Lagrangian Statistics Research Articles

Description of surface transport in the region of the Belizean Barrier Reef based on observations and alternative high-resolution models

The gains from implementing high-resolution versus less costly low-resolution models to describe coastal circulation are not always clear, often lacking statistical evaluation. Here we construct a hierarchy of ocean–atmosphere models operating at multiple scales within a 1 × 1° domain of the Belizean Barrier Reef (BBR). The various components of the atmosphere–ocean models are evaluated with in situ observations of surface drifters, wind and sea surface temperature. First, we compare the dispersion and velocity of 55 surface drifters released in the field in summer 2013 to the dispersion and velocity of simulated drifters under alternative model configurations. Increasing the resolution of the ocean model (from 1/12° to 1/100°, from 1 day to 1h) and atmosphere model forcing (from 1/2° to 1/100°, from 6h to 1h), and incorporating tidal forcing incrementally reduces discrepancy between simulated and observed velocities and dispersion. Next, in trying to understand why the high-resolution models improve prediction, we find that resolving both the diurnal sea-breeze and semi-diurnal tides is key to improving the Lagrangian statistics and transport predictions along the BBR. Notably, the model with the highest ocean–atmosphere resolution and with tidal forcing generates a higher number of looping trajectories and sub-mesoscale coherent structures that are otherwise unresolved. Finally, simulations conducted with this model from June to August of 2013 show an intensification of the velocity fields throughout the summer and reveal a mesoscale anticyclonic circulation around Glovers Reef, and sub-mesoscale cyclonic eddies formed in the vicinity of Columbus Island. This study provides a general framework to assess the best surface transport prediction from alternative ocean–atmosphere models using metrics derived from high frequency drifters’ data and meteorological stations.

EVOLUTION OF THE MAGNETIC FIELD LINE DIFFUSION COEFFICIENT AND NON-GAUSSIAN STATISTICS

ABSTRACT The magnetic field line random walk (FLRW) plays an important role in the transport of energy and particles in turbulent plasmas. For magnetic fluctuations that are transverse or almost transverse to a large-scale mean magnetic field, theories describing the FLRW usually predict asymptotic diffusion of magnetic field lines perpendicular to the mean field. Such theories often depend on the assumption that one can relate the Lagrangian and Eulerian statistics of the magnetic field via Corrsin’s hypothesis, and additionally take the distribution of magnetic field line displacements to be Gaussian. Here we take an ordinary differential equation (ODE) model with these underlying assumptions and test how well it describes the evolution of the magnetic field line diffusion coefficient in 2D+slab magnetic turbulence, by comparisons to computer simulations that do not involve such assumptions. In addition, we directly test the accuracy of the Corrsin approximation to the Lagrangian correlation. Over much of the studied parameter space we find that the ODE model is in fairly good agreement with computer simulations, in terms of both the evolution and asymptotic values of the diffusion coefficient. When there is poor agreement, we show that this can be largely attributed to the failure of Corrsin’s hypothesis rather than the assumption of Gaussian statistics of field line displacements. The degree of non-Gaussianity, which we measure in terms of the kurtosis, appears to be an indicator of how well Corrsin’s approximation works.

Microbubbles and Microparticles are Not Faithful Tracers of Turbulent Acceleration.

We report on the Lagrangian statistics of acceleration of small (sub-Kolmogorov) bubbles and tracer particles with Stokes number St≪1 in turbulent flow. At a decreasing Reynolds number, the bubble accelerations show deviations from that of tracer particles; i.e., they deviate from the Heisenberg-Yaglom prediction and show a quicker decorrelation despite their small size and minute St. Using direct numerical simulations, we show that these effects arise due the drift of these particles through the turbulent flow. We theoretically predict this gravity-driven effect for developed isotropic turbulence, with the ratio of Stokes to Froude number or equivalently the particle drift velocity governing the enhancement of acceleration variance and the reductions in correlation time and intermittency. Our predictions are in good agreement with experimental and numerical results. The present findings are relevant to a range of scenarios encompassing tiny bubbles and droplets that drift through the turbulent oceans and the atmosphere. They also question the common usage of microbubbles and microdroplets as tracers in turbulence research.

