Eulerian Statistics Research Articles

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.

Numerical Considerations for Lagrangian Stochastic Dispersion Models: Eliminating Rogue Trajectories, and the Importance of Numerical Accuracy

When Lagrangian stochastic models for turbulent dispersion are applied to complex atmospheric flows, some type of ad hoc intervention is almost always necessary to eliminate unphysical behaviour in the numerical solution. Here we discuss numerical strategies for solving the non-linear Langevin-based particle velocity evolution equation that eliminate such unphysical behaviour in both Reynolds-averaged and large-eddy simulation applications. Extremely large or ‘rogue’ particle velocities are caused when the numerical integration scheme becomes unstable. Such instabilities can be eliminated by using a sufficiently small integration timestep, or in cases where the required timestep is unrealistically small, an unconditionally stable implicit integration scheme can be used. When the generalized anisotropic turbulence model is used, it is critical that the input velocity covariance tensor be realizable, otherwise unphysical behaviour can become problematic regardless of the integration scheme or size of the timestep. A method is presented to ensure realizability, and thus eliminate such behaviour. It was also found that the numerical accuracy of the integration scheme determined the degree to which the second law of thermodynamics or ‘well-mixed condition’ was satisfied. Perhaps more importantly, it also determined the degree to which modelled Eulerian particle velocity statistics matched the specified Eulerian distributions (which is the ultimate goal of the numerical solution). It is recommended that future models be verified by not only checking the well-mixed condition, but perhaps more importantly by checking that computed Eulerian statistics match the Eulerian statistics specified as inputs.

Flag Statistics from the Ehrhart $h^∗$-Polynomial of Multi-Hypersimplices

It is known that the normalized volume of standard hypersimplices (defined as some slices of the unit hypercube) are the Eulerian numbers. More generally, a recent conjecture of Stanley relates the Ehrhart series of hypersimplices with descents and excedences in permutations. This conjecture was proved by Nan Li, who also gave a generalization to colored permutations. In this article, we give another generalization to colored permutations, using the flag statistics introduced by Foata and Han. We obtain in particular a new proof of Stanley’s conjecture, and some combinatorial identities relating pairs of Eulerian statistics on colored permutations.

A random field approach to the Lagrangian modeling of turbulent transport in vegetated canopies

We present an application of a Lagrangian Stochastic Model (LSM) to turbulent dispersion over complex terrain, where turbulent coherent structures are known to play a crucial role. We investigate the case of a vegetated canopy by using semi-empirical parameterizations of turbulence profiles in the region inside and above a canopy layer. The LSM is based on a 4-dimensional Fokker-Planck (4DFP) equation, which extends the standard Thomson87 Lagrangian approach. The 4DFP model is derived by means of a Random Field description of the turbulent velocity field. The main advantage of this approach is that not only the experimental Eulerian one-point statistics, but also the Eulerian two-point two-time covariance structure can be included explicitly in the LSM. At variance with the standard Thomson87 approach, the 4DFP model allows to consider explicit parameterizations of the turbulent coherent structures as it explicitly includes both spatial and temporal correlation functions. In order to investigate the effect of the turbulent geometrical structure on a scalar concentration profile, we performed numerical simulations with two different covariance parameterizations, the first one isotropic and the second anisotropic. We show that the accumulation of scalars near the ground is due to the anisotropic geometrical properties of the turbulent boundary layer.

