We use direct numerical simulations to examine the onset of stratified turbulence triggered by the zigzag instability recently identified in columnar Taylor–Green vortices (Guo et al. 2024, J. Fluid Mech., vol. 997, A34) and its role in layer formation within the flow. The study focuses on Froude numbers $0.125 \leqslant \textit{Fr} \leqslant 2.0$ and Reynolds numbers ${\textit{Re}}$ ranging from 800 to 3200. The breakdown of the freely evolving vortex array is driven by local density overturns, combining shear and convective mechanisms initiated by the primary zigzag instability. Our results show a linear relationship between the peak buoyancy Reynolds number ${{\textit{Re}}}_b^{\star }$, driven by the zigzag instability, and ${\textit{Re}}\, {\textit{Fr}}^2$. When the flow does not exhibit local shear or convective instability, the value of ${{\textit{Re}}}_b^{\star }$ falls below unity. Both density and momentum layers arise from the zigzag instability: horizontal velocity layers are strong and persistent, while density layers are weaker and more transient. The vertical scale of the mean shear layers increases with ${\textit{Fr}}$ for ${\textit{Fr}} \leqslant 1$, shows weak dependence on ${\textit{Re}}$, and agrees well with the length scale associated with the fastest-growing linear mode of the zigzag instability. Further analysis in the sorted buoyancy coordinate highlights the role of density overturns caused by the zigzag instability in forming buoyancy layers during the transition to turbulence.

Double-diffusive convection can arise when the fluid density is set by multiple species which diffuse at different rates. Different flow regimes are possible depending on the distribution of the diffusing species, including salt fingering and diffusive convection. Flows arising from diffusive convection commonly exhibit step-like density profiles with sharp density interfaces separated by well-mixed layers. The formation of density layers is also seen in stratified turbulence, where a framework based on sorted density coordinates (Winters & D’Asaro 1996 J. Fluid Mech. 317, 179–193) has been used to diagnose layer formation (Zhou et al. 2017 J. Fluid Mech. 823, 198–229; Taylor & Zhou 2017 J. Fluid Mech. 823, R5). In this framework, the evolution of the sorted density profile can be expressed solely in terms of the eddy diffusivity, $\kappa _e$. Here, we use the same framework to diagnose layer formation in two-dimensional simulations of double-diffusive convection. We show that $\kappa _e$ is negative everywhere, with the antidiffusive effect strongest in ‘well-mixed’ layers where a positive diffusion coefficient may be expected. By considering a decomposition of the budget of the square of the Brunt-Väisälä frequency $\partial N^2_*/\partial t$, we demonstrate that the density layers are maintained by fundamentally different processes than in single-component stratified turbulence. In more complicated flows where stratified turbulence and double-diffusive convection can coexist, this framework could provide a method to distinguish between the mechanisms responsible for generating density layers.

Inspired by laboratory experiments showing internal waves generated by a plume impinging upon a stratified fluid layer (Ansong & Sutherland. 2010 J. Fluid Mech. 648, 405–434), we perform large eddy simulations in three dimensions to examine the structure and source of internal waves emanating from the top of a plume that rises vertically into stratification whose strength ranges over two orders of magnitude between different simulations. Provided the plume is sufficiently energetic to penetrate into the stratified layer, internal waves are generated with frequencies in a relatively narrow band moderately smaller than the buoyancy frequency. Through adaptations of ray theory including viscosity and use of dynamic mode decomposition, we show that the waves originate from within the turbulent flow rather than at the turbulent/non-turbulent interface between the fountain top and the surrounding stratified fluid.

Benthic macroalgae (including brown macroalgae or kelp) constitute one of the largest contributors to coastal primary production, but their ability to store and sequester carbon remains uncertain. Here, we use a numerical model of the flow/kelp interactions to study how tidal currents interact with an idealised numerical model of a giant kelp (Macrocystis pyrifera) forest, intending to better understand the potential for kelp growth in nutrient-limited conditions and the export of important tracers such as dissolved organic carbon. We calibrate and test our model using observations of currents within and surrounding a kelp forest in Southern California. By varying the density of kelp in our model, we find that there is a kelp density that maximises the export of tracer released from the kelp forest. Since the tracer advection/diffusion equation is linear with respect to the tracer concentration, the same kelp density corresponds to the maximum uptake for a tracer with a constant far-field concentration. The density at which this maximum occurs coincides with the density typical of natural kelp forests, where kelp growth may be limited by the uptake of dissolved nutrients from the surrounding water. Additionally, the drag induced on the tidal currents by the kelp forest results in a mean circulation through the kelp forest and a mean displacement of the kelp forest canopy.