Influence of compressibility on the Lagrangian statistics of vorticity–strain-rate interactions

The objective of this study is to investigate the influence of compressibility on Lagrangian statistics of vorticity and strain-rate interactions. The Lagrangian statistics are extracted from "almost" time-continuous data sets of direct numerical simulations of compressible decaying isotropic turbulence by employing a cubic spline-based Lagrangian particle tracker. We study the influence of compressibility on Lagrangian statistics of alignment in terms of compressibility parameters-turbulent Mach number, normalized dilatation-rate, and flow topology. In comparison to incompressible turbulence, we observe that the presence of compressibility in a flow field weakens the alignment tendency of vorticity toward the largest strain-rate eigenvector. Based on the Lagrangian statistics of alignment conditioned on dilatation and topology, we find that the weakened tendency of alignment observed in compressible turbulence is because of a special group of fluid particles that have an initially negligible dilatation-rate and are associated with stable-focus-stretching topology.

Concentration effects on turbulence in dilute polymer solutions far from walls.

We report measurements of the modification of turbulence far from any walls by small concentrations of long-chain polymers. We consider a range of statistical properties of the flow, including Eulerian and Lagrangian velocity structure functions, Eulerian acceleration correlation functions, and the relative dispersion of particle pairs. In all cases, we find that the polymer concentration has a strong effect on the extent to which the statistical properties are changed compared to their values in pure water. These effects can be captured by the recently proposed energy flux-balance model (when suitably extended into the time domain for Lagrangian statistics). However, unlike previous measurements, which found that the concentration effect could be completely scaled out, we consistently find that our data collapse onto two different master curves, one for small concentration and one for larger concentration. We suggest that the difference between the two may be related to the onset of interactions among polymer chains, which is likely to be more easily observed at the small Weissenberg numbers we consider here.

Two-particle dispersion in weakly turbulent thermal convection

We present results from a numerical study of particle dispersion in the weakly nonlinear regime of Rayleigh–Bénard convection of a fluid with Prandtl number around unity, where bi-stability between ideal straight convection rolls and weak turbulence in the form of spiral defect chaos exists. While Lagrangian pair statistics has become a common tool for studying fully developed turbulent flows at high Reynolds numbers, we show that key characteristics of mass transport can also be found in convection systems that show no or weak turbulence. Specifically, for short times, we find an interval of t3-scaling of pair dispersion, which we explain quantitatively with the interplay of advection and diffusion. For long times we observe diffusion-like dispersion of particles that becomes independent of the individual particles’ stochastic movements. The spreading rate is found to depend on the degree of spatio-temporal chaos.

Single-particle Lagrangian and structure statistics in kinematically simulated particle-laden turbulent flows

Kinematic simulation (KS) is a means of generating a turbulent-like velocity field, in a manner that enforces a desired input Eulerian energy spectrum. Such models have also been applied in particle-laden flows, due to their ability to enforce spatial organization of the fluid velocity field when simulating the trajectories of individual Lagrangian particles. A critical evaluation of KS is presented; in particular, we examine its ability to reproduce single-particle Lagrangian statistics. Also the ability of KS to reproduce the preferential concentration of inertial particles is examined. Some computational results are presented, in which particles are transported alternatively by (1) turbulence generated by direct numerical simulation (DNS) of the incompressible Navier-Stokes equations, and (2) KS. The effect of unsteadiness formulation in particular is examined. We find that even steady KS qualitatively reproduces the continuity effect, clustering of inertial particles, the elevated dispersion of inertial particles over fluid particles, and the intermittency of Lagrangian velocity signals, but generally not to the same extent as is seen in the DNS.

Lagrangian properties of the entrainment across turbulent/non-turbulent interface layers

Lagrangian statistics obtained from direct numerical simulations of turbulent planar jets and mixing layers are reported for the separation distance between the tracer particles at the outer edge of the turbulent/non-turbulent interface layer, and the entrained fluid particles. In the viscous superlayer (VSL) the mean square particle distance exhibits a ballistic evolution, while the Richardson-like scaling for relative dispersion prevails inside the turbulent sublayer (TSL). The results further support the existence of two different regimes within the interface layer, where small-scale outward enstrophy diffusion governs the entrained particles in the VSL, while inviscid small-scale motions govern the TSL.