Turbulent pair dispersion as a ballistic cascade phenomenology

Since the pioneering work of Richardson in 1926, later refined by Batchelor and Obukhov in 1950, it is predicted that the rate of separation of pairs of fluid elements in turbulent flows with initial separation at inertial scales, grows ballistically first (Batchelor regime), before undergoing a transition towards a super-diffusive regime where the mean-square separation grows as $t^{3}$ (Richardson regime). Richardson empirically interpreted this super-diffusive regime in terms of a non-Fickian process with a scale-dependent diffusion coefficient (the celebrated Richardson’s ‘$4/3$rd’ law). However, the actual physical mechanism at the origin of such a scale dependent diffusion coefficient remains unclear. The present article proposes a simple physical phenomenology for the time evolution of the mean-square relative separation in turbulent flows, based on a scale-dependent ballistic scenario rather than a scale-dependent diffusive. It is shown that this phenomenology accurately retrieves most of the known features of relative dispersion for particles mean-square separation, among others: (i) it is quantitatively consistent with most recent numerical simulations and experiments for mean-square separation between particles (both for the short-term Batchelor regime and the long-term Richardson regime, and for all initial separations at inertial scales); (ii) it gives a simple physical explanation of the origin of the super-diffusive $t^{3}$ Richardson regime which naturally builds itself as an iterative process of elementary short-term scale-dependent ballistic steps; (iii) it shows that the Richardson constant is directly related to the Kolmogorov constant (and eventually to a ballistic persistence parameter); and (iv) in a further extension of the phenomenology, taking into account third-order corrections, it robustly describes the temporal asymmetry between forward and backward dispersion, with an explicit connection to the cascade of energy flux across scales. An important aspect of this phenomenology is that it simply and robustly connects long-term super-diffusive features to elementary short-term mechanisms, and at the same time it connects basic Lagrangian features of turbulent relative dispersion (both at short and long times) to basic Eulerian features of the turbulent field: second-order Eulerian statistics control the growth of separation (both at short and long times) while third-order Eulerian statistics control the temporal asymmetry of the dispersion process, which can then be directly identified as the signature of the energy cascade and associated to well-known exact results as the Karman–Howarth–Monin relation.

Relating Lagrangian and Eulerian horizontal eddy statistics in the surfzone

AbstractConcurrent Lagrangian and Eulerian observations of rotational, low‐frequency (10−4 to 10−2 Hz) surfzone eddies are compared. Surface drifters were tracked for a few hours on each of 11 days at two alongshore uniform beaches. A cross‐shore array of near‐bottom current meters extended from near the shoreline to seaward of the surfzone (typically 100 m wide in these moderate wave conditions). Lagrangian and Eulerian mean alongshore velocities V are similar, with a midsurfzone maximum. Cross‐shore dependent Lagrangian (σL) and Eulerian (σE) rotational eddy velocities, estimated from low‐pass filtered drifter and current meter velocities, respectively, also generally agree. Cross‐shore rotational velocities have a midsurfzone maximum whereas alongshore rotational velocities are distributed more broadly. Daily estimates of the Lagrangian time scale, the time for drifter velocities to decorrelate, vary between 40 and 300 s, with alongshore time scales greater than cross‐shore time scales. The ratio of Lagrangian to apparent Eulerian current meter decorrelation times TL/TA varies considerably, between about 0.5 and 3. Consistent with theory, some of the TL/TA variation is ascribable to alongshore advection and TL/TA is proportional to V/σ, which ranges between about 0.6 and 2.5. Estimates of TL/TA vary between days with similar V/σ suggesting that surfzone Lagrangian particle dynamics vary between days, spanning the range from “fixed‐float” to “frozen‐field” [Lumpkin et al., 2002], although conclusions are limited by the statistical sampling errors in both TL/TA and V/σ.

