Stochastic Dispersion Research Articles

Performance of a Dispersion Model to Estimate Methane Loss from Cattle in Pens

Accurate measurements of enteric methane (CH(4)) emissions from cattle (Bos taurus) are necessary to improve emission coefficients used in national emissions inventories, and to evaluate mitigation strategies. Our study was conducted to evaluate a novel approach that allowed near continuous CH(4) measurement from beef cattle confined in pens. The backward Lagrangian Stochastic (bLS) dispersion technique was used in conjunction with global position system (GPS) information from individual animals, to evaluate CH(4) emissions from pens of cattle. The dispersion technique was compared to estimates of CH(4) production using the SF(6) tracer technique. Sixty growing beef cattle were fed a diet containing 60% barley silage (dry matter basis) supplemented with either barley (Hordeum vulgare L.) grain or corn (Zea mays L.) distillers dried grains. The results show that daily CH(4) emissions were about 7% lower for the dispersion technique than for the tracer technique (185 vs. 199 g CH(4) animal(-1) d(-1)). The precision of the dispersion technique, relative to the SF(6) tracer technique, expressed by the Pearson coefficient was 0.76; the relative accuracy given by the concordance coefficient was 0.69. The bLS dispersion technique was able to detect differences (P < 0.05) due to diet and has the added advantage of measuring the pattern of CH(4) production during the 24-h period, with emissions ranging from 161 to 279 g CH(4) animal(-1) d(-1). Configuring the cattle as point sources resulted in more accurate CH(4) emissions than assuming a uniform area release from the pen surface. The results indicate that the bLS dispersion technique using cattle as point sources can be used to accurately measure enteric CH(4) from cattle and to evaluate the impact of dietary mitigation strategies.

Assessment of the backward Lagrangian Stochastic dispersion technique for continuous measurements of CH 4 emissions

Continuous measurements of methane (CH 4) emissions from agricultural facilities over a period of several days are needed to assess the mitigation effectiveness of management practices. In this study, we examined the feasibility of using an inverse dispersion technique (backward Lagrangian Stochastic model, bLS) for obtaining 15-min averages of CH 4 emissions over a period of 5 days in the Ottawa area. Using two open-path lasers and pan-tilt scanning units, a ground level source was enclosed to measure the CH 4 emissions ( Q) with any wind direction. After application of the recommended data quality criteria screening for low friction velocity and extreme atmospheric stability, the average recovery (ratio of estimates Q bLS to Q) was 1.09 with a standard deviation of 0.45. We observed a tendency in the recovery results toward underestimation during unstable stratification and overestimation during stable stratification. Using a test on developed turbulence instead of a fixed friction velocity threshold improved the accuracy of the emission estimates slightly. An additional data quality criterion based on the standard deviation output of the bLS model led to a significant improvement with a recovery of 1.00 and a standard deviation of 0.30. A strategy for averaging the resulting incomplete dataset is discussed. An assessment of the applicability of this approach to farm-size facilities led to the conclusion that this technique is suitable to determine emissions from realistic CH 4 sources, such as dairy cattle barns, continuously in order to characterize both seasonal and diurnal characteristics of CH 4 emissions.

Employing turbulent and meandering time scales to modeling the contaminants enhanced horizontal dispersion

A Lagrangian stochastic particle dispersion model composed of two coupled Langevin equations, employing a new transport parameterization is used to study the meandering enhanced dispersion in a low wind speed stable atmospheric boundary layer (ABL). The meandering parameterization introduced into the Lagrangian stochastic dispersion model is expressed in terms of a characteristic phenomenological meandering period and of the horizontal local turbulent time scales associated with a fully developed turbulence. The results of this new method are shown to agree with the field observations of Idaho experiments and also with other different meandering dispersion models. The major advance shown throughout this paper is as it follows: For air quality modeling it is highly necessary to include a parameterization that allows a correct description of the dispersion caused by the low-frequency horizontal wind oscillations.

Conditional simulation schemes of rain fields and their application to rainfall–runoff modeling studies in the Sahel