Drifter observations in the summer time Bay of Biscay slope current

During the summer of 2012, 20 surface drifters drogued at 50m depth were deployed on the continental slope to the north of the Bay of Biscay. Initially after release the drifters all crossed the slope, with 14 continuing equatorward, parallel to the slope following an absolute dynamic topography feature and 6 returning to the slope, in an eddy, visible in chlorophyll-a maps. Lagrangian statistics show an anisotropic flow field that becomes less tied to the absolute dynamic topography and increasingly dominated by diffusion and eddy processes. A weaker tie to the absolute dynamic topography allowed for total of 8 of the drifters crossed from the deep water onto the shelf, showing pathways for flow across the slope. A combination of drifter trajectories, absolute dynamic topography and chlorophyll-a concentration maps has been used to show that small anticyclonic eddies, tied to the complex slope topography provide a mechanism for on shelf transport. During the summer, the presence of these eddies can be seen in surface chlorophyll-a maps.

Lagrangian statistics in weakly forced two-dimensional turbulence.

Measurements of Lagrangian single-point and multiple-point statistics in a quasi-two-dimensional stratified layer system are reported. The system consists of a layer of salt water over an immiscible layer of Fluorinert and is forced electromagnetically so that mean-squared vorticity is injected at a well-defined spatial scale ri. Simultaneous cascades develop in which enstrophy flows predominately to small scales whereas energy cascades, on average, to larger scales. Lagrangian correlations and one- and two-point displacements are measured for random initial conditions and for initial positions within topological centers and saddles. Some of the behavior of these quantities can be understood in terms of the trapping characteristics of long-lived centers, the slow motion near strong saddles, and the rapid fluctuations outside of either centers or saddles. We also present statistics of Lagrangian velocity fluctuations using energy spectra in frequency space and structure functions in real space. We compare with complementary Eulerian velocity statistics. We find that simultaneous inverse energy and enstrophy ranges present in spectra are not directly echoed in real-space moments of velocity difference. Nevertheless, the spectral ranges line up well with features of moment ratios, indicating that although the moments are not exhibiting unambiguous scaling, the behavior of the probability distribution functions is changing over short ranges of length scales. Implications for understanding weakly forced 2D turbulence with simultaneous inverse and direct cascades are discussed.

Lagrangian view of time irreversibility of fluid turbulence

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.

A molecular dynamics simulation of the turbulent Couette minimal flow unit

A molecular dynamics simulation of planar Couette flow is presented for the minimal channel in which turbulence structures can be sustained. Evolution over a single breakdown and regeneration cycle is compared to computational fluid dynamics simulations. Qualitative similar structures are observed and turbulent statistics show excellent quantitative agreement. The molecular scale law of the wall is presented in which stick-slip molecular wall-fluid interactions replace the no-slip conditions. The impact of grid resolution is explored and the observed structures are seen to be dependent on averaging time and length scales. The kinetic energy spectra show that a range of scales are present in the molecular system and that spectral content is dependent on the grid resolution employed. The subgrid velocity of the molecules is studied using joint probability density functions, molecular trajectories, diffusion, and Lagrangian statistics. The importance of sub-grid scales, relevance of the Kolmogorov lengthscale, and implications of molecular turbulence are discussed.

Anomalous transport on regular fracture networks: Impact of conductivity heterogeneity and mixing at fracture intersections.

We investigate transport on regular fracture networks that are characterized by heterogeneity in hydraulic conductivity. We discuss the impact of conductivity heterogeneity and mixing within fracture intersections on particle spreading. We show the emergence of non-Fickian transport due to the interplay between the network conductivity heterogeneity and the degree of mixing at nodes. Specifically, lack of mixing at fracture intersections leads to subdiffusive scaling of transverse spreading but has negligible impact on longitudinal spreading. An increase in network conductivity heterogeneity enhances both longitudinal and transverse spreading and leads to non-Fickian transport in longitudinal direction. Based on the observed Lagrangian velocity statistics, we develop an effective stochastic model that incorporates the interplay between Lagrangian velocity correlation and velocity distribution. The model is parameterized with a few physical parameters and is able to capture the full particle transition dynamics.

