Lagrangian Stochastic Research Articles

Stochastic Lagrangian dynamics for charged flows in the E-F regions of ionosphere

We develop a three-dimensional numerical model for the E-F region ionosphere and study the Lagrangian dynamics for plasma flows in this region. Our interest rests on the charge-neutral interactions and the statistics associated with stochastic Lagrangian motion. In particular, we examine the organizing mixing patterns for plasma flows due to polarized gravity wave excitations in the neutral field, using Lagrangian coherent structures (LCS). LCS objectively depict the flow topology—the extracted attractors indicate generation of ionospheric density gradients, due to accumulation of plasma. Using Lagrangian measures such as the finite-time Lyapunov exponents, we locate the Lagrangian skeletons for mixing in plasma, hence where charged fronts are expected to appear. With polarized neutral wind, we find that the corresponding plasma velocity is also polarized. Moreover, the polarized velocity alone, coupled with stochastic Lagrangian motion, may give rise to polarized density fronts in plasma. Statistics of these trajectories indicate high level of non-Gaussianity. This includes clear signatures of variance, skewness, and kurtosis of displacements taking polarized structures aligned with the gravity waves, and being anisotropic.

Connectivity Modeling System: A probabilistic modeling tool for the multi-scale tracking of biotic and abiotic variability in the ocean

Pelagic organisms' movement and motion of buoyant particles are driven by processes operating across multiple, spatial and temporal scales. We developed a probabilistic, multi-scale model, the Connectivity Modeling System (CMS), to gain a mechanistic understanding of dispersion and migration processes in the ocean. The model couples offline a new nested-grid technique to a stochastic Lagrangian framework where individual variability is introduced by drawing particles' attributes at random from specified probability distributions of traits. This allows 1) to track seamlessly a large number of both actively swimming and inertial particles over multiple, independent ocean model domains and 2) to generate ensemble forecasts or hindcasts of the particles' three dimensional trajectories, dispersal kernels, and transition probability matrices used for connectivity estimates. In addition, CMS provides Lagrangian descriptions of oceanic phenomena (advection, dispersion, retention) and can be used in a broad range of oceanographic applications, from the fate of pollutants to the pathways of water masses in the global ocean. Here we describe the CMS modular system where particle behavior can be augmented with specific features, and a parallel module implementation simplifies data management and CPU intensive computations associated with solving for the tracking of millions of active particles. Some novel features include on-the-fly data access of operational hydrodynamic models, individual particle variability and inertial motion, and multi-nesting capabilities to optimize resolution. We demonstrate the performance of the interpolation algorithm by testing accuracy in tracing the flow stream lines in both time and space and the efficacy of probabilistic modeling in evaluating the bio-physical coupling against empirical data. Finally, following recommended practices for the development of community models, we provide an open source code with a series of coupled standalone, optional modules detailed in a user's guide.

Steady plumes in heterogeneous porous formations: A stochastic Lagrangian approach

Key Points Steady concentration (C) field in heterogeneous formations is analyzed Analytical solutions for the mean and variance of C are developed Local scale dispersion exerts a great influence on both C moments

Incorporating Sweeps and Ejections into Lagrangian Stochastic Models of Spore Trajectories Within Plant Canopy Turbulence: Modeled Contact Distributions Are Heavy-Tailed

ABSTRACT The turbulent dispersal of fungal spores within plant canopies is very different from that within atmospheric boundary-layers and closely analogous to dispersal within turbulent mixing-layers. The process is dominated by the presence of large coherent flow structures, high-velocity downdrafts (sweeps) and updrafts (ejections), that punctuate otherwise quiescent flow. Turbulent dispersion within plant canopies is best predicted by Lagrangian stochastic (particle-tracking) models because other approaches (e.g., diffusion models and similarity theory) are either inappropriate or invalid. Nonetheless, attempts to construct such models have not been wholly successful. Accounting for sweeps and ejections has substantially worsened rather than improved model agreement with experimental dispersion data. Here we show how this long-standing difficulty with the formulation of Lagrangian stochastic models can be overcome. The new model is shown to be in good agreement with data from a carefully controlled, well-documented wind-tunnel study of scalar dispersion within plant canopy turbulence. Equally good agreement with this data is obtained using Thomson's (1987) Gaussian model. This bolsters confidence in the application of this simple model to the prediction of spore dispersal within plant canopy turbulence. Contact distributions-the probability distribution function for the distance of viable fungal spore movement until deposition-are predicted to have "heavy" inverse power-law tails. It is known that heavy-tailed contact distributions also characterize the dispersal of spores which pass through the canopy turbulence and enter into the overlying atmospheric boundary-layer. Plant disease epidemics due to the airborne dispersal of fungal spores are therefore predicted to develop as accelerating waves over a vast range of scales-from the within field scale to intercontinental scales. This prediction is consistent with recent analyses of field and historical data for rusts in wheat. Such plant disease epidemics are shown to be governed by space-fractional diffusion equations and by Lévy flights.