We use large-eddy simulations to study the penetration of a buoyant plume carrying a passive tracer into a stably stratified layer with constant buoyancy frequency. Using a buoyancy-tracer volume distribution, we develop a method for objectively partitioning plume fluid in buoyancy-tracer space into three regions, each of which corresponds to a coherent region in physical space. Specifically, we identify a source region where undiluted plume fluid enters the stratified layer, a transport region where much of the transition from undiluted to mixed fluid occurs in the plume cap and an accumulation region corresponding to a radially spreading intrusion. This method enables quantification of different measures of turbulence and mixing within each of the three regions, including potential energy and turbulent kinetic energy dissipation rates, an activity parameter and the instantaneous mixing efficiency. We find that the most intense buoyancy gradients lie in a thin layer at the cap of the penetrating plume. This provides the primary stage of mixing between plume and environment and exhibits a mixing efficiency around 50 %. Newly generated mixtures of environmental and plume fluid join the intrusion and experience relatively weak turbulence and buoyancy gradients. As the intrusion spreads radially, environmental fluid surrounding the intrusion is mixed into the intrusion with moderate mixing efficiency. This dominates the volume of environmental fluid entrained into the region containing plume fluid. However, the ‘strongest’ entrainment, as measured by the specific entrainment rate, is largest in the plume cap, where the most buoyant environmental fluid is entrained.

We investigate the dynamics of a columnar Taylor–Green vortex array under strong stratification, focusing on Froude numbers $0.125\leq Fr \leq 1.0$, with the aim of identifying and understanding the primary instabilities that lead to the vortices’ breakdown. Linear stability analysis reveals that the fastest-growing vertical wavenumber scales with $Fr^{-1}$, while the dimensionless growth rate remains approximately constant. The most unstable eigenmode, identified as the mixed hyperbolic mode by Hattori et al. (J. Fluid Mech., vol. 909, 2021, A4), bears significant similarities to the zigzag instability, first discovered by Billant & Chomaz (J. Fluid Mech., vol. 418, 2000, pp. 167–188). Direct numerical simulations further confirm that the zigzag instability is crucial in amplifying initial random perturbations to finite amplitude, with the flow structure and modal growth rate consistent with the linear stability analysis. In particular, the characteristic vertical length scale of turbulence matches that of the fastest-growing linear mode. These findings underscore the broader relevance of the zigzag instability mechanism beyond its initial discovery in vortex pairs, demonstrating its role in facilitating direct energy transfer from vertically uniform vortical motions to a characteristic vertical length scale proportional to $Fr$ in strongly stratified flows.

We introduce a geometric analysis of turbulent mixing in density-stratified flows based on the alignment of the density gradient in two orthogonal bases that are locally constructed from the velocity gradient tensor. The first basis connects diapycnal mixing to rotation and shearing motions, building on the recent ‘rortex–shear decomposition’ in stratified shear layers (Jiang et al., J. Fluid Mech., vol. 947, 2022, A30), while the second basis connects mixing to the principal axes of the viscous dissipation tensor. Applying this framework to datasets taken in the stratified inclined duct laboratory experiment reveals that density gradients in locations of high shear tend to align preferentially (i) along the direction of minimum dissipation and (ii) normal to the plane spanned by the rortex and shear vectors. The analysis of the local alignment across increasingly turbulent flows offers new insights into the intricate relationship between the density gradient and dissipation, and thus diapycnal mixing.

