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We report on direct numerical simulations of the decay of initially isotropic, homogeneous turbulence subject to the application of stable density stratification. Flows were simulated for three different initial Reynolds numbers, but for the same initial Froude number. We find that the flows pass through three different dynamical regimes as they decay, depending on the local values of the Froude number and activity parameter. These regimes are analogous to those seen in the experimental study of Spedding (J. Fluid Mech., vol. 337, 1997, pp. 283–301) for the wake of a sphere. The flows initially decay with little influence of stratification, up to approximately one buoyancy period, when the local Froude number has dropped below 1. At this point the flows have adjusted to the density stratification, and, if the activity parameter is large enough, begin to decay at a slower rate and spread horizontally at a faster rate, consistent with the predictions of Davidson (J. Fluid Mech., vol. 663, 2010, pp. 268–292) and the scaling arguments of Billant & Chomaz (Phys. Fluids, vol. 13, 2001, pp. 1645–1651). We refer to this second regime as the stratified turbulence regime. As the flows continue to decay, ultimately the activity parameter drops below approximately 1 as viscous effects begin to dominate. In this regime, the flows have become quasi-horizontal, and approximately obey the scaling arguments of Godoy-Diana et al. (J. Fluid Mech., vol. 504, 2004, pp. 229–238).

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 $8192\\times 8192\\times 4096$ grid points. In addition to classifying regions consistently with contour plots of potential enstrophy, our method identifies quiescent regions as regions where $\\unicode[STIX]{x1D716}/\\unicode[STIX]{x1D708}N^{2}\\sim O(1)$ , layers as regions where $\\unicode[STIX]{x1D716}/\\unicode[STIX]{x1D708}N^{2}\\sim O(10)$ and patches as regions where $\\unicode[STIX]{x1D716}/\\unicode[STIX]{x1D708}N^{2}\\sim O(100)$ . Here, $\\unicode[STIX]{x1D716}$ is the dissipation rate of turbulence kinetic energy, $\\unicode[STIX]{x1D708}$ 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.

Classical scaling arguments of Kolmogorov, Oboukhov and Corrsin (KOC) are evaluated for turbulence strongly influenced by stable stratification. The simulations are of forced homogeneous stratified turbulence resolved on up to $8192\\times 8192\\times 4096$ grid points with buoyancy Reynolds numbers of $\\mathit{Re}_{b}=13$ , 48 and 220. A simulation of isotropic homogeneous turbulence with a mean scalar gradient resolved on $8192^{3}$ grid points is used as a benchmark. The Prandtl number is unity. The stratified flows exhibit KOC scaling only for second-order statistics when $\\mathit{Re}_{b}=220$ ; the $4/5$ law is not observed. At lower $\\mathit{Re}_{b}$ , the $-5/3$ slope in the spectra occurs at wavenumbers where the bottleneck effect occurs in unstratified cases, and KOC scaling is not observed in any of the structure functions. For the probability density functions (p.d.f.s) of the scalar and kinetic energy dissipation rates, the lognormal model works as well for the stratified cases with $\\mathit{Re}_{b}=48$ and 220 as it does for the unstratified case. For lower $\\mathit{Re}_{b}$ , the dominance of the vertical derivatives results in the p.d.f.s of the dissipation rates tending towards bimodal. The p.d.f.s of the dissipation rates locally averaged over spheres with radius in the inertial range tend towards bimodal regardless of $\\mathit{Re}_{b}$ . There is no broad scaling range, but the intermittency exponents at length scales near the Taylor length are in the range of $0.25\\pm 0.05$ and $0.35\\pm 0.1$ for the velocity and scalar respectively.

Direct numerical simulations of stratified turbulence are used to test several fundamental assumptions involved in the Osborn, Osborn–Cox, and Thorpe methods commonly used to estimate the turbulent diffusivity from field measurements. The forced simulations in an idealized triply periodic computational domain exhibit characteristic features of stratified turbulence including intermittency and layer formation. When calculated using the volume-averaged dissipation rates from the simulations, the vertical diffusivities inferred from the Osborn and Osborn–Cox methods are within 40% of the value diagnosed using the volume-averaged buoyancy flux for all cases, while the Thorpe-scale method performs similarly well in the simulation with a relatively large buoyancy Reynolds number (Re b ≃ 240) but significantly overestimates the vertical diffusivity in simulations with Re b < 60. The methods are also tested using a limited number of vertical profiles randomly selected from the computational volume. The Osborn, Osborn–Cox, and Thorpe-scale methods converge to their respective estimates based on volume-averaged statistics faster than the vertical diffusivity calculated directly from the buoyancy flux, which is contaminated with reversible contributions from internal waves. When applied to a small number of vertical profiles, several assumptions underlying the Osborn and Osborn–Cox methods are not well supported by the simulation data. However, the vertical diffusivity inferred from these methods compares reasonably well to the exact value from the simulations and outperforms the assumptions underlying these methods in terms of the relative error. Motivated by a recent theoretical development, it is speculated that the Osborn method might provide a reasonable approximation to the diffusivity associated with the irreversible buoyancy flux.

