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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-adjoint-looping 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 investigate fully developed turbulence in stratified plane Couette flows using direct numerical simulations similar to those reported by Deusebio et al. (J. Fluid Mech., vol. 781, 2015, pp. 298–329) expanding the range of Prandtl number $Pr$ examined by two orders of magnitude from 0.7 up to 70. Significant effects of $Pr$ on the heat and momentum fluxes across the channel gap and on the mean temperature and velocity profile are observed. These effects can be described through a mixing length model coupling Monin–Obukhov (M–O) similarity theory and van Driest damping functions. We then employ M–O theory to formulate similarity scalings for various flow diagnostics for the stratified turbulence in the gap interior. The midchannel gap gradient Richardson number $Ri_{g}$ is determined by the length scale ratio $h/L$ , where $h$ is the half-channel gap depth and $L$ is the Obukhov length scale. As $h/L$ approaches very large values, $Ri_{g}$ asymptotes to a maximum characteristic value of approximately 0.2. The buoyancy Reynolds number $Re_{b}\\equiv \\unicode[STIX]{x1D700}/(\\unicode[STIX]{x1D708}N^{2})$ , where $\\unicode[STIX]{x1D700}$ is the dissipation, $\\unicode[STIX]{x1D708}$ is the kinematic viscosity and $N$ is the buoyancy frequency defined in terms of the local mean density gradient, scales linearly with the length scale ratio $L^{+}\\equiv L/\\unicode[STIX]{x1D6FF}_{\\unicode[STIX]{x1D708}}$ , where $\\unicode[STIX]{x1D6FF}_{\\unicode[STIX]{x1D708}}$ is the near-wall viscous scale. The flux Richardson number $Ri_{f}\\equiv -B/P$ , where $B$ is the buoyancy flux and $P$ is the shear production, is found to be proportional to $Ri_{g}$ . This then leads to a turbulent Prandtl number $Pr_{t}\\equiv \\unicode[STIX]{x1D708}_{t}/\\unicode[STIX]{x1D705}_{t}$ of order unity, where $\\unicode[STIX]{x1D708}_{t}$ and $\\unicode[STIX]{x1D705}_{t}$ are the turbulent viscosity and diffusivity respectively, which is consistent with Reynolds analogy. The turbulent Froude number $Fr_{h}\\equiv \\unicode[STIX]{x1D700}/(NU^{\\prime 2})$ , where $U^{\\prime }$ is a turbulent horizontal velocity scale, is found to vary like $Ri_{g}^{-1/2}$ . All these scalings are consistent with our numerical data and appear to be independent of $Pr$ . The classical Osborn model based on turbulent kinetic energy balance in statistically stationary stratified sheared turbulence (Osborn, J. Phys. Oceanogr., vol. 10, 1980, pp. 83–89), together with M–O scalings, results in a parameterization of $\\unicode[STIX]{x1D705}_{t}/\\unicode[STIX]{x1D708}\\sim \\unicode[STIX]{x1D708}_{t}/\\unicode[STIX]{x1D708}\\sim Re_{b}Ri_{g}/(1-Ri_{g})$ . With this parameterization validated through direct numerical simulation data, we provide physical interpretations of these results in the context of M–O similarity theory. These results are also discussed and rationalized with respect to other parameterizations in the literature. This paper demonstrates the role of M–O similarity in setting the mixing efficiency of equilibrated constant-flux layers, and the effects of Prandtl number on mixing in wall-bounded stratified turbulent flows.