Lagrangian Stochastic Model for the Dispersion of Solid-State Radionuclides Released from Uranium Mine Ventilation Shafts

Radionuclides released from uranium mine ventilation shafts would pose radiation exposure to the public and environment. A three-dimensional lagrangian stochastic model has been presented to study the atmospheric dispersion of solid-state radionuclides released from the uranium-bearing mine ventilation shafts. Meteorological conditions and geographical conditions including four downwind velocities (0.5, 1.0, 2.0, 4.0 m/s) and two underlying surface roughness characteristics (0.1 m, 1.0 m) were chose in the study. The radionuclides concentration distributions at various wind speeds and surface roughness were attained to evaluate the pollution in the vicinity of uranium mine ventilation shaft.

Predictive model for scalar concentration fluctuations in and above a model plant canopy

A Lagrangian stochastic (LS) implementation of an interaction by exchange with the conditional mean (IECM) micromixing model is used to estimate concentration fluctuations in plumes dispersing from an in-canopy, localized source into a model plant canopy flow. The sensitivity of the LS-IECM model to the underlying Eulerian flow statistics is investigated by comparing model predictions from simulations driven by two interpolations of the same water-channel flow data. The two simulations showed minor differences in the predicted mean concentration but marked differences in the standard deviation, skewness and kurtosis of concentration. This is shown to be caused by differences in the turbulent kinetic energy (TKE) dissipation rates between the two interpolations and their effects on the IECM model. The LS-IECM model predictions of the first four moments of the scalar concentration field showed fair to good conformance (depending on which TKE dissipation rate is used) with experimental water-channel dispersion data.

Pollen productivity estimates strongly depend on assumed pollen dispersal

Past plant abundance may be reconstructed from pollen data if dispersal distances of pollen and pollen productivities of each taxon are known. Using surface sediment samples from small and medium sized, closed and near circular lakes from lowland Central Europe, we tested the validity of three pollen dispersal models by comparing empirical pollen data from each lake with simulated pollen data derived from applying various pollen dispersal models to vegetation data from rings situated up to 100 km from each site. Pollen assemblages simulated with a Lagrangian stochastic (LS) model best fit real pollen assemblages, simulations with the commonly used Prentice model on pollen dispersal underestimated the amount of pollen arriving from distances larger than 10 km and overestimated the differences in dispersal distances between lighter ( Pinus) and heavier ( Fagus, Picea) pollen grains. The LS model appeared to provide more appropriate simulations. Pollen productivity estimates (PPEs) calculated for the data set showed that the choice of the dispersal model has great impact on the results. If derived with the Prentice model, PPEs for Fagus and Picea are three times higher than with the LS model. Studies on pollen productivities thus need to consider the apparent limitations of the Prentice model. We suggest an alternative approach, which uses simulations instead of the extended R-value model, to calculate PPEs. The approach is flexible in the use of dispersal functions and produced consistent results for two independent data sets from small and medium sized lakes.

Lagrangian Navier–Stokes diffusions on manifolds: Variational principle and stability

We prove a variational principle for stochastic flows on manifolds. It extends V.I. Arnoldʼs description of Lagrangian Euler flows, which are geodesics for the L2 metric on the manifold, to the stochastic case. Here we obtain stochastic Lagrangian flows with mean velocity (drift) satisfying the Navier–Stokes equations. We study the stability properties of such trajectories as well as the evolution in time of the rotation between the underlying particles. The case where the underlying manifold is the two-dimensional torus is described in detail.

