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 $\textit{Re}_{0} = 50$ , an initial bulk Richardson number $\textit{Ri}_{0} = 1/80$ (also corresponding to the initial minimum gradient Richardson number $\textit{Ri}_{{g}}$ ) and a fluid of Prandtl number ${\textit{Pr}} = 1$ 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 $\varGamma _{\chi }$ and dynamically emergent length scales $\varLambda$ 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. $\varLambda _{z} \approx 16$ ) once the horizontal extent of the domain $L_{{h}} \gtrsim 96$ . While this implies a final ${{Re}} \approx 400$ and ${Ri} \approx 0.1$ , we hypothesise that such forced flows ‘tune’ themselves eventually to a state of a gradient Richardson number $\textit{Ri}_{{g}} \lesssim 0.2$ , consistently with several previous studies. Moreover, provided sufficiently extended domains, we observe the emergence of large-scale flow structures with spanwise $\varLambda _{\!y} \approx 50$ and streamwise $\varLambda _{x} \lesssim 115$ . Clearly, these observations demonstrate the marked anisotropy of characteristic emergent length scales, even for such ‘weakly stratified’ forced shear flows. We conjecture that the actual emergent streamwise structures are a vestigial ‘imprint’ in the sheared turbulent flow of the primary linear instability of the converged deepened turbulent shear layer.