Mechanistic modeling of seed dispersal by wind over hilly terrain

Seed dispersal is the main movement mechanism used by plants. The last decade saw rapid progress in understanding the underlying processes, especially for dispersal by wind, in part due to new mechanistic modeling approaches that account for turbulent fluctuations. Yet, current wind dispersal models stop short of explicitly incorporating the effects of landscape topography on the main transporting vector – wind, so that the effects of wind variability over hills on dispersal patterns remain by and large unstudied.A new mechanistic model was developed that combines Eulerian wind statistics derived from a simplified analytical approach of flow over gently sloped forested hills with a Lagrangian seed trajectory model. Model runs were used to explore the effects of seed release location along the hill on dispersal kernels predicted by the new model in relation to their flat-terrain counterparts. The model was parameterized for a Pinus taeda plantation, and a range of seed motion capacities represented by terminal velocity and release height, and realistic topographic variation were then explored.To evaluate model performance, computed kernels were compared to kernel measurements collected in a large flume for spherical ‘seeds’ released near the top of a rod canopy covering gentle cosine hills. The evaluation showed that the model reproduced the key experimental differences in dispersal patterns for releases at the hill crest and bottom.The simulations revealed several novel findings. For seeds released within the canopy, both median and 99th percentile dispersal distances on the hill upwind side were up to two times longer than on flat terrain for the same motion capacity. Seeds released on the lee side traveled mostly toward the hill crest – following the local within-the-canopy wind direction. This direction was contrary to the ‘regional’ wind direction set by the flow conditions above the canopy. There, the directionality of the long-distance dispersal was additionally dependent on uplifting probability, affected by seed motion capacity.It was demonstrated that neglecting the effects of even gentle topography in mechanistic seed dispersal models can lead to biased estimates of dispersal distances and directionality on hilly terrain. These results are pertinent to plant population demography, connectivity and spread on hills. More broadly, the approach developed here can be extended to movement of pollen and various airborne organisms over hills.

Flow structures and their contribution to turbulent dispersion in a randomly packed porous bed based on particle image velocimetry measurements

An experimental study was undertaken to explore the evolution of flow structures and their characteristics within a randomly packed porous bed with particular attention to evaluating turbulent scalar dispersion. A low aspect ratio bed of 4.67 (bed width to spherical solid phase particle diameter) with fluid phase refractive index matched to that of the solid phase was used in order to obtain time resolved two component particle image velocimetry data. Results are based on detailed velocity vector maps obtained at selected pores near the bed center. Pore, or large scale, regions that are associated with the mean flow were identified based on Reynolds decomposed velocity fields, while smaller scale structures embedded within pore scale regions were identified and quantified by combining large eddy scale decomposition and swirling strength analysis. The velocity maps collected in distinctive pore geometries showed presence of three types of flow regions that display very different mean flow conditions, described as regions with tortuous channel like flow, high fluid momentum jet like regions, and low fluid momentum recirculating regions. The major portion of pore space is categorized as tortuous channel flow. Time series of instantaneous velocity field maps were used to identify mean and turbulent flow structures based on their spatial scales in the different regions. Even though regions exhibit varied Eulerian statistics, they show very similar eddy characteristics such as spinning rate and number density. The integral scale eddy structures show nearly a linear rate of increase in their rotation rate with increasing pore Reynolds number, indicating a linear decrease in their time scales. The convective velocities of these eddies are shown to reach an asymptotic limit at high pore Reynolds numbers, unique for each flow region. Detailed Eulerian statistics for the identified flow regions are presented and are used to predict mechanical dispersion through the use of estimated Lagrangian statistics. Contributions from each of the flow regions are presented and the recirculating regions are shown to contribute most to the overall longitudinal dispersion, whereas the tortuous channel regions contribute most to the transverse dispersion. The overall dispersion estimates agree well with global data in the limit of high Schmitt number.

Lagrangian statistics across the turbulent-nonturbulent interface in a turbulent plane jet

Lagrangian statistics from millions of particles are used to study the turbulent entrainment mechanism in a direct numerical simulation of a turbulent plane jet at Re(λ) ≈ 110. The particles (tracers) are initially seeded at the irrotational region of the jet near the turbulent shear layer and are followed as they are drawn into the turbulent region across the turbulent-nonturbulent interface (TNTI), allowing the study of the enstrophy buildup and thereby characterizing the turbulent entrainment mechanism in the jet. The use of Lagrangian statistics following fluid particles gives a more correct description of the entrainment mechanism than in previous works since the statistics in relation to the TNTI position involve data from the trajectories of the entraining fluid particles. The Lagrangian statistics for the particles show the existence of a velocity jump and a characteristic vorticity jump (with a thickness which is one order of magnitude greater than the Kolmogorov microscale), in agreement with previous results using Eulerian statistics. The particles initially acquire enstrophy by viscous diffusion and later by enstrophy production, which becomes "active" only deep inside the turbulent region. Both enstrophy diffusion and production near the TNTI differ substantially from inside the turbulent region. Only about 1% of all particles find their way into pockets of irrotational flow engulfed into the turbulent shear layer region, indicating that "engulfment" is not significant for the present flow, indirectly suggesting that the entrainment is largely due to "nibbling" small-scale mechanisms acting along the entire TNTI surface. Probability density functions of particle positions suggests that the particles spend more time crossing the region near the TNTI than traveling inside the turbulent region, consistent with the particles moving tangent to the interface around the time they cross it.