Effect of chemical degradation on fluxes of reactive compounds – a study with a stochastic Lagrangian transport model

Abstract. In the analyses of VOC fluxes measured above plant canopies, one usually assumes the flux above canopy to equal the exchange at the surface. Thus one assumes the chemical degradation to be much slower than the turbulent transport. We used a stochastic Lagrangian transport model in which the chemical degradation was described as first order decay in order to study the effect of the chemical degradation on above canopy fluxes of chemically reactive species. With the model we explored the sensitivity of the ratio of the above canopy flux to the surface emission on several parameters such as chemical lifetime of the compound, friction velocity, stability, and canopy density. Our results show that friction velocity and chemical lifetime affected the loss during transport the most. The canopy density had a significant effect if the chemically reactive compound was emitted from the forest floor. We used the results of the simulations together with oxidant data measured during HUMPPA-COPEC-2010 campaign at a Scots pine site to estimate the effect of the chemistry on fluxes of three typical biogenic VOCs, isoprene, α-pinene, and β-caryophyllene. Of these, the chemical degradation had a major effect on the fluxes of the most reactive species β-caryophyllene, while the fluxes of α-pinene were affected during nighttime. For these two compounds representing the mono- and sesquiterpenes groups, the effect of chemical degradation had also a significant diurnal cycle with the highest chemical loss at night. The different day and night time loss terms need to be accounted for, when measured fluxes of reactive compounds are used to reveal relations between primary emission and environmental parameters.

First-Order Inconsistencies Caused by Rogue Trajectories

A theoretical requirement of the Interaction by Exchange with the Conditional Mean (IECM) micromixing model is that the mean concentration field produced by it must be consistent with the mean concentration field produced by a traditional Lagrangian stochastic (LS) marked particle model. We examine the violation of this requirement that occurs in a coupled LS-IECM model when unrealistically high particle velocities occur. No successful strategy was found to mitigate the effects of these rogue trajectories. It is our hope that this work will provide renewed impetus for investigation into rogue trajectories and methods to eliminate them from LS models.

Longitudinal Dispersion in River Flows Characterized by Random Large-Scale Bed Irregularities: First-Order Analytical Solution

AbstractA stochastic Lagrangian approach is proposed for the analytical derivation of a longitudinal dispersion coefficient that accounts for both transverse and longitudinal flow field variability in straight-axis open channels characterized by longitudinal large-scale bed heterogeneity, that is, by longitudinal bed irregularities over a wide range of representative lengths, from the simple grain roughness to large undulations (up to order of kilometers), possibly related to topographical discontinuities or to extended and inhomogeneous depositional processes carried out by natural or anthropogenic agents. The resulting dimensionless expression, involving the average Chezy coefficient and the ratios of river width and longitudinal heterogeneity correlation length to the average flow depth, is obtained at the first order in the depth fluctuations as a time-dependent function given by the sum of three distinct components. The first main component is related to the transverse velocity distribution and would...

Turbulent dispersion of a gas tracer in a nocturnal atmospheric flow

A Lagrangian turbulent dispersion model has been coupled with the Regional Atmospheric Modelling System (RAMS) to predict the transport, dispersion and concentration of a gas tracer released into a stable atmosphere (nocturnal drainage flow) of the Anderson Springs Valley near San Francisco, California. The mean wind velocities governed by the topography of the region and the surface layer fluxes of momentum and heat into the stable atmosphere, were calculated by the RAMS code. The atmospheric transport and dispersion of a gas tracer was simulated in non-homogeneous turbulence conditions by using a Lagrangian Stochastic Particle Model. A Moving Least Squares technique was used to calculate the tracer gas concentration. The vertical profiles of the mean wind speed and the wind direction, as well as the tracer gas concentration, were compared with the field measurements for the fourth night of the ASCOT experiment (the night between 19 and 20 September 1980) carried out in the Anderson Springs Valley, California, USA The computed profiles for wind direction and wind speed agreed well with the field data. Although at some locations the model predicted strong winds which transported the particle to larger distances downwind, the overall pattern of concentration distribution generated by the model compared quite well with the observations. It was concluded that the concentration patterns tend to follow the topography when the synoptic winds are weak.

