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The transport of matter by turbulent flows plays an important role, in particular in a geophysical context. Here, we study the relative movement of groups of two (pairs) and four (tetrahedra) Lagrangian particles using direct numerical simulations of the stably-stratified Boussinsesq equations, with Brunt-V\"ais\"al\"a frequency $N$ and Coriolis parameter $f$. We cover regimes close to homogeneous isotropic turbulence, to flows dominated by stratification and rotation, keeping fixed the ratio $N/f = 5$. The flows studied are anisotropic, so the relative motion between two particles depends not only on the initial separation between the particles, but also on their orientation with respect to the vertical axis. In all cases considered, we demonstrate that the relative particle motion differs depending on whether dispersion is considered forward or backwards in time, although the asymmetry becomes less pronounced when stratification and rotation increase. On the other hand, the strong fluctuations in the dispersion between two particles become more extreme when $N$ and $f$ increase. We also find evidence for the formation of shear layers, which become more pronounced as $N$ and $f$ become larger. Finally, we show that the irreversibility on the dispersion of a set of particles forming initially a regular tetrahedron becomes weaker when the influence of stratification and rotation increase, a property that we relate to that of the perceived rate-of-strain tensor.

Turbulence in stratified and rotating turbulent flows is characterized by an interplay between waves and eddies, resulting in continuous exchanges between potential and kinetic energy. Here, we study how these processes affect the turbulent energy cascade from large to small scales, which manifests itself by an irreversible evolution of the relative kinetic energy between two tracer particles. We find that when $r_0$, the separation between particles, is below a characteristic length $\ell_t$, potential energy is on average transferred to kinetic energy, reducing time irreversibility, and conversely when $r_0 > \ell_t$. Our study reveals that the scale $\ell_t$ coincides with the buoyancy length scale $L_B$ over a broad range of configurations until a transitional wave-dominated regime is reached.

We discuss a model for directed percolation in which the flux of material along each bond is a dynamical variable. The model includes a physically significant limiting case where the total flux of material is conserved. We show that the distribution of fluxes is asymptotic to a power law at small fluxes. We give an implicit equation for the exponent, in terms of probabilities characterising site occupations. In one dimension the site occupations are exactly independent, and the model is exactly solvable. In two dimensions, the independent-occupation assumption gives a good approximation. We explore the relationship between this model and traditional models for directed percolation., Comment: 30 pages, 16 figures

We investigate the role of pressure, via its Hessian tensor $\mathbf{H}$, on amplification of vorticity and strain-rate and contrast it with other inviscid nonlinear mechanisms. Results are obtained from direct numerical simulations of isotropic turbulence with Taylor-scale Reynolds number in the range $140-1300$. Decomposing $\mathbf{H}$ into local isotropic ($\mathbf{H}^{\rm I}$) and nonlocal deviatoric ($\mathbf{H}^{\rm D}$) components reveals that $\mathbf{H}^{\rm I}$ depletes vortex stretching (VS), whereas $\mathbf{H}^{\rm D}$ enables it, with the former slightly stronger. The resulting inhibition is significantly weaker than the nonlinear mechanism which always enables VS. However, in regions of intense vorticity, identified using conditional statistics, contribution from $\mathbf{H}$ dominates over nonlinearity, leading to overall depletion of VS. We also observe near-perfect alignment between vorticity and the eigenvector of $\mathbf{H}$ corresponding to the smallest eigenvalue, which conforms with well-known vortex-tubes. We discuss the connection between this depletion, essentially due to (local) $\mathbf{H}^{\rm I}$, and recently identified self-attenuation mechanism [Buaria et al. {\em Nat. Commun.} 11:5852 (2020)], whereby intense vorticity is locally attenuated through inviscid effects. In contrast, the influence of $\mathbf{H}$ on strain-amplification is weak. It opposes strain self-amplification, together with VS, but its effect is much weaker than VS. Correspondingly, the eigenvectors of strain and $\mathbf{H}$ do not exhibit any strong alignments. For all results, the dependence on Reynolds number is very weak. In addition to the fundamental insights, our work provides useful data and validation benchmarks for future modeling endeavors, for instance in Lagrangian modeling of velocity gradient dynamics, where conditional $\mathbf{H}$ is explicitly modeled., Comment: 14 pages, 10 figures

We derive from first principles analytic relations for the second and third order moments of the velocity gradient mij = dui/dxj in compressible turbulence, which generalize known relations in incompressible flows. These relations, although derived for homogeneous flows, hold approximately for a mixing layer. We also discuss how to apply these relations to determine all the second and third moments of the velocity gradient experimentally., Comment: 10 pages, 5 figures

The transfer of kinetic energy from large to small scales is a hallmark of turbulent flows. Yet, a precise mechanistic description of this transfer, which is expected to occur via an energy cascade, is still missing. Several conceptually simple configurations with vortex tubes have been proposed as a testing ground to understand the energy cascade. Here, we focus on incompressible flows and compare the energy transfer occurring in a statistically steady homogeneous isotropic turbulent (HIT) flow with the generation of fine-scale motions in configurations involving vortex tubes. We start by filtering the velocity field in bands of wavenumbers distributed logarithmically, which allows us to study energy transfer in Fourier space and also visualize the energy cascade in real space. In the case of a statistically steady HIT flow at a moderate Reynolds number, our numerical results do not reveal any significant correlation between regions of intense energy transfers and vorticity or strain, filtered in corresponding wavenumber bands, nor any simple self-similar process. In comparison, in the transient turbulent flow obtained from the interaction between two antiparallel vortex tubes, we observe a qualitatively simpler organization of the intense structures, as well as of the energy transfer. The process leading to vortex reconnection via flattening of interacting vortex cores in two intense ribbons of vorticity, also appears as qualitatively different from the physics of HIT. Our results indicate that the specific properties of the transient flows affect the way energy is transferred, and may not be representative of HIT.