A study on the motion characteristics and their impact on the wind measurement post-processing of the GPS dropwindsonde Part I: effects of the wind-finding equations

Since Global Positioning System dropwindsonde wind measurements are taken in neither a conventional Eulerian framework nor a perfect Lagrangian framework, proper post-processing is necessary to derive conventional Eulerian wind statistics from dropwindsonde measurements. In this study, the equations governing the dropwindsonde motions were first linearized and then analytically solved to explore the effects of the wind-finding equations introduced by Hock and Franklin (Bull Amer Meteor Soc 80:407–420, 1999) when retrieving horizontal winds from dropwindsonde measurements. It is found that the wind-finding equations are essential for calculating turbulence statistics from dropwindsonde measurements. To confirm, numerical simulations of dropwindsonde motions in a pseudo-stochastic wind field were further conducted based on the full nonlinear dropwindsonde motion-governing equations. Based on the simulation results, the present paper showed the effects of the wind-finding equations and the companion paper discussed the impacts of various post-processing/composition schemes on both the mean and turbulent wind statistics composited from pseudo-dropwindsonde measurements.

Lagrangian and Eulerian Statistics of Pipe Flows Measured with 3D-PTV at Moderate and High Reynolds Numbers

Three-dimensional particle tracking velocimetry (3D-PTV) measurements have provided accurate Eulerian and Lagrangian high-order statistics of velocity and acceleration fluctuations and correlations at Reynolds number 10,300, based on the bulk velocity and the pipe diameter. Spatial resolution required in the analysis method and number of correlation samples required for Lagrangian and Eulerian statistics have been quantified. Flaws in a previously published analyzing method have been overcome. Furthermore, new experimental solutions are presented to facilitate similar measurements at Reynolds numbers of 15,000 and beyond. The Lagrangian velocity correction functions are used to determine the Kolmogorov constant.

Eulerian and Lagrangian Statistics in the South China Sea as Deduced from Surface Drifters

Abstract Eulerian and Lagrangian statistics of the surface flows over the South China Sea (SCS) are investigated based on observations of satellite-tracked Lagrangian drifters of the Surface Velocity Program. Although the dataset contains the most recent observations up to 2011, the spatial and temporal distributions of the data are quite inhomogeneous. The Luzon Strait as well as the northern and western parts of the SCS are heavily sampled by drifters whereas the internal SCS is less sampled. The overall Eulerian mean circulation derived from all drifter observations resembles the wintertime circulations because of the seasonal sampling bias toward wintertime. It is found that the calculation of the diffusivity may be misled by the low-frequency temporal variabilities such as annual cycle, which could be removed using the Gauss–Markov decomposition. In the regions around Luzon Strait, typical values of diffusivity, time scale, and length scale are 3.7–11.9 × 107 cm2 s−1, 0.8–1.9 days, and 16.4–44.4 km, respectively. In the regions around Hainan Island and the western boundary of the SCS, the typical estimated diffusivity and length scale are slightly smaller, that is, 2.3–6.9 × 107 cm2 s−1 and 12.8–33.7 km while the time scale (0.8–2.0 days) is approximately the same as that in the regions around Luzon Strait. Although these regions are close to the coast, the “flux versus gradient” relationship is still valid in these coastal regions.