Direct numerical simulation studies of Lagrangian intermittency in turbulence

Lag-averaged Lagrangian statistics from direct numerical simulations over a range of Reynolds numbers are analyzed to test the predictions of the Lagrangian Refined Similarity Hypothesis (LRSH). The analysis uses the Lagrangian integral time scale to scale the lag since it is the natural time scale to reveal trends and scaling with Reynolds number. Both the velocity difference and the dissipation rate probability density functions (PDFs) collapse across inertial sub-range and diffusive scales for approximately the same values of the scaled lag, and in the zero lag limit are independent of the lag and depend only on the Reynolds number. These findings are consistent with the LRSH. The velocity difference PDFs are characterized by stretched exponential tails, while the dissipation rate PDFs for small lags have a log normal core with power law tails at both large and small values of the dissipation rate. The velocity structure functions show inertial sub-range similarity scaling with Reynolds number which extends to smaller scales with increasing Reynolds number. Estimates of the scaling exponents obtained are consistent with those from previous studies. They tend to saturate at a value of about two for high order moments. Non-dimensional acceleration moments show a striking power law dependence on Reynolds number from which novel estimates of the scaling exponents have been determined. Similarity scaling is much more elusive to demonstrate in the dissipation rate moments. The data are consistent with, but do not confirm, the Oboukhov relationship connecting velocity structure functions and dissipation rate moments on inertial sub-range scales.

Time irreversibility of the statistics of a single particle in compressible turbulence.

We investigate time irreversibility from the point of view of a single particle in Burgers turbulence. Inspired by the recent work for incompressible flows [Xu et al., Proc. Natl. Acad. Sci. USA 111, 7558 (2014)], we analyze the evolution of the kinetic energy for fluid markers and use the fluctuations of the instantaneous power as a measure of time irreversibility. For short times, starting from a uniform distribution of markers, we find the scaling 〈[E(t)-E(0)](n)〉∝t and 〈p(n)〉∝Re(n-1) for the power as a function of the Reynolds number. Both observations can be explained using the "flight-crash" model, suggested by Xu et al. Furthermore, we use a simple model for shocks that reproduces the moments of the energy difference, including the pre-factor for 〈E(t)-E(0)〉. To complete the single-particle picture for Burgers we compute the moments of the Lagrangian velocity difference and show that they are bifractal. This arises in a similar manner to the bifractality of Eulerian velocity differences. In the above setting, time irreversibility is directly manifest as particles eventually end up in shocks. We additionally investigate time irreversibility in the long-time limit when all particles are located inside shocks and the Lagrangian velocity statistics are stationary. We find the same scalings for the power and energy differences as at short times and argue that this is also a consequence of rare "flight-crash" events related to shock collisions.

Lagrangian velocity and acceleration statistics of fluid and inertial particles measured in pipe flow with 3D particle tracking velocimetry

Three-dimensional particle tracking velocimetry (3D-PTV) has been applied to particle-laden pipe flow at Reynolds number 10,300, based on the bulk velocity and the pipe diameter. The volume fraction of the inertial particles was equal to 1.4×10−5. Lagrangian velocity and acceleration statistics were determined both for tracers and for inertial particles with Stokes number equal to 2.3, based on the particle relaxation time and the viscous time scale. The decay of Lagrangian velocity and acceleration correlation functions was measured both for the fluid and for the dispersed phase at various radial positions. The decay of Lagrangian velocity correlations is faster for inertial particles than for flow tracers, whereas the decay of Lagrangian acceleration correlations is about 25% slower for inertial particles than for flow tracers. Further differences between inertial and tracer particles are found in velocity fluctuations evaluated for both positive and negative time lags. The asymmetry in time of velocity cross-correlations is more pronounced for inertial particles. Quadrant analysis revealed another difference still near the wall: ejection and sweep events are less frequent for inertial particles than for tracers.