Building upon the intrinsic properties of Navier-Stokes dynamics, namely the prevalence of intense vortical structures and the interrelationship between vorticity and strain rate, we propose a simple framework to quantify the extreme events and the smallest scales of turbulence. We demonstrate that our approach is in excellent agreement with the best available data from direct numerical simulations of isotropic turbulence, with Taylor-scale Reynolds number up to 1300. We additionally highlight a shortcoming of prevailing intermittency models due to their disconnection from observed correlation between vorticity and strain. Our work accentuates the importance of this correlation as a crucial step in developing an accurate understanding of intermittency in turbulence., Comment: 6 pages, 4 figures

The head-on collision of two vortex rings can produce diverse phenomena: a tiara of secondary rings, vortex sheets which flatten and interact iteratively, or the violent disintegration of the rings into a turbulent cloud. The outcome of the interaction is determined by the nature of the instability affecting two impinging vortex rings. Here, we carry out a systematic study to determine the dominant instability as a function of the parameters of the problem. To this end, we numerically simulate the head on collision of vortex rings with circulation Reynolds numbers between $1000$ and $3500$ and varying slenderness ratios $\Lambda=a/R$ ranging from $\Lambda=0.1$ to $0.35$, with $a$ the core radius and $R$ the ring radius. By studying the temporal evolution of the energy and viscous dissipation, we elucidate the role azimuthal instabilities play in determining what the outcomes of the collision are. We then compare these collisions to the head-on impact of a vortex ring on a free-slip and a no-slip wall. The free-slip wall imposes a mirror symmetry, which impedes certain instabilities and at sufficiently large Reynolds numbers leads to the formation of a half-tiara of vortices. Impact against a no-slip wall results in the process where a secondary vortex ring is formed after the ejection of the resulting boundary layer. When the Reynolds number is above a certain threshold, which increases with $\Lambda$, the vortices disintegrate through azimuthal instabilities, resulting in a turbulent cloud.

Intense fluctuations of energy dissipation rate in turbulent flows result from the self-amplification of strain rate via a quadratic nonlinearity, with contributions from vorticity (via the vortex stretching mechanism) and the pressure Hessian tensor, which we analyze here using direct numerical simulations of isotropic turbulence in periodic domains of up to $12288^3$ grid points, and Taylor-scale Reynolds numbers in the range $140-1300$. We extract the statistics of various terms involved in amplification of strain and additionally condition them on the magnitude of strain. We find that strain is overall self-amplified by the quadratic nonlinearity, and depleted via vortex stretching; whereas pressure Hessian acts to redistribute strain fluctuations towards the mean-field and thus depleting intense strain. Analyzing the intense fluctuations of strain in terms of its eigenvalues reveals that the net amplification is solely produced by the third eigenvalue, resulting in strong compressive action. In contrast, the self-amplification terms acts to deplete the other two eigenvalues, whereas vortex stretching acts to amplify them, both effects canceling each other almost perfectly. The effect of the pressure Hessian for each eigenvalue is qualitatively similar to that of vortex stretching, but significantly weaker in magnitude. Our results conform with the familiar notion that intense strain is organized in sheet-like structures, which are in the vicinity of, but never overlap with regions of intense vorticity due to fundamental differences in their amplifying mechanisms., Comment: 13 pages, 7 figures

We observe the emergence of strong vertical drafts in direct numerical simulations of the Boussinesq equations in a range of parameters of geophysical interest. These structures, which appear intermittently in space and time, generate turbulence and enhance kinetic and potential energy dissipation, providing a possible explanation for the observed variability of the local energy dissipation in the bulk of oceanic flows, and the modulation of its probability distribution function. We show how, due to the extreme drafts, in runs with Froude numbers observable in geophysical scenarios, roughly 10% of the domain flow can account for up to 50% of the global volume dissipation, reminiscent of estimates based on oceanic models., Comment: 16 pages, 6 figures

The nonlinear and nonlocal coupling of vorticity and strain-rate constitutes a major hindrance in understanding the self-amplification of velocity gradients in turbulent fluid flows. Utilizing highly-resolved direct numerical simulations of isotropic turbulence in periodic domains of up to $12288^3$ grid points, and Taylor-scale Reynolds number $R_\lambda$ in the range $140-1300$, we investigate this non-locality by decomposing the strain-rate tensor into local and non-local contributions obtained through Biot-Savart integration of vorticity in a sphere of radius $R$. We find that vorticity is predominantly amplified by the non-local strain coming beyond a characteristic scale size, which varies as a simple power-law of vorticity magnitude. The underlying dynamics preferentially align vorticity with the most extensive eigenvector of non-local strain. The remaining local strain aligns vorticity with the intermediate eigenvector and does not contribute significantly to amplification; instead it surprisingly attenuates intense vorticity, leading to breakdown of the observed power-law and ultimately also the scale-invariance of vorticity amplification, with important implications for prevailing intermittency theories., Comment: 6 pages, 6 figures

