Turbulent disruption of density staircases in stratified shear flows

Abstract

Formation of step-like 'density staircase' distributions induced by stratification and turbulence has been widely studied and can be explained by the 'instability' of a sufficiently strongly stably stratified turbulent flow due to the decrease of the turbulent density flux with increasing stratification via the 'Phillips mechanism'. However, such density staircases are not often observed in ocean interiors, except in regions where double diffusion processes are important, leading to thermohaline staircases. Using reduced order models for the evolution of velocity and density gradients, we analyse staircase formation in stratified and sheared turbulent flows. Under the assumption of inertial scaling $\epsilon \sim U^3/L$ for the kinetic energy dissipation rate $\epsilon$, where $U$ and $L$ are characteristic velocity and length scales, we determine ranges of bulk Richardson numbers $\text{Ri}_{b}$ and turbulent Prandtl numbers $\text{Pr}_{T}$ for which staircases can potentially form and show that the Phillips mechanism only survives in the limit of sufficiently small turbulent Prandtl numbers. For relevant oceanic parameters, a range of turbulent Prandtl numbers above which the system is not prone to staircases is found to be $\text{Pr}_T \simeq 0.5 - 0.7$. Since several studies indicate that the turbulent Prandtl number in stably stratified turbulence and in ocean interiors is usually above this threshold, this result supports the empirical observation that staircases are not favoured in ocean interiors in the presence of ambient turbulence. We also show that our analysis is robust to other scalings for $\epsilon$ (such as the more strongly stratified scaling $\epsilon \sim U^{2}N$, where $N$ is a characteristic value of the buoyancy frequency), supporting our results in both shear-dominated and buoyancy-dominated turbulent regimes as well as in weakly and strongly stratified regimes.