The issue of the physical mechanism(s) that control the efficiency with which the density field in stably stratified fluid is mixed by turbulent processes has remained enigmatic. Similarly enigmatic has been an explanation of the numerical value of ∼0.2, which is observed to characterize this efficiency experimentally. We review recent work on the turbulence transition in stratified parallel flows that demonstrates that this value is not only numerically predictable but also that it is expected to be a nonmonotonic function of the Richardson number that characterizes preturbulent stratification strength. This value of the mixing efficiency appears to be characteristic of the late-time behavior of the turbulent flow that develops after an initially laminar shear flow has undergone the transition to turbulence through an intermediate instability of Kelvin-Helmholtz type.

We investigate the detailed nature of the ‘mixing transition’ through which turbulence may develop in both homogeneous and stratified free shear layers. Our focus is upon the fundamental role in transition, and in particular the associated ‘mixing’ (i.e. small-scale motions which lead to an irreversible increase in the total potential energy of the flow) that is played by streamwise vortex streaks, which develop once the primary and typically two-dimensional Kelvin–Helmholtz (KH) billow saturates at finite amplitude.Saturated KH billows are susceptible to a family of three-dimensional secondary instabilities. In homogeneous fluid, secondary stability analyses predict that the stream-wise vortex streaks originate through a ‘hyperbolic’ instability that is localized in the vorticity braids that develop between billow cores. In sufficiently strongly stratified fluid, the secondary instability mechanism is fundamentally different, and is associated with convective destabilization of the statically unstable sublayers that are created as the KH billows roll up.We test the validity of these theoretical predictions by performing a sequence of three-dimensional direct numerical simulations of shear layer evolution, with the flow Reynolds number (defined on the basis of shear layer half-depth and half the velocity difference) Re = 750, the Prandtl number of the fluid Pr = 1, and the minimum gradient Richardson number Ri(0) varying between 0 and 0.1. These simulations quantitatively verify the predictions of our stability analysis, both as to the spanwise wavelength and the spatial localization of the streamwise vortex streaks. We track the nonlinear amplification of these secondary coherent structures, and investigate the nature of the process which actually triggers mixing. Both in stratified and unstratified shear layers, the subsequent nonlinear amplification of the initially localized streamwise vortex streaks is driven by the vertical shear in the evolving mean flow. The two-dimensional flow associated with the primary KH billow plays an essentially catalytic role. Vortex stretching causes the streamwise vortices to extend beyond their initially localized regions, and leads eventually to a streamwise-aligned collision between the streamwise vortices that are initially associated with adjacent cores.It is through this collision of neighbouring streamwise vortex streaks that a final and violent finite-amplitude subcritical transition occurs in both stratified and unstratified shear layers, which drives the mixing process. In a stratified flow with appropriate initial characteristics, the irreversible small-scale mixing of the density which is triggered by this transition leads to the development of a third layer within the flow of relatively well-mixed fluid that is of an intermediate density, bounded by narrow regions of strong density gradient.

Turbulent mixing produced by breaking of internal waves plays an important role in setting the patterns of downwelling and upwelling of deep dense waters and thereby helps sustain the global deep ocean overturning circulation. A key parameter used to characterize turbulent mixing is its efficiency, defined here as the fraction of the energy available to turbulence that is invested in mixing. Efficiency is conventionally approximated by a constant value near one sixth. Here we show that efficiency varies significantly in the abyssal ocean and can be as large as approximately one third in density stratified regions near topographic features. Our results indicate that variations in efficiency exert a first‐order control over the rate of overturning of the lower branch of the meridional overturning circulation.