Summary In regions characterized by a great inter-annual variability or by decadal-scale changes of the rainfall regime, the simulation of long series of rainfall events is an efficient way to explore the runoff fluctuations or modifications resulting from this rainfall variability. In a context of great uncertainty regarding the Sahelian rainfall regime in a changing climate, a coherent stochastic framework is presented here to produce high spatial resolution rain fields in order to force a rainfall–runoff model and to perform sensitivity analyses. The focus of the paper is on the comparison of various conditioning methods reflecting the various types of information available for the study of past situations (data from rain gage and satellite) as well as of future scenarios (outputs of atmospheric models). Various types of rainfall simulations are performed over a 13 year period, using four levels of conditioning information obtained from a 15 gauge network covering a 60 × 60 km2 region. These simulations are then used as inputs to a Hortonian rainfall–runoff model. The simulation relevance is first assessed by studying the simulated rain field series (event time-step mean characteristics, seasonal cycle and inter-annual variability) in comparison with reference rain fields estimated by kriging. This shows that the conditioning of the simulations, even by a minimal information provided by a unique station, is of great relevance for constraining the stochastic dispersion and thus to retrieve the rainfall variability at the considered scales. Significant differences are reported between runoff obtained by the different types of created rain fields, one of the most noticeable being that runoff obtained from kriging is 25% lower than runoff obtained from point conditional simulations. The results confirm the sensitivity of Hortonian hydrological systems to rainfall intensity and particularly point out the importance of representing realistic spatial rainfall patterns to force hydrological models.

The Lagrangian stochastic model for estimating footprint and water vapor fluxes over inhomogeneous surfaces

This study investigated a two-dimensional Lagrangian stochastic dispersion model for estimating water vapor fluxes and footprint over homogeneous and inhomogeneous surfaces. Over the homogeneous surface, particle trajectories were computed from a 2-D Lagrangian model forced by Eulerian velocity statistics determined by Monin-Obukhov similarity theory (MOST). For an inhomogeneous surface, the velocity and atmospheric stability profiles were computed using a second-order Eulerian closure model, and these local profiles were then used to drive the Lagrangian model. The model simulations were compared with water vapor flux measurements carried out above an irrigated bare soil site and an irrigated potato site. The inhomogeneity involved a step change in surface roughness, humidity, and temperature. Good agreement between eddy-correlation-measured and Lagrangian-model-predicted water vapor fluxes was found for both sites. Hence, this analysis demonstrates the practical utility of second-order closure models in conjunction with Lagrangian analysis to estimate the scalar footprint in planar inhomogeneous flows.

On Non-Existence of a One Factor Interest Rate Model for Volatility Averaged Generalized Fong-Vasicek Term Structures

The purpose of this paper is to study the generalized Fong - Vasicek two-factor interest rate model with stochastic volatility. In this model the dispersion of the stochastic short rate (square of volatility) is assumed to be stochastic as well and it follows a non-negative process with volatility proportional to the square root of dispersion. The drift of the stochastic process for the dispersion is assumed to be in a rather general form including, in particular, linear function having one root (yielding the original Fong - Vasicek model or a cubic like function having three roots (yielding a generalized Fong - Vasicek model for description of the volatility clustering). We consider averaged bond prices with respect to the limiting distribution of stochastic dispersion. The averaged bond prices depend on time and current level of the short rate like it is the case in many popular one-factor interest rate model including in particular the Vasicek and Cox - Ingersoll-Ross model. However, as a main result of this paper we show that there is no such one-factor model yielding the same bond prices as the averaged values described above.

Approaches to measuring fluxes of methane and nitrous oxide between landscapes and the atmosphere

The theory, applications, strengths and weaknesses of approaches commonly used for measuring trace gas fluxes are reviewed. Chambers, representing the smallest scale (∼1 m2), are the most common tools. Their operating principle is simple, they can be highly sensitive, the cost can be low and field requirements small. Problems include leaks, stickiness of some gases, inhibition of fluxes through concentration build-up, pressure effects and spatial and temporal variability in gas fluxes. Mass balance techniques are suitable for small, defined source areas, typically tens to thousands of square metres in extent. Emissions are calculated from the difference in the rates at which the gas is carried into a control volume above the source area by the wind and carried out. The required primary data are profiles of gas concentration on the downwind boundaries as well as the wind speed profile, the wind direction and the upwind background gas concentration. They have been used to measure gas emissions from landfills, treated fields and small animal herds. Circular test areas make the method independent of wind direction. A newly developed technique based on a backward Lagrangian stochastic dispersion model is also applicable to small, well-defined source areas of any shape. The surface flux is calculated form measurements of atmospheric turbulence and stability and the gas concentration at any height downwind. Implementation of the method is aided greatly by a software package WindTrax. The combination provides a powerful new tool for measuring gas emissions from treated areas and intensive animal production systems. Finally, techniques suitable for measuring gas emissions on large landscape scales (ha) are discussed. Eddy covariance is the micrometeorologist’s preferred technique for this scale. The method uses fast response anemometers and gas sensors to make direct measurements of the vertical gas flux at a point, several times a second. However, it is not feasible for many trace gases for a variety of reasons. These are discussed. Relaxed eddy accumulation is an alternative technique that retains the attraction of eddy covariance by providing a direct point measurement. It removes the need for a fast response gas sensor by substituting for it a fast solenoid valve sampling system. Flux–gradient methods are in more common use. Fluxes are calculated as the product of an eddy diffusivity and the vertical concentration gradient of the gas or the product of a transfer coefficient and the difference in gas concentration between two heights. Assumptions of the method and precautions in its application are discussed.