The stratified inclined duct (SID) sustains an exchange flow in a long, gently sloping duct as a model for continuously forced density-stratified flows such as those found in estuaries. Experiments have shown that the emergence of interfacial waves and their transition to turbulence as the tilt angle is increased appears to be linked to a threshold in the exchange flow rate given by inviscid two-layer hydraulics. We uncover these hydraulic mechanisms by (i) using recent direct numerical simulations (DNS) providing full flow data in the key flow regimes (Zhu et al., J. Fluid Mech., vol. 969, 2023, A20), (ii) averaging these DNS into two layers, and (iii) using an inviscid two-layer shallow-water and instability theory to diagnose interfacial wave behaviour and provide physical insight. The laminar flow is subcritical and stable throughout the duct and hydraulically controlled at the ends of the duct. As the tilt is increased, the flow becomes supercritical everywhere and unstable to long waves. An internal jump featuring stationary waves first appears near the centre of the duct, then leads to larger-amplitude travelling waves, and to stronger jumps, wave breaking and intermittent turbulence at the largest tilt angle. Long waves described by the (nonlinear) shallow-water equation are interpreted locally as linear waves on a two-layer parallel base flow described by the Taylor–Goldstein equation. This link helps us to interpret long-wave instability and contrast it with short-wave (e.g. Kelvin–Helmholtz) instability. Our results suggest a transition to turbulence in SID through long-wave instability relying on vertical confinement by the top and bottom walls.

The stratified inclined duct (SID) experiment consists of a zero-net-volume exchange flow in a long tilted rectangular duct, which allows the study of realistic stratified shear flows with sustained internal forcing. We present the first three-dimensional direct numerical simulations (DNS) of SID to explore the transitions between increasingly turbulent flow regimes first described by Meyer & Linden (J. Fluid Mech., vol. 753, 2014, pp. 242–253). We develop a numerical set-up that faithfully reproduces the experiments and sustains the flow for arbitrarily long times at minimal computational cost. We recover the four qualitative flow regimes found experimentally in the same regions of parameter space: laminar flow, waves, intermittent turbulence and fully developed turbulence. We find good qualitative and quantitative agreement between DNS and experiments and highlight the added value of DNS to complement experimental diagnostics and increase our understanding of the transition to turbulence, both temporally (laminar/turbulent cycles) and parametrically (as the tilt angle of the duct and the Reynolds number are increased). These results demonstrate that numerical studies of SID – and deeper integration between simulations and experiments – have the potential to lead to a better understanding of stratified turbulence.

Buoyant material such as microplastics accumulate near the ocean surface in regions with convergent surface currents where they can be harmful to marine life. Here, we use large eddy simulations to investigate the transport and accumulation of buoyant material in a turbulent ocean mixed layer under combined wind and convection forcing. We model non-inertial buoyant particles with a combination of buoyant tracers and Lagrangian surface particles, which allows us to explore a wide range of particle buoyancies. Surface cooling drives convection, and under this regime persistent convective vortices form that trap buoyant particles, leading to large concentrations. Despite their small size, the convective vortices exhibit a bias towards cyclonic vorticity that has not been reported previously. Based on an analysis of Lagrangian trajectories, the average time that a particle spends inside a convective vortex is long enough for planetary vorticity to become important and further vortex stretching causes an exponential increase in vorticity. When wind forcing is included, there is a transition from convective cells to longitudinal wind rolls with three distinct flow patterns observed under weak, moderate and strong wind forcing. For sufficiently weak winds, convective vortices survive but are less effective at trapping buoyant material. Under strong wind forcing, convective vortices no longer exist, but some clustering occurs in regions of high speed associated with longitudinal wind rolls. We quantify the degree of clustering using the Gini coefficient and find that clustering is strongly influenced by the relative size of the friction and convective velocities and the particle buoyancy.

Motivated by observations of turbulence in the strongly stratified ocean thermocline, we use direct numerical simulations to investigate the interaction of a sinusoidal shear flow and a large-amplitude internal gravity wave. Despite strong nonlinearities in the flow and a lack of scale separation, we find that linear ray-tracing theory is qualitatively useful in describing the early development of the flow as the wave is refracted by the shear. Consistent with the linear theory, the energy of the wave accumulates in regions of negative mean shear where we observe evidence of convective and shear instabilities. Streamwise-aligned convective rolls emerge the fastest, but their contribution to irreversible mixing is dwarfed by shear-driven billow structures that develop later. Although the wave strongly distorts the buoyancy field on which these billows develop, the mixing efficiency of the subsequent turbulence is similar to that arising from Kelvin–Helmholtz instability in a stratified shear layer. We run simulations at Reynolds numbers Re of 5000 and 8000, and vary the initial amplitude of the internal gravity wave. For high values of initial wave amplitude, the results are qualitatively independent of $Re$. Smaller initial wave amplitudes delay the onset of the instabilities, and allow for significant laminar diffusion of the internal wave, leading to reduced turbulent activity. We discuss the complex interaction between the mean flow, internal gravity wave and turbulence, and its implications for internal wave-driven mixing in the ocean.