Understanding how turbulence leads to the enhanced irreversible transport of heat and other scalars such as salt and pollutants in density-stratified fluids is a fundamental and central problem in geophysical and environmental fluid dynamics. This review discusses recent research activity directed at improving community understanding, modeling, and parameterization of the subtle interplay between energy conversion pathways, instabilities, turbulence, external forcing, and irreversible mixing in density-stratified fluids. The conceptual significance of various length scales is highlighted, and in particular, the importance is stressed of overturning or scouring in the formation and maintenance of layered stratifications, i.e., robust density distributions with relatively deep and well-mixed regions separated by relatively thin interfaces of substantially enhanced density gradient. Expected final online publication date for the Annual Review of Fluid Mechanics, Volume 53 is January 7, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

We use a variational formulation incorporating the full Navier–Stokes equations to identify initial perturbations with finite kinetic energy ${E}_{0} $ which generate the largest gain in perturbation kinetic energy at some time $T$ later for plane Couette flow. Using the flow geometry originally used by Butler & Farrell (Phys. Fluids A, vol. 4, 1992, pp. 1637–1650) to identify the linear transient optimal perturbations for ${E}_{0} \ensuremath{\rightarrow} 0$ and incorporating $T$ as part of the optimization procedure, we show how the addition of nonlinearity smoothly changes the result as ${E}_{0} $ increases from zero until a small but finite ${E}_{c} $ is reached. At this point, the variational algorithm is able to identify an initial condition of completely different form which triggers turbulence – called the minimal seed for turbulence. If instead $T$ is fixed at some asymptotically large value, as suggested by Pringle, Willis & Kerswell (J. Fluid Mech., vol. 703, 2012, pp. 415–443), a fundamentally different ‘final’ optimal perturbation emerges from our algorithm above some threshold initial energy ${E}_{f} \in (0, {E}_{c} )$ which shows signs of localization. This nonlinear optimal perturbation clearly approaches the structure of the minimal seed as ${E}_{0} \ensuremath{\rightarrow} { E}_{c}^{\ensuremath{-} } $, although for ${E}_{0} \lt {E}_{c} $, its maximum gain over all time intervals is always less than the equivalent maximum gain for the ‘quasi-linear optimal perturbation’, i.e. the finite-amplitude manifestation of the underlying linear optimal perturbation. We also consider a wider flow geometry recently studied by Monokrousos et al. (Phys. Rev. Lett., vol. 106, 2011, 134502) and present evidence that the critical energy for transition ${E}_{c} $ they found by using total dissipation over a time interval as the optimizing functional is recovered using energy gain at a fixed target time as the optimizing functional, with the same associated minimal seed emerging. This emphasizes that the precise form of the functional does not appear to be important for identifying ${E}_{c} $ provided it takes heightened values for turbulent flows, as postulated by Pringle, Willis & Kerswell (J. Fluid Mech., vol. 703, 2012, pp. 415–443). All our results highlight the irrelevance of the linear energy gain optimal perturbation for predicting or describing the lowest-energy flow structure which triggers turbulence.