Non-Gaussian statistics of large-scale fields are routinely observed in data from atmospheric and oceanic campaigns and global models. Recent direct numerical simulations (DNSs) showed that large-scale intermittency in stably stratified flows is due to the emergence of sporadic, extreme events in the form of bursts in the vertical velocity and the temperature. This phenomenon results from the interplay between waves and turbulent motions, affecting mixing. We provide evidence of the enhancement of the classical small-scale (or internal) intermittency due to the emergence of large-scale drafts, connecting large- and small-scale bursts. To this aim we analyze a large set of DNSs of the stably stratified Boussinesq equations over a wide range of values of the Froude number ($Fr\approx 0.01-1$). The variation of the buoyancy field kurtosis with $Fr$ is similar to (though with smaller values than) the kurtosis of the vertical velocity, both showing a non-monotonic trend. We present a mechanism for the generation of extreme vertical drafts and vorticity enhancements which follows from the exact equations for field gradients., Comment: 7 pages, 4 figures

At high Reynolds number, the interaction between two vortex tubes leads to intense velocity gradients, which are at the heart of fluid turbulence. This vorticity amplification comes about through two different instability mechanisms of the initial vortex tubes, assumed anti-parallel and with a mirror plane of symmetry. At moderate Reynolds number, the tubes destabilize via a Crow instability, with the nonlinear development leading to strong flattening of the cores into thin sheets. These sheets then break down into filaments which can repeat the process. At higher Reynolds number, the instability proceeds via the elliptical instability, producing vortex tubes that are perpendicular to the original tube directions. In this work, we demonstrate that these same transition between Crow and Elliptical instability occurs at moderate Reynolds number when we vary the initial angle $\beta$ between two straight vortex tubes. We demonstrate that when the angle between the two tubes is close to $\pi/2$, the interaction between tubes leads to the formation of thin vortex sheets. The subsequent breakdown of these sheets involves a twisting of the paired sheets, followed by the appearance of a localized cloud of small scale vortex structures. At smaller values of the angle $\beta$ between the two tubes, the breakdown mechanism changes to an elliptic cascade-like mechanism. Whereas the interaction of two vortices depends on the initial condition, the rapid formation of fine-scales vortex structures appears to be a robust feature, possibly universal at very high Reynolds numbers.

Turbulent fluid flows are ubiquitous in nature and technology, and are mathematically described by the incompressible Navier-Stokes equations (INSE). A hallmark of turbulence is spontaneous generation of intense whirls, resulting from amplification of the fluid rotation-rate (vorticity) by its deformation-rate (strain). This interaction, encoded in the non-linearity of INSE, is non-local, i.e., depends on the entire state of the flow, constituting a serious hindrance in turbulence theory and in establishing regularity of INSE. Here, we unveil a novel aspect of this interaction, by separating strain into local and non-local contributions utilizing the Biot-Savart integral of vorticity in a sphere of radius R. Analyzing highly-resolved numerical turbulent solutions to INSE, we find that when vorticity becomes very large, the local strain over small R surprisingly counteracts further amplification. This uncovered self-attenuation mechanism is further shown to be connected to local Beltramization of the flow, and could provide a direction in establishing the regularity of INSE., Comment: 8 pages, 3 figures

Preterm labor is the leading cause of neonatal morbidity and mortality and has attracted research efforts from many scientific areas. The inter-relationship between uterine contraction and the underlying electrical activities makes uterine electrohysterogram (EHG) a promising direction for preterm detection and prediction. Due the scarcity of EHG signals, especially those of preterm patients, synthetic algorithms are applied to create artificial samples of preterm type in order to remove prediction bias towards term, at the expense of a reduction of the feature effectiveness in machine-learning based automatic preterm detecting. To address such problem, we quantify the effect of synthetic samples (balance coefficient) on features' effectiveness, and form a general performance metric by utilizing multiple feature scores with relevant weights that describe their contributions to class separation. Combined with the activation/inactivation functions that characterizes the effect of the abundance of training samples in term and preterm prediction precision, we obtain an optimal sample balance coefficient that compromise the effect of synthetic samples in removing bias towards the majority and the side-effect of reducing features' importance. Substantial improvement in prediction precision has been achieved through a set of numerical tests on public available TPEHG database, and it verifies the effectiveness of the proposed method.

An essential ingredient of turbulent flows is the vortex stretching mechanism, which emanates from the non-linear interaction of vorticity and strain-rate tensor and leads to formation of extreme events. We analyze the statistical correlations between vorticity and strain rate by using a massive database generated from very well resolved direct numerical simulations of forced isotropic turbulence in periodic domains. The grid resolution is up to $12288^3$, and the Taylor-scale Reynolds number is in the range $140-1300$. In order to understand the formation and structure of extreme vorticity fluctuations, we obtain statistics conditioned on enstrophy (vorticity-squared). The magnitude of strain, as well as its eigenvalues, is approximately constant when conditioned on weak enstrophy; whereas they grow approximately as power laws for strong enstrophy, which become steeper with increasing $R_\lambda$. We find that the well-known preferential alignment between vorticity and the intermediate eigenvector of strain tensor is even stronger for large enstrophy, whereas vorticity shows a tendency to be weakly orthogonal to the most extensive eigenvector (for large enstrophy). Yet the dominant contribution to the production of large enstrophy events arises from the most extensive eigendirection, the more so as $R_\lambda$ increases. Nevertheless, the stretching in intense vorticity regions is significantly depleted, consistent with the kinematic properties of weakly-curved tubes in which they are organized. Further analysis reveals that intense enstrophy is primarily depleted via viscous diffusion, though viscous dissipation is also significant. Implications for modeling are nominally addressed as appropriate., Comment: 17 pages, 8 figures