The first demonstration of laser action in ruby was made in 1960 by T. H. Maiman of Hughes Research Laboratories, USA. Many laboratories worldwide began the search for lasers using different materials, operating at different wavelengths. In the UK, academia, industry and the central laboratories took up the challenge from the earliest days to develop these systems for a broad range of applications. This historical review looks at the contribution the UK has made to the advancement of the technology, the development of systems and components and their exploitation over the last 60 years.

Turbulent mixing exerts a significant influence on many physical processes in the ocean. In a stably stratified Boussinesq fluid, this irreversible mixing describes the conversion of available potential energy (APE) to background potential energy (BPE). In some settings the APE framework is difficult to apply and approximate measures are used to estimate irreversible mixing. For example, numerical simulations of stratified turbulence often use triply periodic domains to increase computational efficiency. In this set-up, however, BPE is not uniquely defined and the method of Winters et al. (J. Fluid Mech., vol. 289, 1995, pp. 115–128) cannot be directly applied to calculate the APE. We propose a new technique to calculate APE in periodic domains with a mean stratification. By defining a control volume bounded by surfaces of constant buoyancy, we can construct an appropriate background buoyancy profile $b_\ast (z,t)$ and accurately quantify diapycnal mixing in such systems. This technique also permits the accurate calculation of a finite-amplitude local APE density in periodic domains. The evolution of APE is analysed in various turbulent stratified flow simulations. We show that the mean dissipation rate of buoyancy variance $\chi$ provides a good approximation to the mean diapycnal mixing rate, even in flows with significant variations in local stratification. When quantifying measures of mixing efficiency in transient flows, we find significant variation depending on whether laminar diffusion of a mean flow is included in the kinetic energy dissipation rate. We discuss how best to interpret these results in the context of quantifying diapycnal diffusivity in real oceanographic flows.

We report an idealized numerical study of a melting and freezing solid adjacent to a turbulent, buoyancy-affected shear flow, in order to improve our understanding of topography generation by phase changes in the environment. We use the phase-field method to dynamically couple the heat equation for the solid with the Navier–Stokes equations for the fluid. We investigate the evolution of an initially flat and horizontal solid boundary overlying a pressure-driven turbulent flow. We assume a linear equation of state for the fluid and change the sign of the thermal expansion coefficient, such that the background density stratification is either stable, neutral or unstable. We find that channels aligned with the direction of the mean flow are generated spontaneously by phase changes at the fluid–solid interface. Streamwise vortices in the fluid, the interface topography and the temperature field in the solid influence each other and adjust until a statistical steady state is obtained. The crest-to-trough amplitude of the channels is larger than approximately 10$\delta _{\nu }$ in all cases, with $\delta _{\nu }$ the viscous length scale, but is much larger and more persistent for an unstable stratification than for a neutral or stable stratification. This happens because a stable stratification makes the cool melt fluid buoyant such that it shields the channel from further melting, whereas an unstable stratification makes the cool melt fluid sink, inducing further melting by rising hot plumes. The statistics of flow velocities and melt rates are investigated, and we find that channels and keels emerging in our simulations do not significantly change the mean drag coefficient.

Cholinergic deficits are a hallmark of Alzheimer’s disease (AD) and Lewy body dementia (LBD). The nucleus basalis of Meynert (NBM) provides the major source of cortical cholinergic input; studying its functional connectivity might, therefore, provide a tool for probing the cholinergic system and its degeneration in neurodegenerative diseases. Forty-six LBD patients, 29 AD patients, and 31 healthy age-matched controls underwent resting-state functional magnetic resonance imaging (fMRI). A seed-based analysis was applied with seeds in the left and right NBM to assess functional connectivity between the NBM and the rest of the brain. We found a shift from anticorrelation in controls to positive correlations in LBD between the right/left NBM and clusters in right/left occipital cortex. Our results indicate that there is an imbalance in functional connectivity between the NBM and primary visual areas in LBD, which provides new insights into alterations within a part of the corticopetal cholinergic system that go beyond structural changes.