Interaction of a converging spherical shock wave with isotropic turbulence

Simulations of converging spherical shock waves propagating through a region of compressible isotropic turbulence are carried out. Both converging and reflected phases of the shock are studied. Effect of the reflected phase of the shock is found to be quite different from the expanding shock in the Taylor blast wave-turbulence interaction problem. Vorticity and turbulent kinetic energy are amplified due to passage of the shock. Similar to the latter problem, the vorticity-dilatation term is primarily responsible for the observed behavior. This is confirmed through Eulerian and Lagrangian statistics. Transverse vorticity amplification is compared with linear planar shock-turbulence theory. The smallest eddies, represented by the Kolmogorov scale, decrease in size after passing through the converging shock and this is shown to be related to a decrease in kinematic viscosity and increase in dissipation behind the converging shock. Distortion of the shock due to turbulence is also investigated and quantified. Turbulence also affects maximum compression achieved at the point of shock reflection, when the shock radius is at a minimum. This decrease in compression is quantified by comparing with pure shock simulations.

Conditional Eulerian and Lagrangian velocity increment statistics of fully developed turbulent flow

Conditional statistics of homogeneous isotropic turbulent flow is investigated by means of high Reynolds number direct numerical simulations performed with 20483 collocation points. Eulerian as well as Lagrangian velocity increment statistics under several conditions are analyzed and compared. In agreement with experimental data, longitudinal probability density functions P(δl||u|εl) conditioned on a scale-averaged energy dissipation rate are close to Gaussian distributions over all scales within the inertial range of scales. Also transverse increments conditioned on either the dissipation rate or the square of the vorticity have quasi-Gaussian probability density functions (PDFs). Concerning Lagrangian statistics we found that conditioning on a trajectory averaged energy dissipation rate ετ significantly reduces the scale dependence of the increment PDFs P(δτui|ετ). By means of dimensional arguments we propose a novel condition for Lagrangian increments which is shown to reduce even more the flatness of the corresponding PDFs and thus intermittency in the inertial range of scales. The conditioned Lagrangian PDF corresponding to the smallest increment considered is reasonably well described by a prediction for the PDF of acceleration of a non-intermittent velocity field. Conditioned structure functions show approximately K41 scaling with a larger scaling range than the unconditioned ones.

Large Scale Accumulation Patterns of Inertial Particles in Wall-Bounded Turbulent Flow

Turbulent internal flow in channel and pipe geometry with a diluted second phase of inertial particles is studied numerically. Direct numerical simulations (DNS) are performed at moderate Reynolds number (Reτ ≈ 200) in pipe and two channels—a smaller one similar in size to previous studies and a 3 × 3-times larger one—and Eulerian statistics pertaining to the particle concentration are evaluated. This simulation box constitutes the largest domain used for particle-laden flows so far. The resulting two-point correlations of the particle concentration show that in the smaller channel the particles organize in thin, streamwise elongated patterns which are very regular and long. The spanwise spacing of these structures is 120 and 160 plus units for the channel and pipe, respectively. Only in the larger box, the streamwise extent is long enough for the particle streaks to decorrelate, thus allowing the particles to move more freely. The influence of the box size on the characteristics of the turbophoresis is clearly shown; a 10% increase of the near-wall correlation is observed for particles with Stokes number St + = 50. It is thus shown that the box dimensions are an important factor in correctly assessing the motion of inertial particles, and their relation to the underlying velocity field. In addition the binning size effects on the correlation statistics of particle concentration are exploited. In particular the spanwise correlation peak values appear very sensitive to the adopted binning size, although the position of these peaks is found almost independent. Hence to allow a significant comparison between data of different configurations it is necessary to adopt the same binning spacing in inner variable.

Statistics on staircase tableaux, eulerian and mahonian statistics

We give a simple bijection between some staircase tableaux and tables of inversion. Some nice properties of the bijection allows us to define some q-Eulerian polynomials related to the staircase tableaux. We also give a combinatorial interpretation of these q-Eulerian polynomials in terms of permutations. Nous proposons une bijection simple entre certains tableaux escalier et les tables d'inversion. Cette bijection nous permet de montrer que les statistiques Euleriennes et Mahoniennes sont naturelles sur les tableaux escalier. Nous définissons des polynômes q-Eulériens et en donnons une interprétation combinatoire.

