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

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 $Λ_{x} \lesssim 115$. …

💡 Research Summary

The paper investigates the statistically steady state of a forced, stably stratified shear flow using three‑dimensional direct numerical simulations (DNS) with a spectral‑element code. The authors initialize the flow with a bulk Reynolds number Re₀ = 50, a bulk Richardson number Ri₀ = 1/80 (≈0.0125), Prandtl number Pr = 1, and a response time t_r = 100 for the volumetric forcing that continuously relaxes the velocity and buoyancy fields toward the prescribed tanh profiles uₓ₀(z)=tanh(z) and b₀(z)=tanh(R₀z). The initial configuration is linearly unstable to a primary Kelvin‑Helmholtz instability (KHI); after the KHI breaks down, secondary instabilities generate turbulence that is sustained indefinitely by the forcing.

A systematic study of horizontal domain sizes (Lₓ, L_y) reveals that when the horizontal extent L_h ≳ 96 (in units of the initial shear‑layer half‑depth), the shear‑layer half‑depth converges to a finite value d ≈ 8, corresponding to a vertical large‑scale length Λ_z ≈ 16. At this point the bulk Reynolds number has grown to Re ≈ 400 and the bulk Richardson number to Ri ≈ 0.1. Moreover, the flow self‑organises such that the gradient Richardson number Rig settles at values ≤ 0.2, in line with previous observations that turbulence in stratified shear flows tends to maintain Rig below the Miles‑Howard critical value of 0.25.

The emergent large‑scale structures are strongly anisotropic: the streamwise length scale Λₓ is ≤ 115, the spanwise scale Λ_y ≈ 50, and the vertical scale Λ_z ≈ 16. These scales are identified through spatial correlation analyses and are shown to converge only when the domain is sufficiently wide; smaller domains artificially suppress the development of the spanwise and streamwise structures, leading to biased statistics. The authors also compute the flux coefficient Γ_χ, which quantifies the ratio of buoyancy flux to kinetic‑energy dissipation, and find that Γ_χ stabilises only for the large‑domain cases, indicating that irreversible mixing rates become independent of domain size only when the flow can support the full set of emergent structures.

The paper proposes a physical mechanism for the observed “tuning” of the flow: the volumetric forcing continuously injects momentum and buoyancy in a way that drives the mean shear‑to‑stratification ratio toward a quasi‑equilibrium. As the shear layer thins, the flow remains marginally stable, and large‑scale wave‑like structures appear in the horizontal directions, providing a pathway for energy transfer from the mean shear to turbulent motions while keeping Rig near the critical threshold.

In summary, the study demonstrates that forced stratified shear turbulence exhibits a self‑regulated state characterized by (i) a finite shear‑layer depth independent of initial conditions, (ii) strongly anisotropic large‑scale structures that require horizontal domains of order 100 times the initial shear‑layer half‑depth for convergence, and (iii) a gradient Richardson number that settles below 0.2. These findings have direct implications for the parameterisation of mixing in geophysical models, suggesting that domain‑size effects must be carefully considered when representing sustained stratified shear turbulence in large‑scale climate and ocean simulations.