Rhythmogenesis, which is critical for many biological functions, involves a transition to coherent activity through cell-cell communication. In the absence of centralized coordination by specialized cells (pacemakers), competing oscillating clusters impede this global synchrony. We show that spatial symmetry-breaking through a frequency gradient results in the emergence of localized wave sources driving system-wide activity. Such gradients, arising through heterogeneous inter-cellular coupling, may explain directed rhythmic activity during labor in the uterus despite the absence of pacemakers., Comment: 6 pages, 3 figures

Geophysical fluid flows are predominantly turbulent and often strongly affected by the Earth's rotation, as well as by stable density stratification. Using direct numerical simulations of forced Boussinesq equations, we study the influence of these effects on the motion of fluid particles, focusing on cases where the frequencies associated with rotation and stratification (RaS), $N$ and $f$ respectively, are held at a fixed ratio $N/f=5$. As the intensity of RaS increases, a sharp transition is observed between a regime dominated by eddies to a regime dominated by waves, which can also be seemingly described by simply comparing the time scale $1/N$ and $\tau_\eta$ (the Kolmogorov time scale). We perform a detailed study of Lagrangian statistics of acceleration, velocity and related quantities in the two regimes. The flow anisotropy induces a clear difference between particle motion in the horizontal and vertical directions. In the regime $N\tau_\eta <1$, acceleration statistics in both horizontal and vertical directions, exhibit well known characteristics of isotropic turbulence. In contrast for $N\tau_\eta >1$, they are directly influenced by imposed RaS. The Lagrangian velocity statistics exhibit visible anisotropy for all runs; nevertheless the degree of anisotropy becomes very strong in the regime $N\tau_\eta >1$. We find that in the regime $N\tau_\eta <1$, rotation enhances the mean displacement of particles in horizontal planes at short times, but inhibits them at longer times. This inhibition of horizontal displacement becomes stronger for $N\tau_\eta >1$, with no clear diffusive behavior. Displacements in the vertical direction are always inhibited. The inhibition becomes extremely strong when $N\tau_\eta >1$, with the particles almost being trapped horizontally., Comment: 24 pages, 10 figures

The essence of turbulent flow is the conveyance of energy through the formation, interaction, and destruction of eddies over a wide range of spatial scales--from the largest scales where energy is injected, down to the smallest scales where it is dissipated through viscosity. Currently, there is no mechanistic framework that captures how the interactions of vortices drive this cascade. We show that iterations of the elliptical instability, arising from the interactions between counter-rotating vortices, lead to the emergence of turbulence. We demonstrate how the nonlinear development of the elliptical instability generates an ordered array of antiparallel secondary filaments. The secondary filaments mutually interact, leading to the formation of even smaller tertiary filaments. In experiments and simulations, we observe two and three iterations of this cascade, respectively. Our observations indicate that the elliptical instability could be one of the fundamental mechanisms by which the turbulent cascade develops., Comment: 8 Page Manuscript, 5 Figures, 18 Page Supplemental, 10 Supplemental Figures, and 11 Supplemental Movies

How non-spherical particles orient as they settle in a flow has important practical implications in a number of scientific and engineering problems. In a quiescent fluid, a slowly settling particle orients so that it settles with its broad side first. This is an effect of the torque due to convective inertia of the fluid set in motion by the settling particle, which maximises the drag experienced by the particle. Turbulent flows tend to randomise the particle orientation. Recently the settling of non-spherical particles in turbulence was analysed neglecting the effect of convective fluid inertia, but taking into account the effect of the turbulent fluid-velocity gradients on the particle orientation. These studies reached the opposite conclusion, namely that a rod settles preferentially with its tip first, wheras a disk settles with its edge first, therefore minimizing the drag on the particle. Here, we consider both effects, the convective inertial torque as well as the torque due to fluctuating velocity gradients, and ask under which circumstances either one or the other dominate. To this end we estimate the ratio of the magnitudes of the two torques. Our estimates suggest that the fluid-inertia torque prevails in high-Reynolds number flows. In this case non-spherical particles are expected to settle with a maximal drag. But when the Reynolds number is small then the torque due to fluid-velocity gradients may dominate, causing the particle to settle with its broad side first.

Fully turbulent flows are characterized by intermittent formation of very localized and intense velocity gradients. These gradients can be orders of magnitude larger than their typical value and lead to many unique properties of turbulence. Using direct numerical simulations of the Navier-Stokes equations with unprecedented small-scale resolution, we characterize such extreme events over a significant range of turbulence intensities, parameterized by the Taylor-scale Reynolds number ($R_\lambda$). Remarkably, we find the strongest velocity gradients to empirically scale as $\tau_K^{-1} R_\lambda^{\beta}$, with $\beta \approx 0.775 \pm 0.025$, where $\tau_K$ is the Kolmogorov time scale (with its inverse, $\tau_K^{-1}$, being the {r.m.s.} of velocity gradient fluctuations). Additionally, we observe velocity increments across very small distances $r \le \eta$, where $\eta$ is the Kolmogorov length scale, to be as large as the {r.m.s.} of the velocity fluctuations. Both observations suggest that the smallest length scale in the flow behaves as $\eta R_\lambda^{-\alpha}$, with $\alpha = \beta - \frac{1}{2}$, which is at odds with predictions from existing phenomenological theories. We find that extreme gradients are arranged in vortex tubes, such that strain conditioned on vorticity grows on average slower than vorticity, approximately as a power law with an exponent $\gamma < 1$, which weakly increases with $R_\lambda$. Using scaling arguments, we get $\beta=(2-\gamma)^{-1}$, which suggests that $\beta$ would also slowly increase with $R_\lambda$. We conjecture that approaching the limit of infinite $R_\lambda$, the flow is overall smooth, with intense velocity gradients over scale $ \eta R_\lambda^{-1/2}$, corresponding to $\beta = 1$., Comment: 14 pages, 8 figures