Stochastic Lagrangian trajectory model for drifting objects in the ocean

The prediction of drifting object trajectories in the ocean is a complex problem plagued with uncertainties. This problem is usually solved simulating the possible trajectories based on wind and advective numerical and/or instrumental data in real time, which are incorporated into Lagrangian trajectory models. However, both data and Lagrangian models are approximations of reality and when comparing trajectory data collected from drifter exercises with respect to Lagrangian models results, they differ considerably. This paper introduces a stochastic Lagrangian trajectory model that allows quantifying the uncertainties related to: (i) the wind and currents numerical and/or instrumental data, and (ii) the Lagrangian trajectory model. These uncertainties are accounted for within the model through random model parameters. The quantification of these uncertainties consists in an estimation problem, where the parameters of the probability distribution functions of the random variables are estimated based on drifter exercise data. Particularly, it is assumed that estimated parameters maximize the likelihood of our model to reproduce the trajectories from the exercise. Once the probability distribution parameters are estimated, they can be used to simulate different trajectories, obtaining location probability density functions at different times. The advantage of this method is that it allows: (i) site specific calibration, and (ii) comparing uncertainties related to different wind and currents predictive tools. The proposed method is applied to data collected during the DRIFTER Project (eranet AMPERA, VI Programa Marco), showing very good predictive skills.

Stochastic Lagrangian flows on some compact manifolds

We investigate ponctual as well as L 2 distances of some stochastic processes with values in the group of homeomorphisms of a compact manifold including processes modelling time evolution of fluids. These processes are associated with operators of the form Laplace–Beltrami plus a first-order term. Several constructions are presented, in particular via coupling methods, the corresponding behaviour of the distance depending on the construction and on the drift properties.

Lagrangian models of reactive transport in heterogeneous porous media with uncertain properties

We consider multi-component reactive transport in heterogeneous porous media with uncertain hydraulic and chemical properties. This parametric uncertainty is quantified by treating relevant flow and transport parameters as random fields, which renders the governing equations stochastic. We adopt a stochastic Lagrangian framework to replace a three-dimensional advection–reaction transport equation with a one-dimensional equation for solute travel times. We derive approximate expressions for breakthrough curves and their temporal moments. To illustrate our general theory, we consider advective transport of dissolved species undergoing an irreversible bimolecular reaction.

Measuring Trace Gas Emission from Multi-Distributed Sources Using Vertical Radial Plume Mapping (VRPM) and Backward Lagrangian Stochastic (bLS) Techniques

Two micrometeorological techniques for measuring trace gas emission rates from distributed area sources were evaluated using a variety of synthetic area sources. The vertical radial plume mapping (VRPM) and the backward Lagrangian stochastic (bLS) techniques with an open-path optical spectroscopic sensor were evaluated for relative accuracy for multiple emission-source and sensor configurations. The relative accuracy was calculated by dividing the measured emission rate by the actual emission rate; thus, a relative accuracy of 1.0 represents a perfect measure. For a single area emission source, the VRPM technique yielded a somewhat high relative accuracy of 1.38 ± 0.28. The bLS technique resulted in a relative accuracy close to unity, 0.98 ± 0.24. Relative accuracies for dual source emissions for the VRPM and bLS techniques were somewhat similar to single source emissions, 1.23 ± 0.17 and 0.94 ± 0.24, respectively. When the bLS technique was used with vertical point concentrations, the relative accuracy was unacceptably low.