Recently, direct numerical simulations (DNS) of stably stratified turbulence have shown that as the Prandtl number ($Pr$) is increased from 1 to 7, the mean turbulent potential energy dissipation rate (TPE-DR) drops dramatically, while the mean turbulent kinetic energy dissipation rate (TKE-DR) increases significantly. Through an analysis of the equations governing the fluctuating velocity and density gradients we provide a mechanistic explanation for this surprising behaviour and test the predictions using DNS. We show that the mean density gradient gives rise to a mechanism that opposes the production of fluctuating density gradients, and this is connected to the emergence of ramp cliffs. The same term appears in the velocity gradient equation but with the opposite sign, and is the contribution from buoyancy. This term is ultimately the reason why the TPE-DR reduces while the TKE-DR increases with increasing $Pr$. Our analysis also predicts that the effects of buoyancy on the smallest scales of the flow become stronger as $Pr$ is increased, and this is confirmed by our DNS data. A consequence of this is that the standard buoyancy Reynolds number does not correctly estimate the impact of buoyancy at the smallest scales when $Pr$ deviates from 1, and we derive a suitable alternative parameter. Finally, an analysis of the filtered gradient equations reveals that the mean density gradient term changes sign at sufficiently large scales, such that buoyancy acts as a source for velocity gradients at small scales, but as a sink at large scales.

Whole blood (WB) transcriptomics offers a minimal-invasive method to assess patients’ immune system. This study aimed to identify transcriptional patterns in WB associated with clinical outcomes in patients treated with immune checkpoint inhibitors (ICIs). We performed RNA-sequencing on pre-treatment WB samples from 145 patients with advanced cancer. Additionally, we compiled a separate dataset of 14,085 WB transcriptomes from diverse health backgrounds from public repositories and applied consensus-independent component analysis (c-ICA) to identify transcriptional components (TCs). The biological processes represented by these TCs were elucidated using gene set enrichment analysis. The activity of the TCs was then quantified in the 145 WB profiles and analyzed for associations with tumor response, progression-free survival, and overall survival using univariate and multivariate analyses in a permutation framework. RNA-sequencing variant calling was performed, and the activity of the TCs was assessed in specific cell lineages using a single-cell immune cell atlas of the human hematopoietic system. c-ICA on 14,085 WB transcriptomes identified 1262 distinct TCs representing various cellular processes. Of these, 18 TCs were associated with ≥ 1 outcome parameter, with three specifically linked to tumor response. Top genes in these three TCs included CCHCR1 , TCF19 , LTA , DDX39B , and PPP 1R18 . RNA-sequencing variant calling and single-cell transcriptome projections revealed associations between these four TCs and germline variants. These findings support the potential of the identified WB-based transcriptional patterns to complement tumor characteristics in predictive and prognostic models for improved patient stratification.

In freely decaying stably stratified turbulent flows, numerical evidence shows that the horizontal displacement of Lagrangian tracers is diffusive while the vertical displacement converges towards a stationary distribution, as shown numerically by Kimura & Herring (J. Fluid Mech., vol. 328, 1996, pp. 253–269). Here, we develop a stochastic model for the vertical dispersion of Lagrangian tracers in stably stratified turbulent flows that aims to replicate and explain the emergence of a stationary probability distribution for the vertical displacement of such tracers. More precisely, our model is based on the assumption that the dynamical evolution of the tracers results from the competing effects of buoyancy forces that tend to bring a vertically perturbed fluid parcel (carrying tracers) to its equilibrium position and turbulent fluctuations that tend to disperse tracers. When the density of a fluid parcel is allowed to change due to molecular diffusion, a third effect needs to be taken into account: irreversible mixing. Indeed, ‘mixing’ dynamically and irreversibly changes the equilibrium position of the parcel and affects the buoyancy force that ‘stirs’ it on larger scales. These intricate couplings are modelled using a stochastic resetting process (Evans & Majumdar, Phys. Rev. Lett., vol. 106, issue 16, 2011, 160601) with memory. More precisely, Lagrangian tracers in stratified turbulent flows are assumed to follow random trajectories that obey a Brownian process. In addition, their stochastic paths can be reset to a given position (corresponding to the dynamically changing equilibrium position of a density structure containing the tracers) at a given rate. Scalings for the model parameters as functions of the molecular properties of the fluid and the turbulent characteristics of the flow are obtained by analysing the dynamics of an idealised density structure. Even though highly idealised, the model has the advantage of being analytically solvable. In particular, we show the emergence of a stationary distribution for the vertical displacement of Lagrangian tracers. We compare the predictions of this model with direct numerical simulation data at various Prandtl numbers $Pr$, the ratio of kinematic viscosity to molecular diffusion.