Many physical processes result in very uneven, apparently random, distributions of matter, characterized by fluctuations of the local density over orders of magnitude. The density of matter in the sparsest regions can have a power-law distribution, with an exponent that we term the lacunarity exponent. We discuss a mechanism which explains the wide occurrence of these power laws, and give analytical expressions for the exponent in some simple models., Comment: 5 pages, 4 figures

When vortex rings collide head-on at high enough Reynolds numbers, they ultimately annihilate through a violent interaction which breaks down their cores into a turbulent cloud. We experimentally show that this very strong interaction, which leads to the production of fluid motion at very fine scales, uncovers direct evidence of a novel iterative cascade of instabilities in a bulk fluid. When the coherent vortex cores approach each other, they deform into tent-like structures, and the mutual strain causes them to locally flatten into extremely thin vortex sheets. These sheets then break down into smaller secondary vortex filaments, which themselves rapidly flatten and break down into even smaller tertiary filaments. By performing numerical simulations of the full Navier-Stokes equations, we also resolve one iteration of this instability and highlight the subtle role that viscosity must play in the rupturing of a vortex sheet. The concurrence of this observed iterative cascade of instabilities over various scales with those of recent theoretical predictions could provide a new mechanistic framework in which the evolution of turbulent flows can be examined in real-time as a series of discrete dynamic instabilities., Comment: 11 pages, 9 figures, 6 videos

Phoresis, the drift of particles induced by scalar gradients in a flow, can result in an effective compressibility, bringing together or repelling particles from each other. Here, we ask whether this effect can affect the transport of particles in a turbulent flow. To this end, we study how the dispersion of a cloud of phoretic particles is modified when injected in the flow, together with a blob of scalar, whose effect is to transiently bring particles together, or push them away from the center of the blob. The resulting phoretic effect can be quantified by a single dimensionless number. Phenomenological considerations lead to simple predictions for the mean separation between particles, which are consistent with results of direct numerical simulations. Using the numerical results presented here, as well as those from previous studies, we discuss quantitatively the experimental consequences of this work and the possible impact of such phoretic mechanisms in natural systems., Comment: 17 pages, 6 figures

This paper analyses a model for the intensity of distribution for rays propagating without absorption in a random medium. The random medium is modelled as a dynamical map. After $N$ iterations, the intensity is modelled as a sum $S$ of ${\cal N}$ contributions from different trajectories, each of which is a product of $N$ independent identically distributed random variables $x_k$, representing successive focussing or de-focussing events. The number of ray trajectories reaching a given point is assumed to proliferate exponentially: ${\cal N}=\Lambda^N$, for some $\Lambda>1$. We investigate the probability distribution of $S$. We find a phase transition as parameters of the model are varied. There is a phase where the fluctuations of $S$ are suppressed as $N\to \infty$, and a phase where the $S$ has large fluctuations, for which we provide a large deviation analysis., Comment: 14 pages, 3 figures

Intrinsic instability of trajectories characterizes chaotic dynamical systems. We report here that trajectories can exhibit a surprisingly high degree of stability, over a very long time, in a chaotic dynamical system. We provide a detailed quantitative description of this effect for a one-dimensional model of inertial particles in a turbulent flow using large-deviation theory. Specifically, the determination of the entropy function for the distribution of finite-time Lyapunov exponents reduces to the analysis of a Schr\"odinger equation, which is tackled by semi-classical methods., Comment: 6 pages, 4 figures

The expected universality of small-scale properties of turbulent flows implies isotropic properties of the velocity gradient tensor in the very large Reynolds number limit. Using direct numerical simulations, we determine the tensors formed by n = 2 and 3 velocity gradients at a single point in turbulent homogeneous shear flows, and in the log-layer of a turbulent channel flow, and we characterize the departure of these tensors from the corresponding isotropic prediction. Specifically, we separate the even components of the tensors, invariant under reflexion with respect to all axes, from the odd ones, which identically vanish in the absence of shear. Our results indicate that the largest deviation from isotropy comes from the odd component of the third velocity gradient correlation function, especially from the third moment of the derivative along the normal direction of the streamwise velocity component. At the Reynolds numbers considered (R_lambda ~ 140), we observe that these second and third order correlation functions are significantly larger in turbulent channel flows than in homogeneous shear flow. Overall, our work demonstrates that a mean shear leads to relatively simple structure of the velocity gradient tensor. How isotropy is restored in the very large Reynolds limit remains to be understood., Comment: 14 pages, 2 figures. To appear in Phys. Review Fluids

We present results from moist convection in a mixture of pressurized sulfur hexa-flouride (liquid and vapor) and helium (gas) to model the wet and dry components of the earth's atmosphere. To allow for homogeneous nucleation, we operate the experiment close to critical conditions. We report on the nucleation of microdroplets in the wake of large cold liquid drops falling through the supersaturated atmosphere and show that the homogeneous nucleation is caused by isobaric cooling of the saturated sulfur hexaflouride vapor. Our results carry over to atmospheric clouds: falling hail and cold rain drops may enhance the heterogeneous nucleation of microdroplets in their wake under supersaturated atmospheric conditions. We also observed that under appropriate conditions settling microdroplets form a rather stable horizontal cloud layer, which separates regions of super and sub critical saturation., Comment: 6 pages, 3 figures