Regional Source Identification Using Lagrangian Stochastic Particle Dispersion and HYSPLIT Backward-Trajectory Models

ABSTRACT The main objective of this study was to investigate the capabilities of the receptor-oriented inverse mode Lagrangian Stochastic Particle Dispersion Model (LSPDM) with the 12-km resolution Mesoscale Model 5 (MM5) wind field input for the assessment of source identification from seven regions impacting two receptors located in the eastern United States. The LSPDM analysis was compared with a standard version of the Hybrid Single-Particle Lagrangian Integrated Trajectory (HYSPLIT) single-particle backward-trajectory analysis using inputs from MM5 and the Eta Data Assimilation System (EDAS) with horizontal grid resolutions of 12 and 80 km, respectively. The analysis included four 7-day summertime events in 2002; residence times in the modeling domain were computed from the inverse LSPDM runs and HYPSLIT-simulated backward trajectories started from receptor-source heights of 100, 500, 1000, 1500, and 3000 m. Statistics were derived using normalized values of LSPDM- and HYSPLIT-predicted residence times versus Community Multiscale Air Quality model-predicted sulfate concentrations used as baseline information. From 40 cases considered, the LSPDM identified first- and second-ranked emission region influences in 37 cases, whereas HYSPLIT-MM5 (HYSPLIT-EDAS) identified the sources in 21 (16) cases. The LSPDM produced a higher overall correlation coefficient (0.89) compared with HYSPLIT (0.55–0.62). The improvement of using the LSPDM is also seen in the overall normalized root mean square error values of 0.17 for LSPDM compared with 0.30–0.32 for HYSPLIT. The HYSPLIT backward trajectories generally tend to underestimate near-receptor sources because of a lack of stochastic dispersion of the backward trajectories and to overestimate distant sources because of a lack of treatment of dispersion. Additionally, the HYSPLIT backward trajectories showed a lack of consistency in the results obtained from different single vertical levels for starting the backward trajectories. To alleviate problems due to selection of a backward-trajectory starting level within a large complex set of 3-dimensional winds, turbulence, and dispersion, results were averaged from all heights, which yielded uniform improvement against all individual cases. IMPLICATIONS Backward-trajectory analysis is one of the standard procedures for determining the spatial locations of possible emission sources affecting given receptors, and it is frequently used to enhance receptor modeling results. This analysis simplifies some of the relevant processes such as pollutant dispersion, and additional methods have been used to improve receptor-source relationships. A methodology of inverse Lagrangian stochastic particle dispersion modeling was used in this study to complement and improve standard backward-trajectory analysis. The results show that inverse dispersion modeling can identify regional sources of haze in national parks and other regions of interest.

Comparing Two Implementations of a Micromixing Model. Part II: Canopy Flow

The sequential particle micromixing model (SPMMM) is used to estimate concentration fluctuations in plumes dispersing into a canopy flow. SPMMM uses the familiar single-particle Lagrangian stochastic (LS) trajectory framework to pre-calculate the required conditional mean concentrations, which are then used by an interaction by exchange with the conditional mean (IECM) micromixing model to predict the higher-order fluctuations of the scalar concentration field. The predictions are compared with experimental wind-tunnel dispersion data for a neutrally stratified canopy flow, and with a previously reported implementation using simultaneous particle trajectories. The two implementations of the LS–IECM model are shown to be largely consistent with one another and are able to simulate dispersion in a canopy flow with fair to good accuracy.

Comparing Two Implementations of a Micromixing Model. Part I: Wall Shear-Layer Flow

A Lagrangian stochastic (LS) micromixing model is used for estimating concentration fluctuations in plumes of a passive, non-reactive tracer dispersing from elevated and ground-level compact sources into a neutral wall shear-layer flow. SPMMM (for sequential particle micromixing model) implements the familiar IECM (interaction by exchange with the conditional mean) micromixing scheme. The parametrization of the scalar micromixing time scale is identical to that proposed in a previously reported LS–IECM model (Cassiani et al., Atmos Environ 39:1457–1469, 2005a). However, while SPMMM is mathematically equivalent to the previously reported model, it differs in its numerical implementation: SPMMM releases N independent particles sequentially, whereas the previously reported model releases N independent particles simultaneously. In both implementations, the trajectories of the N particles are governed by single-point velocity statistics. The sequential particle implementation is computationally efficient, but cannot be applied to the case of reacting species. Results from both implementations are compared to experimental wind-tunnel dispersion data and to each other.

Lagrangian Navier–Stokes flows: a stochastic model

We associate stochastic Lagrangian flows with Navier–Stokes (deterministic) velocities and show their unstable behaviour.