An Interesting New Mahonian Permutation Statistic

The standard algorithm for generating a random permutation gives rise to an obvious permutation statistic DIS that is readily seen to be Mahonian. We give evidence showing that it is not equal to any previously published statistic. Nor does its joint distribution with the standard Eulerian statistics des and exc appear to coincide with any known Euler-Mahonian pair. A general construction of Skandera yields an Eulerian partner eul such that (eul, DIS) is equidistributed with (des, MAJ). However eul itself appears not to be a known Eulerian statistic. Several ideas for further research on this topic are listed.

Inertial range Eulerian and Lagrangian statistics from numerical simulations of isotropic turbulence

We present a study of Eulerian and Lagrangian statistics from a high-resolution numerical simulation of isotropic and homogeneous turbulence using the FLASH code, with an estimated Taylor microscale Reynolds number of around 600. Statistics are evaluated over a data set with 18563 spatial grid points and with 2563 = 16.8 million particles, followed for about one large-scale eddy turnover time. We present data for the Eulerian and Lagrangian structure functions up to the tenth order. We analyze the local scaling properties in the inertial range. The Eulerian velocity field results show good agreement with previous data and confirm the puzzling differences previously found between the scaling of the transverse and the longitudinal structure functions. On the other hand, accurate measurements of sixth-and-higher-order Lagrangian structure functions allow us to highlight some discrepancies from earlier experimental and numerical results. We interpret this result in terms of a possible contamination from the viscous scale, which may have affected estimates of the scaling properties in previous studies. We show that a simple bridge relation based on a multifractal theory is able to connect scaling properties of both Eulerian and Lagrangian observables, provided that the small differences between intermittency of transverse and longitudinal Eulerian structure functions are properly considered.

Improving SCIPUFF Dispersion Forecasts with NWP Ensembles

Abstract The relationships between atmospheric transport and dispersion (AT&D) plume uncertainty and uncertainties in the transporting wind fields are investigated using the Second-Order Closure, Integrated Puff (SCIPUFF) AT&D model driven by numerical weather prediction (NWP) meteorological fields. Modeled contaminant concentrations for episode 1 of the 1983 Cross-Appalachian Tracer Experiment (CAPTEX-83) are compared with recorded ground-level concentrations of the inert tracer gas C7F14. This study evaluates a Taylor-diffusion-based parameterization of dispersion uncertainty for SCIPUFF that uses Eulerian meteorological ensemble velocity statistics and a Lagrangian integral time scale as input. These values are diagnosed from NWP ensemble data. Individual simulations of the tracer release fail to reproduce some of the monitored surface concentrations of the tracer. The plumes that are predicted using the uncertainty model in SCIPUFF are broader, improving the overlap between the predicted and observed results. Augmenting the meteorological input to SCIPUFF with meteorological ensemble-uncertainty parameters therefore provides both a better estimate of the expected plume location and the relative uncertainties in the predicted concentrations than single deterministic forecasts. These results suggest that this new parameterization of NWP wind field uncertainty for dispersion may provide more sophisticated information that may benefit emergency response and decision making.

The Lefschetz property for barycentric subdivisions of shellable complexes

We show that an ‘almost strong Lefschetz’ property holds for the barycentric subdivision of a shellable complex. From this we conclude that for the barycentric subdivision of a Cohen-Macaulay complex, the $h$-vector is unimodal, peaks in its middle degree (one of them if the dimension of the complex is even), and that its $g$-vector is an $M$-sequence. In particular, the (combinatorial) $g$-conjecture is verified for barycentric subdivisions of homology spheres. In addition, using the above algebraic result, we derive new inequalities on a refinement of the Eulerian statistics on permutations, where permutations are grouped by the number of descents and the image of $1$.