations to understand how statistically steady stratified turbulence self-organizes into large-scale anisotropic structures. The forcing sustains turbulence while relaxing toward initial base profiles, enabling a controlled study of emergent vertical and horizontal length scales. The simulations reveal a finite turbulent-shear-layer depth $\Lambda_z \approx 16$, and emergent horizontal scales $\Lambda_y \approx 50$ and $\Lambda_x \approx 75$–$115$, with strong anisotropy $\Lambda_z < \Lambda_y < \Lambda_x$; convergence of these scales requires very extended domains (e.g., $L_h \gtrsim 96$). The flow tends to a self-tuned state with typical midmixing-zone Richardson values $\rm Ri_g \lesssim 0.2$, and the most unstable background-KHI wavelength remains closely linked to the emergent streamwise scale, suggesting a persistent imprint of primary instability even in sustained turbulence. These results have practical implications for domain sizing in simulations of shear-driven mixing and bear on geophysical contexts where forced stratified turbulence occurs, including the need to consider very large horizontal extents to capture converged statistics and pattern formation.

Abstract

Although stably stratified shear flows, where the base velocity shear is quasi-continuously forced externally, arise in many geophysically and environmentally relevant circumstances, the emergent dynamics of their ensuing statistically steady stratified turbulence is still an open question. We address this phenomenon in a series of three-dimensional direct numerical simulations using spectral element methods. We consider a forced, stably stratified shear flow with an initial bulk Reynolds number $\ReO = 50$, an initial bulk Richardson number $\RiO = 1/80$ (also corresponding to the initial minimum gradient Richardson number $\Rig$), and a fluid of Prandtl number $\Pr = 1$ in horizontally extended domains. Although the initial configuration is unstable to a primary Kelvin-Helmholtz instability, the ensuing turbulence is sustained by continuously relaxing the resulting flow back towards the initial profiles of streamwise velocity and buoyancy. We study statistical as well as structural aspects of the final statistically steady flows, including the flux coefficient $\Gchi$ and dynamically emergent length scales $Λ$ associated with the large-scale dynamics, respectively. Despite the ongoing stirring and mixing, we find that the shear layer half-depth converges to a finite value of $d \approx 8$ (i.e., $Λ_{z} \approx 16$) once the horizontal extent of the domain $\Gh \gtrsim 96$. While this implies a final $\Re \approx 400$ and $\Ri \approx 0.1$, we hypothesise that such forced flows \enquote{tune} themselves eventually to a state of a gradient Richardson number $\Rig \lesssim 0.2$, consistently with several previous studies. Moreover, provided sufficiently extended domains, we observe the emergence of large-scale flow structures with spanwise $Λ_{y} \approx 50$ and streamwise $Λ_{x} \lesssim 115$. ...

Figures (11)

• Figure 1: Configuration of the forced stratified shear flows considered here. While the initial buoyancy profile $b_{0}$ is statically stable, the imposed initial velocity profile $u_{x, 0}$ may induce flow instabilities that lead to a transition to turbulence. A continuous forcing $f_{\Phi}$ may inject the required energy to sustain this turbulence, ensuring statistically stationary dynamics.
• Figure 2: Temporal evolution of forced, stratified shear flows. (a -- e) The flow is prone to a primary KHI, leading eventually to "overturning billows" and streamwise mergers. A continuous forcing sustains the induced turbulence for arbitrarily long times. During this evolution of the flow, (f -- h) the interface broadens before reaching a statistically stationary depth. Note that for this relatively small $\mathrm{Re}_{0}$, as shown in (g), molecular diffusion of the shear layer and density interface dominates the development of the primary instability until the depths of the shear layer and density interface have approximately doubled. In this figure, $L_{\mathrm{h}} = 128$ while panels (a -- e) visualise $b \left( x, y = 0, z, t \right)$ with the colour bar matching figure \ref{['fig:emerging_horizontally_extended_dynamics']} (l, p).
• Figure 3: Sustained turbulent interactions. (a) Despite statistical stationarity, the deepening of the density interface can be affected by the horizontal extent of the domain $L_{\mathrm{h}} \equiv L_{x} = L_{y}$ before it converges eventually. This deepening tends to stabilise the emergent flow, resulting in a significantly increased minimum value of $0.074$ of (b) the average late $\mathrm{Ri}_{\textrm{g}}$. Nevertheless, we find $\mathrm{Ri}_{\textrm{g}} \leq 0.25$ (with this canonical value being marked with a vertical line) throughout the turbulent "mixing zone". The resulting associated mixing in this zone is underlined by high amplitudes in (c) the stabilising vertical buoyancy advection (i.e. the buoyancy flux) $B$ and (d) the dissipation rates of kinetic energy and scaled buoyancy variance, $\varepsilon_{u}$ and $\chi$.
• Figure 4: Impact of the horizontal extent of the domain on the mixing. The flow topology -- comprising the (a) interface (half-) depths, (b) final emergent (bulk) Reynolds number and Richardson number, as well as averages across the midplane and across the entire mixing zone of (c) gradient Richardson number, (d) buoyancy flux, (e) kinetic energy dissipation and (f) buoyancy variance dissipation -- only converges for horizontally extended domains, $L_{\mathrm{h}} \gtrsim L_{\mathrm{h, crit}} = 96$. Vertical solid lines indicate the temporal standard deviation. All panels share the same abscissa.
• Figure 5: Statistics of mixing properties at midplane. Statistical distributions of the (a) kinetic energy dissipation rate $\varepsilon_{u}$, (b) scaled buoyancy variance dissipation rate $\chi$, and (c) local flux coefficient $\Gamma_{\chi}$ are affected by insufficient extents of the domain but converge eventually. The grey dashed vertical line marks the canonical value $\Gamma_{\chi} = 0.2$. Note the emergent scaling for $\Gamma_{\chi}$ for extreme mixing events, and the marked difference of the high tails of the PDFs of $\varepsilon_{u}$ and $\chi$.