We investigate statistical properties of trails formed by a random process incorporating aggregation, fragmentation, and diffusion. In this stochastic process, which takes place in one spatial dimension, two neighboring trails may combine to form a larger one and also, one trail may split into two. In addition, trails move diffusively. The model is defined by two parameters which quantify the fragmentation rate and the fragment size. In the long-time limit, the system reaches a steady state, and our focus is the limiting distribution of trail weights. We find that the density of trail weight has power-law tail $P(w) \sim w^{-\gamma}$ for small weight $w$. We obtain the exponent $\gamma$ analytically, and find that it varies continuously with the two model parameters. The exponent $\gamma$ can be positive or negative, so that in one range of parameters small-weight tails are abundant, and in the complementary range, they are rare., Comment: 8 pages, 8 figures

Chaos is widely understood as being a consequence of sensitive dependence upon initial conditions. This is the result of an instability in phase space, which separates trajectories exponentially. Here, we demonstrate that this criterion should be refined. Despite their overall intrinsic instability, trajectories may be very strongly convergent in phase space over extremely long periods, as revealed by our investigation of a simple chaotic system (a realistic model for small bodies in a turbulent flow). We establish that this strong convergence is a multi-facetted phenomenon, in which the clustering is intense, widespread and balanced by lacunarity of other regions. Power laws, indicative of scale-free features, characterise the distribution of particles in the system. We use large-deviation and extreme-value statistics to explain the effect. Our results show that the interpretation of the 'butterfly effect' needs to be carefully qualified. We argue that the combination of mixing and clustering processes makes our specific model relevant to understanding the evolution of simple organisms. Lastly, this notion of 'convergent chaos', which implies the existence of conditions for which uncertainties are unexpectedly small, may also be relevant to the valuation of insurance and futures contracts., Comment: 9 pages, 7 figures

We consider a particle moving with equation of motion $\dot x=f(t)$, where $f(t)$ is a random function with statistics which are independent of $x$ and $t$, with a finite drift velocity $v=\langle f\rangle$ and in the presence of a reflecting wall. Far away from the wall, translational invariance implies that the stationary probability distribution is $P(x)\sim \exp(\alpha x)$. A classical example of a problem of this type is sedimentation equilibrium, where $\alpha$ is determined by temperature. In this work we do not introduce a thermal reservoir and $\alpha$ is determined from the equation of motion. We consider a general approach to determining $\alpha$ which is not always in agreement with Einstein's relation between the mean velocity and the diffusion coefficient. We illustrate our results with a model inspired by the Boltzmann equation., Comment: 5 pages, 1 figure

Collisions between particles suspended in a fluid play an important role in many physical processes. As an example, collisions of microscopic water droplets in clouds are a necessary step in the production of macroscopic raindrops. Collisions of dust grains are also conjectured to be important for planet formation in the gas surrounding young stars, and also to play a role in the dynamics of sand storms. In these processes, collisions are favoured by fast turbulent motions. Here we review recent advances in the understanding of collisional aggregation due to turbulence. We discuss the role of fractal clustering of particles, and caustic singularities of their velocities. We also discuss limitations of the Smoluchowski equation for modelling these processes. These advances lead to a semi-quantitative understanding on the influence of turbulence on collision rates, and point to deficiencies in the current understanding of rainfall and planet formation., Comment: Review article, 36 pages, 9 figures, Ann. Rev. Cond. Matter, in press

In three-dimensional turbulent flows energy is supplied at large scales and cascades down to the smallest scales where viscosity dominates. The flux of energy through scales implies the generation of small scales from larger ones, which is the fundamental reason for the irreversibility of the dynamics of turbulent flows. As we showed recently, this irreversibility manifests itself by an asymmetry of the probability distribution of the instantaneous power $p$ of the forces acting on fluid elements. In particular, the third moment of $p$ was found to be negative. Yet, a physical connection between the irreversibility manifested in the distribution of $p$ and the energy flux or small-scale generation in turbulence has not been established. Here, with analytical calculations and support from numerical simulations of fully developed turbulence, we connect the asymmetry in the power distribution, {\it i.e.}, the negative value of $\langle p^3 \rangle$, to the generation of small scales, or more precisely, to the amplification (stretching) of vorticity in turbulent flows. Our result is the first step towards a quantitative understanding of the origin of the irreversibility observed at the level of individual Lagrangian trajectories in turbulent flows., Comment: 13 pages, 8 figures

A turbulent flow is maintained by an external supply of kinetic energy, which is eventually dissipated into heat at steep velocity gradients. The scale at which energy is supplied greatly differs from the scale at which energy is dissipated, the more so as the turbulent intensity (the Reynolds number) is larger. The resulting energy flux over the range of scales, intermediate between energy injection and dissipation, acts as a source of time irreversibility. As it is now possible to follow accurately fluid particles in a turbulent flow field, both from laboratory experiments and from numerical simulations, a natural question arises: how do we detect time irreversibility from these Lagrangian data? Here we discuss recent results concerning this problem. For Lagrangian statistics involving more than one fluid particle, the distance between fluid particles introduces an intrinsic length scale into the problem. The evolution of quantities dependent on the relative motion between these fluid particles, including the kinetic energy in the relative motion, or the configuration of an initially isotropic structure can be related to the equal-time correlation functions of the velocity field, and is therefore sensitive to the energy flux through scales, hence to the irreversibility of the flow. In contrast, for single-particle Lagrangian statistics, the most often studied velocity structure functions cannot distinguish the "arrow of time." Recent observations from experimental and numerical simulation data, however, show that the change of kinetic energy following the particle motion, is sensitive to time-reversal. We end the survey with a brief discussion of the implication of this line of work., Comment: accepted for publication in Science China - Physics, Mechanics & Astronomy