We study stratified turbulence in plane Couette flow using direct numerical simulations. Two external dimensionless parameters control the dynamics, the Reynolds number $\\mathit{Re}=Uh/{\\it\\nu}$ and the bulk Richardson number $\\mathit{Ri}=g{\\it\\alpha}_{V}Th/U^{2}$ , where $U$ and $T$ are half the velocity and temperature difference between the two walls respectively, $h$ is the half channel depth, ${\\it\\nu}$ is the kinematic viscosity and $g{\\it\\alpha}_{V}$ 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 $\\mathit{Re}{-}\\mathit{Ri}$ 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- $\\mathit{Re}$ low- $\\mathit{Ri}$ intermittent regime and the high- $\\mathit{Re}$ high- $\\mathit{Ri}$ intermittent regime. At low- $\\mathit{Re}$ low- $\\mathit{Ri}$ , turbulent regions span the entire gap, whereas at high- $\\mathit{Re}$ high- $\\mathit{Ri}$ , 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- $\\mathit{Re}$ Couette flows appears for $L^{+}<200$ , where $L^{+}=Lu_{{\\it\\tau}}/{\\it\\nu}$ , with $L$ being the Monin–Obukhov length scale, $L=u_{{\\it\\tau}}^{3}/C_{{\\it\\kappa}}q_{w}$ , $q_{w}$ the wall heat flux, $C_{{\\it\\kappa}}$ the von Kármán constant and $u_{{\\it\\tau}}=\\sqrt{{\\it\\tau}_{w}/{\\it\\rho}_{0}}$ the friction velocity determined from the wall shear stress ${\\it\\tau}_{w}$ , where ${\\it\\rho}_{0}$ is the constant background density. We also consider the mixing as quantified by various versions of the flux Richardson number $\\mathit{Ri}_{f}$ , 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, $\\mathit{Ri}_{f}$ 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 $\\mathit{Re}{-}\\mathit{Ri}$ plane.

We study direct numerical simulations of turbulence arising from the interaction of an initial background shear, a linear background stratification and an external body force. In each simulation the turbulence produced is spatially intermittent, with dissipation rates varying over orders of magnitude in the vertical. We focus analysis on the statistically quasi-steady states achieved by applying large-scale body forcing to the domain, and compare flows forced by internal gravity waves with those forced by vertically uniform vortical modes. By considering the turbulent energy budgets for each simulation, we find that the injection of potential energy from the wave forcing permits a reversal in the sign of the mean buoyancy flux. This change in the sign of the buoyancy flux is associated with large, convective density overturnings, which in turn lead to more efficient mixing in the wave-forced simulations. The inhomogeneous dissipation in each simulation allows us to investigate localised correlations between the kinetic and potential energy dissipation rates. These correlations lead us to the conclusion that an appropriate definition of an instantaneous mixing efficiency, $\\unicode[STIX]{x1D702}(t):=\\unicode[STIX]{x1D712}/(\\unicode[STIX]{x1D712}+\\unicode[STIX]{x1D700})$ (where $\\unicode[STIX]{x1D700}$ and $\\unicode[STIX]{x1D712}$ are the volume-averaged turbulent viscous dissipation rate and fluctuation density variance destruction rate respectively) in the wave-forced cases is independent of an appropriately defined local turbulent Froude number, consistent with scalings proposed for low Froude number stratified turbulence.