Lagrangian statistics in isotropic turbulent flows with deterministic and stochastic forcing schemes

Artificial input of energy into the flow is necessary to create and maintain a statistically stationary isotropic turbulence for sampling in studying the statistics. Due to the nonlinear coupling among different Fourier modes through the triadic interaction, whether or not various forcing schemes affect the statistics in turbulence is an important and open question. We present detailed comparison of Lagrangian statistics of fluids particles in forced isotropic turbulent flows in 1283, 2563, and 5123 simulations, with Taylor-scale Reynolds numbers in the range of 64–171, using a deterministic and a stochastic forcing scheme, respectively. Several Lagrangian statistics are compared, such as velocity and acceleration autocorrelations, and moments of Lagrangian velocity increments. The differences in the Lagrangian statistics obtained from the two forcing schemes are shown to be small, indicating that the isotropic forcing schemes used have little effects on the Lagrangian statistics in the isotropic turbulence.

Characteristics of the Near-Surface Currents in the Indian Ocean as Deduced from Satellite-Tracked Surface Drifters. Part II: Lagrangian Statistics

AbstractLagrangian statistics of the surface circulation in the Indian Ocean (IO) are investigated using drifter observations during 1985–2013. The methodology isolates the influence of low-frequency variations and horizontal shear of mean flow. The estimated Lagrangian statistics are spatially inhomogeneous and anisotropic over the IO basin, with values of ~6–85 × 107 cm2 s−1 for diffusivity, ~2–7 days for integral time scale, and ~33–223 km for length scale. Large diffusivities (>20 × 107 cm2 s−1) occur in the central-eastern equatorial IO and the eastern African coast. Small diffusivities (~6–8 × 107 cm2 s−1) appear in the subtropical gyre of the southern IO and the southeastern Arabian Sea. The equatorial IO has the largest zonal diffusivity (~85 × 107 cm2 s−1), corresponding to the largest time scale (~7 days) and length scale (~223 km), while the eastern coast of Somalia has the largest meridional diffusivity (~31 × 107 cm2 s−1). The minor component of the Lagrangian length scale is approximately equal to the first baroclinic Rossby radius (R1) at midlatitudes (R1 ~ 30–50 km), while the major component equals R1 in the equatorial region (R1 > 80 km). The periods of the energetic eddy-containing bands in the IO in Lagrangian spectra range from several days to a couple of months, where anticyclones dominate. A significant result is that the drifter-derived diffusivities asymptote to constant values in relatively short time lags (~10 days) for some subregions of the IO if they are correctly calculated. This is an important contribution to the ongoing debate regarding drifter-based diffusivity estimates with relatively short Lagrangian velocity time series versus tracer-based estimates.

Weak–strong clustering transition in renewing compressible flows

AbstractWe investigate the statistical properties of Lagrangian tracers transported by a time-correlated compressible renewing flow. We show that the preferential sampling of the phase space performed by tracers yields significant differences between the Lagrangian statistics and its Eulerian counterpart. In particular, the effective compressibility experienced by tracers has a non-trivial dependence on the time correlation of the flow. We examine the consequence of this phenomenon on the clustering of tracers, focusing on the transition from the weak- to the strong-clustering regime. We find that the critical compressibility at which the transition occurs is minimum when the time correlation of the flow is of the order of the typical eddy turnover time. Further, we demonstrate that the clustering properties in time-correlated compressible flows are non-universal and are strongly influenced by the spatio-temporal structure of the velocity field.

Structures and Lagrangian statistics of the Taylor–Green dynamo

The evolution of a Taylor–Green forced magnetohydrodynamic system showing dynamo activity is analyzed via direct numerical simulations. The statistical properties of the velocity and magnetic fields in Eulerian and Lagrangian coordinates are found to change between the kinematic, nonlinear and saturated regime. Fluid element (tracer) trajectories change from chaotic quasi-isotropic (kinematic phase) to mean magnetic field aligned (saturated phase). The probability density functions (PDFs) of the magnetic field change from strongly non-Gaussian in the kinematic to quasi-Gaussian PDFs in the saturated regime so that their flatness give a precise handle on the definition of the limiting points of the three regimes. Also the statistics of the kinetic and magnetic fluctuations along fluid trajectories changes. All this goes along with a dramatic increase of the correlation time of the velocity and magnetic fields experienced by tracers, significantly exceeding one turbulent large-eddy turn-over time. A remarkable consequence is an intermittent scaling regime of the Lagrangian magnetic field structure functions at unusually long time scales.