We consider a dynamical system which has a stable attractor and which is perturbed by an additive noise. Under some quite typical conditions, the fluctuations from the attractor are intermittent and have a probability distribution with power-law tails. We show that this results from a stochastic cascade of amplification of fluctuations due to transient periods of instability. The exponent of the power-law is interpreted as a negative fractal dimension., Comment: 5 pages, 4 figures

The spontaneous emergence of contraction-inducing electrical activity in the uterus at the beginning of labor remains poorly understood, partly due to the seemingly contradictory observation that isolated uterine cells are not spontaneously active. It is known, however, that the expression of gap junctions increases dramatically in the approach to parturition, which results in a significant increase in inter-cellular electrical coupling. In this paper, we build upon previous studies of the activity of electrically excitable smooth muscle cells (myocytes) and investigate the mechanism through which the coupling of these cells to electrically passive cells results in the generation of spontaneous activity in the uterus. Using a recently developed, realistic model of uterine muscle cell dynamics, we investigate a system consisting of a myocyte coupled to passive cells. We then extend our analysis to a simple two-dimensional lattice model of the tissue, with each myocyte being coupled to its neighbors, as well as to a random number of passive cells. We observe that different dynamical regimes can be observed over a range of gap junction conductances: at low coupling strength, the activity is confined to cell clusters, while the activity for high coupling may spread across the entire tissue. Additionally, we find that the system supports the spontaneous generation of spiral wave activity. Our results are both qualitatively and quantitatively consistent with observations from in vitro experiments. In particular, we demonstrate that an increase in inter-cellular electrical coupling, for realistic parameter values, strongly facilitates the appearance of spontaneous action potentials that may eventually lead to parturition.

In three-dimensional turbulent flows, the flux of energy from large to small scales breaks time symmetry. We show here that this irreversibility can be quantified by following the relative motion of several Lagrangian tracers. We find by analytical calculation, numerical analysis and experimental observation that the existence of the energy flux implies that, at short times, two particles separate temporally slower forwards than backwards, and the difference between forward and backward dispersion grows as $t^3$. We also find the geometric deformation of material volumes, surrogated by four points spanning an initially regular tetrahedron, to show sensitivity to the time-reversal with an effect growing linearly in $t$. We associate this with the structure of the strain rate in the flow., Comment: 5 pages, 4 figures

The use of simplified models of turbulent flows provides an appealing possibility to study the collision rate of turbulent suspensions, especially in conditions relevant to astrophysics, which require large time scale separations. To check the validity of such approaches, we used a direct numerical simulation (DNS) velocity field, which satisfies the Navier-Stokes equations (although it neglects the effect of the suspended particles on the flow field), and a kinematic simulation (KS) velocity field, which is a random field designed so that its statistics are in accord with the Kolmogorov theory for fully-developed turbulence. In the limit where the effects of particle inertia (characterised by the Stokes number) are negligible, the collision rates from the two approaches agree. As the Stokes number St increases, however, we show that the DNS collision rate exceeds the KS collision rate by orders of magnitude. We propose an explanation for this phenomenon and explore its consequences. We discuss the collision rate $R$ for particles in high Reynolds number flows at large Stokes number, and present evidence that $R\propto \sqrt{{\rm St}}$., Comment: 9 pages, 3 figures

The breaking of detailed balance, the symmetry between forward and backward probability transition between two states, is crucial to understand irreversible systems. In hydrodynamic turbulence, a far-from equilibrium system, we observe a strong manifestation of the breaking of detailed balance by following the evolution of the kinetic energy of individual fluid elements. We found in all the flows that we have investigated that fluid elements decelerate faster than they accelerate, giving rise to negative third moment of energy increments, independently of space dimensionality. The exchange of energy between fluid elements however is fundamentally different in two and three dimensions. While pressure forces do not provide net energy change to slow or fast particles in two dimensions, they tend to transfer energy from {\it slow} to {\it fast} particles in three dimensions, possibly implying a runaway of energy., Comment: Updated figures 1d & 4a with better converged data

In turbulent suspensions, collision rates determine how rapidly particles coalesce or react with each other. To determine the collision rate, many numerical studies rely on the 'Ghost Collision Approximation' (GCA), which simply records how often pairs of point particles come within a threshold distance. In many applications, the suspended particles stick (or in the case of liquid droplets, coalesce) upon collision, and it is the frequency of first contact which is of interest. If a pair of 'ghost' particles undergoes multiple collisions, the GCA may overestimate the true collision rate. Here, using fully resolved Direct Numerical Simulations of turbulent flows at moderate Reynolds number (R_\lambda = 130), we investigate the prevalence and properties of multiple collisions. We demonstrate that the GCA leads to a systematic overestimate of the collision rate, which is of the order of 15% when the particle inertia is small, and slowly decreases when inertia increases. We investigate the probability P(N) for a given pair of ghost particles colliding N times. We find P(N)=\beta \alpha^N for N_>1, where \alpha and \beta are coefficients which depend upon the particle inertia. This result is used to explain the discrepancy between the GCA and the true collision rates. We also investigate the statistics of the times that ghost particles remain in contact. We show that the probability density function of the contact time is different for the first collision. The difference is explained by the effect of caustics in the phase space of the suspended particles., Comment: 14 pages, 12 figures