We consider numerically the transition to turbulence and associated mixing in stratified shear flows with initial velocity distribution $\\overline{U}(z,0)\\,\\boldsymbol{e}_{x}=U_{0}\\,\\boldsymbol{e}_{x}\\tanh (z/d)$ and initial density distribution $\\overline{\\unicode[STIX]{x1D70C}}(z,0)=\\unicode[STIX]{x1D70C}_{0}[1-\\tanh (z/\\unicode[STIX]{x1D6FF})]$ away from a hydrostatic reference state $\\unicode[STIX]{x1D70C}_{r}\\gg \\unicode[STIX]{x1D70C}_{0}$ . When the ratio $R=d/\\unicode[STIX]{x1D6FF}$ of the characteristic length scales over which the velocity and density vary is equal to one, this flow is primarily susceptible to the classic well-known Kelvin–Helmholtz instability (KHI). This instability, which typically manifests at finite amplitude as an array of elliptical vortices, strongly ‘overturns’ the density interface of strong initial gradient, which nevertheless is the location of minimum initial gradient Richardson number $Ri_{g}(0)=Ri_{b}=g\\unicode[STIX]{x1D70C}_{0}d/\\unicode[STIX]{x1D70C}_{r}U_{0}^{2}$ , where $Ri_{g}(z)=-([g/\\unicode[STIX]{x1D70C}_{r}]\\,\\text{d}\\overline{\\unicode[STIX]{x1D70C}}/\\text{d}z)/(\\text{d}\\overline{U}/\\text{d}z)^{2}$ and $Ri_{b}$ is a bulk Richardson number. As is well known, at sufficiently high Reynolds numbers ( $Re$ ), the primary KHI induces a vigorous but inherently transient burst of turbulence and associated irreversible mixing localised in the vicinity of the density interface, leading to a relatively well-mixed region bounded by stronger density gradients above and below. We explore the qualitatively different behaviour that arises when $R\\gg 1$ , and so the density interface is sharp, with $Ri_{g}(z)$ now being maximum at the density interface $Ri_{g}(0)=RRi_{b}$ . This flow is primarily susceptible to Holmboe wave instability (HWI) (Holmboe, Geophys. Publ., vol. 24, 1962, pp. 67–113), which manifests at finite amplitude in this symmetric flow as counter-propagating trains of elliptical vortices above and below the density interface, thus perturbing the interface so as to exhibit characteristically cusped interfacial waves which thereby ‘scour’ the density interface. Unlike previous lower- $Re$ experimental and numerical studies, when $Re$ is sufficiently high the primary HWI becomes increasingly more three-dimensional due to the emergence of shear-aligned secondary convective instabilities. As $Re$ increases, (i) the growth rate of secondary instabilities appears to saturate and (ii) the perturbation kinetic energy exhibits a $k^{-5/3}$ spectrum for sufficiently large length scales that are influenced by anisotropic buoyancy effects. Therefore, at sufficiently high $Re$ , vigorous turbulence is triggered that also significantly ‘scours’ the primary density interface, leading to substantial irreversible mixing and vertical transport of mass above and below the (robust) primary density interface. Furthermore, HWI produces a markedly more long-lived turbulence event compared to KHI at a similarly high $Re$ . Despite their vastly different mechanics (i.e. ‘overturning’ versus ‘scouring’) and localisation, the mixing induced by KHI and HWI is comparable in both absolute terms and relative efficiency. Our results establish that, provided the flow Reynolds number is sufficiently high, shear layers with sharp density interfaces and associated locally high values of the gradient Richardson number may still be sites of substantial and efficient irreversible mixing.

We consider turbulence in a stratified ‘Kolmogorov’ flow, driven by horizontal shear in the form of sinusoidal body forcing in the presence of an imposed background linear stable stratification in the third direction. This flow configuration allows the controlled investigation of the formation of coherent structures, which here organise the flow into horizontal layers by inclining the background shear as the strength of the stratification is increased. By numerically converging exact steady states from direct numerical simulations of chaotic flow, we show, for the first time, a robust connection between linear theory predicting instabilities from infinitesimal perturbations to the robust finite-amplitude nonlinear layered state observed in the turbulence. We investigate how the observed vertical length scales are related to the primary linear instabilities and compare to previously considered examples of shear instability leading to layer formation in other horizontally sheared flows.