We study stratified turbulence in plane Couette flow using direct numerical simulations. Two external dimensionless parameters control the dynamics, the Reynolds number Re = Uh/ν and the bulk Richardson number Ri = gα V Th/U 2 , where U and T are half the velocity and temperature difference between the two walls respectively, h is the half channel depth, ν is the kinematic viscosity and gα V is the buoyancy parameter. We focus on spatio-temporal intermittency due to stratification and we explore the boundary between fully developed turbulence and intermittent flow in the Re-Ri plane. The structures populating the intermittent flow regime show coexistence between laminar and turbulent patches, and we demonstrate that there are qualitative differences between the previously studied low-Re low-Ri intermittent regime and the high-Re high-Ri intermittent regime. At low-Re low-Ri, turbulent regions span the entire gap, whereas at high-Re high-Ri, turbulence is confined vertically with complex dynamics arising from interacting turbulent layers. Consistent with a previous investigation of Flores & Riley (Boundary-Layer Meteorol., vol. 129 (2), 2010, pp. 241-259), we present evidence suggesting that intermittency in the asymptotic regime of high-Re Couette flows appears for L + < 200, where L + = Lu τ /ν, with L being the Monin-Obukhov length scale, L = u 3 τ /C κ q w , q w the wall heat flux, C κ the von Kármán constant and u τ = √ τ w /ρ 0 the friction velocity determined from the wall shear stress τ w , where ρ 0 is the constant background density. We also consider the mixing as quantified by various versions of the flux Richardson number Ri f , defined as the ratio of the conversion rate from kinetic to potential energy to the turbulent kinetic energy injection rate due to shear. We investigate how laminar and turbulent regions separately contribute to the overall mixing. Remarkably, we find that although fluctuations are greatly suppressed in the laminar regions, Ri f does not change significantly compared with its value in turbulent regions. As we observe a tight coupling between the mean temperature and velocity fields, we demonstrate that both Monin-Obukhov self-similarity theory (Monin & Obukhov, Contrib. Geophys. Inst. Acad. Sci. USSR, vol. 151, 1954, pp. 163-187) and the explicit algebraic model of Lazeroms et al. (J. Fluid Mech., vol. 723, 2013, pp. 91-125) predict the mean profiles well. We thus use these models to trace out the boundary between fully developed turbulence and intermittency in the Re-Ri plane.

We consider the nonlinear optimisation of the mixing of a passive scalar, initially arranged in two layers, in a two-dimensional plane Poiseuille flow at finite Reynolds and Péclet numbers, below the linear instability threshold. We use a nonlinear-adjointlooping approach to identify optimal perturbations leading to maximum time-averaged energy as well as maximum mixing in a freely evolving flow, measured through the minimisation of either the passive scalar variance or the so-called mix-norm, as defined by Mathew, Mezić & Petzold (Physica D, vol. 211, 2005, pp. 23-46). We show that energy optimisation appears to lead to very weak mixing of the scalar field whereas the optimal mixing initial perturbations, despite being less energetic, are able to homogenise the scalar field very effectively. For sufficiently long time horizons, minimising the mix-norm identifies optimal initial perturbations which are very similar to those which minimise scalar variance, demonstrating that minimisation of the mix-norm is an excellent proxy for effective mixing in this finite-Péclet-number bounded flow. By analysing the time evolution from initial perturbations of several optimal mixing solutions, we demonstrate that our optimisation method can identify the dominant underlying mixing mechanism, which appears to be classical Taylor dispersion, i.e. shear-augmented diffusion. The optimal mixing proceeds in three stages. First, the optimal mixing perturbation, energised through transient amplitude growth, transports the scalar field across the channel width. In a second stage, the mean flow shear acts to disperse the scalar distribution leading to enhanced diffusion. In a final third stage, linear relaxation diffusion is observed. We also demonstrate the usefulness of the developed variational framework in a more realistic control case: mixing optimisation by prescribed streamwise velocity boundary conditions.

We present a new robust method for identifying three dynamically distinct regions in a stratified turbulent flow, which we characterise as quiescent flow, intermittent layers, and turbulent patches. The method uses the cumulative filtered distribution function of the local density gradient to identify each region. We apply it to data from direct numerical simulations of homogeneous stratified turbulence, with unity Prandtl number, resolved on up to 8192×8192×4096 grid points. In addition to classifying regions consistently with contour plots of potential enstrophy, our method identifies quiescent regions as regions where /νN 2 ∼ O(1), layers as regions where /νN 2 ∼ O(10), and patches as regions where /νN 2 ∼ O(100). Here is the dissipation rate of turbulence kinetic energy, ν is the kinematic viscosity, and N is the (overall) buoyancy frequency. By far the highest local dissipation and mixing rates, and the majority of dissipation and mixing, occur in patch regions, even when patch regions occupy only 5% of the flow volume. We conjecture that treating stratified turbulence as an instantaneous assemblage of these different regions in varying proportions may explain some of the apparently highly scattered flow dynamics and statistics previously reported in the literature.