Turbulence facilitates collisions between particles suspended in a turbulent flow. Two effects have been proposed which can enhance the collision rate at high turbulence intensities: 'preferential concentration' (a clustering phenomenon) and the 'sling effect' (arising from the formation of caustic folds in the phase-space of the suspended particles). We have determined numerically the collision rate of small heavy particles as a function of their size and densities. The dependence on particle densities reveals that the enhancement by turbulence of the collision rate of particles with significant inertia is due almost entirely to the sling effect., Comment: 5 pages, 4 figures

Three particles floating on a fluid surface define a triangle. The aim of this paper is to characterise the shape of the triangle, defined by two of its angles, as the three vertices are subject to a complex or turbulent motion. We consider a simple class of models for this process, involving a combination of a random strain of the fluid and Brownian motion of the particles. Following D. G. Kendall, we map the space of triangles to a sphere, whose equator corresponds to degenerate triangles with colinear vertices, with equilaterals at the poles. We map our model to a diffusion process on the surface of the sphere and find an exact solution for the shape distribution. Whereas the action of the random strain tends to make the shape of the triangles infinitely elongated, in the presence of a Brownian diffusion of the vertices, the model has an equilibrium distribution of shapes. We determine here exactly this shape distribution in the simple case where the increments of the strain are diffusive., Comment: 16 pages, 1 figure

Coherent oscillatory activity can arise spontaneously as a result of increased coupling in a system of excitable and passive cells, each being quiescent in isolation. This can potentially explain the appearance of spontaneous rhythmic contractions in the pregnant uterus close to term. We investigate the transition to periodic activity using a model system comprising a lattice of excitable cells, each being coupled to a variable number of passive cells whose distribution defines a quenched realization (replica) of spatial disorder. Close to the transition between quiescent state and sustained oscillations in the system we observe large fluctuations between different replicas induced by variations in the local density of passive cells around an excitable cell. We demonstrate that the disorder-induced fluctuations can be described in terms of a simple scaling relation which involves the strength of coupling between excitable cells, the mean passive cell density, as well as the logarithm of the system size. Our results can be interpreted as suggesting that larger organs will have greater variability in the onset of persistent activity., Comment: 12 pages, 10 figures

We describe the structure and dynamics of turbulence by the scale-dependent perceived velocity gradient tensor as supported by following four tracers, i.e. fluid particles, that initially form a regular tetrahedron. We report results from experiments in a von K\'arm\'an swirling water flow and from numerical simulations of the incompressible Navier-Stokes equation. We analyze the statistics and the dynamics of the perceived rate of strain tensor and vorticity for initially regular tetrahedron of size $r_0$ from the dissipative to the integral scale. Just as for the true velocity gradient, at any instant, the perceived vorticity is also preferentially aligned with the intermediate eigenvector of the perceived rate of strain. However, in the perceived rate of strain eigenframe fixed at a given time $t=0$, the perceived vorticity evolves in time such as to align with the strongest eigendirection at $t=0$. This also applies to the true velocity gradient. The experimental data at the higher Reynolds number suggests the existence of a self-similar regime in the inertial range. In particular, the dynamics of alignment of the perceived vorticity and strain can be rescaled by $t_0$, the turbulence time scale of the flow when the scale $r_0$ is in the inertial range. For smaller Reynolds numbers we found the dynamics to be scale dependent.

Synchronized oscillations are of critical functional importance in many biological systems. We show that such oscillations can arise without centralized coordination in a disordered system of electrically coupled excitable and passive cells. Increasing the coupling strength results in waves that lead to coherent periodic activity, exhibiting cluster, local and global synchronization under different conditions. Our results may explain the self-organized transition in a pregnant uterus from transient, localized activity initially to system-wide coherent excitations just before delivery., Comment: 5 pages, 4 figures

The paper considers random motion of a point on the surface of a sphere, in the case where the angular velocity is determined by an Ornstein-Uhlenbeck process. The solution is fully characterized by only one dimensionless number, the persistence angle, which is the typical angle of rotation of the object during the correlation time of the angular velocity. We first show that the two-dimensional case is exactly solvable. When the persistence angle is large, a series for the correlation function has the surprising property that its sum varies much more slowly than any of its individual terms. In three dimensions we obtain asymptotic forms for the correlation function, in the limits where the persistence angle is very small and very large. The latter case exhibits a complicated transient, followed by a much slower exponential decay. The decay rate is determined by the solution of a radial Schrodinger equation in which the angular momentum quantum number takes an irrational value, namely j=(sqrt{17}-1)/2. Possible applications of the model to objects tumbling in a turbulent environment are discussed.

The motion of a large, neutrally buoyant, particle, freely advected by a turbulent flow is determined experimentally. We demonstrate that both the translational and angular accelerations exhibit very wide probability distributions, a manifestation of intermittency. The orientation of the angular velocity with respect to the trajectory, as well as the translational acceleration conditioned on the spinning velocity provide evidence of a lift force acting on the particle., Comment: 4 page, 4 figures

We study the 6-dimensional dynamics -- position and orientation -- of a large sphere advected by a turbulent flow. The movement of the sphere is recorded with 2 high-speed cameras. Its orientation is tracked using a novel, efficient algorithm; it is based on the identification of possible orientation `candidates' at each time step, with the dynamics later obtained from maximization of a likelihood function. Analysis of the resulting linear and angular velocities and accelerations reveal a surprising intermittency for an object whose size lies in the integral range, close to the integral scale of the underlying turbulent flow.