We consider the nonlinear optimisation of irreversible mixing induced by an initial finite amplitude perturbation of a statically stable density-stratified fluid with kinematic viscosity $\\unicode[STIX]{x1D708}$ and density diffusivity $\\unicode[STIX]{x1D705}$ . The initial diffusive error function density distribution varies continuously so that $\\unicode[STIX]{x1D70C}\\in [\\bar{\\unicode[STIX]{x1D70C}}-\\unicode[STIX]{x1D70C}_{0}/2,\\bar{\\unicode[STIX]{x1D70C}}+\\unicode[STIX]{x1D70C}_{0}/2]$ . A constant pressure gradient is imposed in a plane two-dimensional channel of depth $2h$ . We consider flows with a finite Péclet number $Pe=U_{m}h/\\unicode[STIX]{x1D705}=500$ and Prandtl number $Pr=\\unicode[STIX]{x1D708}/\\unicode[STIX]{x1D705}=1$ , and a range of bulk Richardson numbers $Ri_{b}=g\\unicode[STIX]{x1D70C}_{0}h/(\\bar{\\unicode[STIX]{x1D70C}}U^{2})\\in [0,1]$ where $U_{m}$ is the maximum flow speed of the laminar parallel flow, and $g$ is the gravitational acceleration. We use the constrained variational direct-adjoint-looping (DAL) method to solve two optimisation problems, extending the optimal mixing results of Foures et al. (J. Fluid Mech., vol. 748, 2014, pp. 241–277) to stratified flows, where the irreversible mixing of the active scalar density leads to a conversion of kinetic energy into potential energy. We identify initial perturbations of fixed finite kinetic energy which maximise the time-averaged perturbation kinetic energy developed over a finite time interval, and initial perturbations that minimise the value (at a target time, chosen to be $T=10$ ) of a ‘mix-norm’ as first introduced by Mathew et al. (Physica D, vol. 211, 2005, pp. 23–46), further discussed by Thiffeault (Nonlinearity, vol. 25, 2012, pp. 1–44) and shown by Foures et al. (2014) to be a computationally efficient and robust proxy for identifying perturbations that minimise the long-time variance of a scalar distribution. We demonstrate, for all bulk Richardson numbers considered, that the time-averaged kinetic-energy maximising perturbations are significantly suboptimal at mixing compared to the mix-norm minimising perturbations, and also that minimising the mix-norm remains (for density-stratified flows) a good proxy for identifying perturbations which minimise the variance at long times. Although increasing stratification reduces the mixing in general, mix-norm minimising optimal perturbations can still trigger substantial mixing for $Ri_{b}\\lesssim 0.3$ . By considering the time evolution of the kinetic energy and potential energy reservoirs, we find that such perturbations lead to a flow which, through Taylor dispersion, very effectively converts perturbation kinetic energy into ‘available potential energy’, which in turn leads rapidly and irreversibly to thorough and efficient mixing, with little energy returned to the kinetic-energy reservoirs.

We consider a passive zero-mean scalar field organised into two layers of different concentrations in a three-dimensional plane channel flow subjected to a constant along-stream pressure gradient. We employ a nonlinear direct-adjoint-looping method to identify the optimal initial perturbation of the velocity field with given initial energy which yields ‘maximal’ mixing by a target time horizon, where maximal mixing is defined here as the minimisation of the spatially integrated variance of the concentration field. We verify in three-dimensional flows the conjecture by Foures et al. (J. Fluid Mech., vol. 748, 2014, pp. 241–277) that the initial perturbation which maximises the time-averaged energy gain of the flow leads to relatively weak mixing, and is qualitatively different from the optimal initial ‘mixing’ perturbation which exploits classical Taylor dispersion. We carry out the analysis for two different Reynolds numbers ( $Re=U_{m}h/\\unicode[STIX]{x1D708}=500$ and $Re=3000$ , where $U_{m}$ is the maximum flow speed of the unperturbed flow, $h$ is the channel half-depth and $\\unicode[STIX]{x1D708}$ is the kinematic viscosity of the fluid) demonstrating that this key finding is robust with respect to the transition to turbulence. We also identify the initial perturbations that minimise, at chosen target times, the ‘mix-norm’ of the concentration field, i.e. a Sobolev norm of negative index in the class introduced by Mathew et al. (Physica D, vol. 211, 2005, pp. 23–46). We show that the ‘true’ variance-based mixing strategy can be successfully and practicably approximated by the mix-norm minimisation since we find that the mix-norm-optimal initial perturbations are far less sensitive to changes in the target time horizon than their optimal variance-minimising counterparts.