Budgets of turbulent kinetic energy (TKE) and turbulent potential energy (TPE) at different scales $\\ell$ in sheared, stably stratified turbulence are analysed using a filtering approach. Competing effects in the flow are considered, along with the physical mechanisms governing the energy fluxes between scales, and the budgets are used to analyse data from direct numerical simulation at buoyancy Reynolds number $Re_b=O(100)$. The mean TKE exceeds the TPE by an order of magnitude at the large scales, with the difference reducing as $\\ell$ is decreased. At larger scales, buoyancy is never observed to be positive, with buoyancy always converting TKE to TPE. As $\\ell$ is decreased, the probability of locally convecting regions increases, though it remains small at scales down to the Ozmidov scale. The TKE and TPE fluxes between scales are both downscale on average, and their instantaneous values are correlated positively, but not strongly so, and this occurs due to the different physical mechanisms that govern these fluxes. Moreover, the contributions to these fluxes arising from the sub-grid fields are shown to be significant, in addition to the filtered scale contributions associated with the processes of strain self-amplification, vortex stretching and density gradient amplification. Probability density functions (PDFs) of the $Q,R$ invariants of the filtered velocity gradient are considered and show that as $\\ell$ increases, the sheared-drop shape of the PDF becomes less pronounced and the PDF becomes more symmetric about $R=0$.

We present a deep probabilistic convolutional neural network (PCNN) model for predicting local values of small-scale mixing properties in stratified turbulent flows, namely the dissipation rates of turbulent kinetic energy and density variance, $\\varepsilon$ and $\\chi$. Inputs to the PCNN are vertical columns of velocity and density gradients, motivated by data typically available from microstructure profilers in the ocean. The architecture is designed to enable the model to capture several characteristic features of stratified turbulence, in particular the dependence of small-scale isotropy on the buoyancy Reynolds number $Re_b:=\\varepsilon /(\\nu N^2)$, where $\\nu$ is the kinematic viscosity and $N$ is the background buoyancy frequency, the correlation between suitably locally averaged density gradients and turbulence intensity and the importance of capturing the tails of the probability distribution functions of values of dissipation. Empirically modified versions of commonly used isotropic models for $\\varepsilon$ and $\\chi$ that depend only on vertical derivatives of density and velocity are proposed based on the asymptotic regimes $Re_b\\ll 1$ and $Re_b\\gg 1$, and serve as an instructive benchmark for comparison with the data-driven approach. When trained and tested on a simulation of stratified decaying turbulence which accesses a range of turbulent regimes (associated with differing values of $Re_b$), the PCNN outperforms assumptions of isotropy significantly as $Re_b$ decreases, and additionally demonstrates improvements over the fitted empirical models. A differential sensitivity analysis of the PCNN facilitates a comparison with the theoretical models and provides a physical interpretation of the features enabling it to make improved predictions.

We investigate the effects of the turbulent dynamic range on active scalar mixing in stably stratified turbulence by adapting the theoretical passive scalar modelling arguments of Beguier, Dekeyser & Launder (1978) (Phys. Fluids, vol. 21 (3), pp. 307–310) and demonstrating their usefulness through consideration of the results of direct numerical simulations of statistically stationary homogeneous stratified and sheared turbulence. By analysis of inertial and inertial–convective subrange scalings, we show that the relationship between the active scalar and turbulence time scales is predicted by the ratio of the Kolmogorov and Obukhov–Corrsin constants, provided mean flow parameters permit the two subrange scalings to be appropriate approximations. We use the resulting relationship between time scales to parameterise an appropriate turbulent mixing coefficient $\\varGamma \\equiv \\chi /\\epsilon$, defined here as the ratio of available potential energy ($E_p$) and turbulent kinetic energy ($E_k$) dissipation rates. With the analysis presented here, we show that $\\varGamma$ can be estimated by $E_p,E_k$ and a universal constant provided an appropriate Reynolds number is sufficiently high. This large Reynolds number regime appears here to occur at $ {{Re_b}} \\equiv \\epsilon / \\nu N^{2} \\gtrapprox 300$ where $\\nu$ is the kinematic viscosity and $N$ is the characteristic buoyancy frequency. We propose a model framework for irreversible diapycnal mixing with robust theoretical parametrisation and asymptotic behaviour in this high-$ {{Re_b}}$ limit.