Turbulent mixing plays a major role in enabling the large-scale ocean circulation. The accuracy of mixing rates estimated from observations depends on our understanding of basic fluid mechanical processes underlying the nature of turbulence in a stratified fluid. Several of the key assumptions made in conventional mixing parameterizations have been increasingly scrutinized in recent years, primarily on the basis of adequately high resolution numerical simulations. We add to this evidence by compiling results from a suite of numerical simulations of the turbulence generated through stratified shear instability processes. We study the inherently intermittent and time-dependent nature of wave-induced turbulent life cycles and more specifically the tight coupling between inherently anisotropic scales upon which small-scale isotropic turbulence grows. The anisotropic scales stir and stretch fluid filaments enhancing irreversible diffusive mixing at smaller scales. We show that the characteristics of turbulent mixing depend on the relative time evolution of the Ozmidov length scale $L_{O}$ compared to the so-called Thorpe overturning scale $L_{T}$ which represents the scale containing available potential energy upon which turbulence feeds and grows. We find that when $L_{T}\\sim L_{O}$ , the mixing is most active and efficient since stirring by the largest overturns becomes ‘optimal’ in the sense that it is not suppressed by ambient stratification. We argue that the high mixing efficiency associated with this phase, along with observations of $L_{O}/L_{T}\\sim 1$ in oceanic turbulent patches, together point to the potential for systematically underestimating mixing in the ocean if the role of overturns is neglected. This neglect, arising through the assumption of a clear separation of scales between the background mean flow and small-scale quasi-isotropic turbulence, leads to the exclusion of an highly efficient mixing phase from conventional parameterizations of the vertical transport of density. Such an exclusion may well be significant if the mechanism of shear-induced turbulence is assumed to be representative of at least some turbulent events in the ocean. While our results are based upon simulations of shear instability, we show that they are potentially more generic by making direct comparisons with $L_{T}-L_{O}$ data from ocean and lake observations which represent a much wider range of turbulence-inducing physical processes.

We employ direct numerical simulation to investigate the efficiency of diapycnal mixing by shear-induced turbulence in stably stratified free shear layers for flows with bulk Richardson numbers in the range $0. 12\\leq R{i}_{0} \\leq 0. 2$ and Reynolds number $Re= 6000$ . We show that mixing efficiency depends non-monotonically upon $R{i}_{0} $ , peaking in the range 0.14–0.16, which coincides closely with the range in which both the buoyancy flux and the dissipation rate are maximum. By detailed analyses of the energetics of flow evolution and the underlying dynamics, we show that the existence of high mixing efficiency in the range $0. 14\\lt R{i}_{0} \\lt 0. 16$ is due to the emergence of a large number of small-scale instabilities which do not exist at lower Richardson numbers and are stabilized at high Richardson numbers. As discussed in Mashayek & Peltier (J. Fluid Mech., vol. 725, 2013, pp. 216–261), the existence of such a well-populated ‘zoo’ of secondary instabilities at intermediate Richardson numbers and the subsequent high mixing efficiency is realized only if the Reynolds number is higher than a critical value which is generally higher than that achievable in laboratory settings, as well as that which was achieved in the majority of previous numerical studies of shear-induced stratified turbulence. We furthermore show that the primary assumptions upon which the widely employed Osborn (J. Phys. Oceanogr. vol. 10, 1980, pp. 83–89) formula is based, as well as its counterparts and derivatives, which relate buoyancy flux to dissipation rate through a (constant) flux coefficient ( $\\Gamma $ ), fail at higher Richardson numbers provided that the Reynolds number is sufficiently high. Specifically, we show that the assumptions of fully developed, stationary, and isotropic turbulence all break down at high Richardson numbers. We show that the breakdown of these assumptions occurs most prominently at Richardson numbers above that corresponding to the maximum mixing efficiency, a fact that highlights the importance of the non-monotonicity of the dependence of mixing efficiency upon Richardson number, which we establish to be characteristic of stratified shear-induced turbulence. At high $R{i}_{0} $ , the lifecycle of the turbulence is composed of a rapidly growing phase followed by a phase of rapid decay. Throughout the lifecycle, there is considerable exchange of energy between the small-scale turbulence and larger coherent structures which survive the various stages of flow evolution. Since shear instability is one of the most prominent mechanisms for turbulent dissipation of energy at scales below hundreds of metres and at various depths of the ocean, our results have important implications for the inference of turbulent diffusivities on the basis of microstructure measurements in the oceanic environment.