Turbulence dissipation rate derivation for meandering occurrences in a stable planetary boundary layer

Abstract. A new formulation for the turbulence dissipation rate ε occurring in meandering conditions has been presented. The derivation consists of a MacLaurin series expansion of a lateral dispersion parameter that represents cases in which turbulence and oscillatory movements associated to the meandering events coexist. The new formulation presents the identical physical premises contained in the classical and largely used one, but the new formulation derived from meandering situations is expressed in terms of the loop parameter m that controls the absolute value of the negative lobe in the meandering autocorrelation function. Therefore, the m magnitude regulates the turbulence dissipation rate. This dissipation rate decreases for cases in which turbulence and low frequency horizontal wind oscillations coexist and increases for a fully developed turbulence. Furthermore, a statistical comparison to observed concentration data shows that the alternative relation for the turbulent dissipation rate occurring in situations of meandering enhanced dispersion is suitable for applications in Lagrangian Stochastic dispersion models.

Polarization and frequency disentanglement of photons via stochastic polarization mode dispersion

We investigate the quantum decoherence of frequency and polarization variables of photons via polarization mode dispersion in optical fibers. By observing the analogy between the propagation equation of the field and the Schr\odinger equation, we develop a master equation under Markovian approximation and analytically solve for the field density matrix. We identify distinct decay behaviors for the polarization and frequency variables for single-photon and two-photon states. For the single-photon case, purity functions indicate that complete decoherence for each variable is possible only for infinite fiber length. For entangled two-photon states passing through separate fibers, entanglement associated with each variable can be completely destroyed after characteristic finite propagation distances. In particular, we show that frequency disentanglement is independent of the initial polarization status. For propagation of two photons in a common fiber, the evolution of a polarization singlet state is addressed. We show that while complete polarization disentanglement occurs at a finite propagation distance, frequency entanglement could survive at any finite distance for Gaussian states.

Turbulent statistical characteristics associated to the north wind phenomenon in southern Brazil with application to turbulent diffusion

A parameterization for the transport processes in a shear driven planetary boundary layer (PBL) has been established employing turbulent statistical quantities measured during the north wind phenomenon in southern Brazil. Therefore, observed one-dimensional turbulent energy spectra are compared with a spectral model based on the Kolmogorov arguments. The good agreement obtained from this comparison leads to well defined formulations for the turbulent velocity variance, local decorrelation time scale and eddy diffusivity. Furthermore, for vertical regions in which the wind shear forcing is relevant, the eddy diffusivity derived from the north wind data presents a similar profile to those obtained from the non-extensive statistical mechanics theory. Finally, a validation for the present parameterization has been accomplished, using a Lagrangian stochastic dispersion model. The Prairie Grass data set, which presents high mean wind speed, is simulated. The analysis developed in this study shows that the turbulence parameterization constructed from wind data for north wind flow cases is able to describe the diffusion in a high wind speed, shear-dominated PBL.

Estimation of the Lagrangian Kolmogorov constant from Eulerian measurements for distinct Reynolds number with application to pollution dispersion model

An expression that allows the determination of the Lagrangian Kolmogorov structure function constant C 0 from the knowledge of the constant C S associated to the second order Eulerian velocity structure function and the γ numerical coefficient relating the Eulerian spectra constant to the Lagrangian one, is suggested. Experimental data from both wind tunnel and atmospheric boundary layer observation were used to determine C S. The study agrees with the universal nature of the Eulerian C S constant for fully developed turbulence. The analysis leads to an estimate of C 0≅5.09±0.93, in agreement with previous studies. This estimated value has been employed in a Lagrangian stochastic dispersion model to simulate the observed concentration obtained from classical diffusion experiments.

Evaluation of backward and forward Lagrangian footprint models in the surface layer

We ran a Lagrangian stochastic (LS) dispersion model in both forward-in-time and backward-in-time ways to derive footprints. Three Eulerian analytical footprint models were compared with this Lagrangian model for a wide atmospheric stability range. Despite some differences among the three analytical footprint models, their results generally agreed. Results from the forward LS simulations agreed well with the analytical solutions for both concentration footprints and flux footprints, if turbulent parameters were properly prescribed. Quantitative equivalence between the forward and backward Lagrangian footprint estimates was demonstrated. However, concentration footprint derived by backward LS simulation can be seriously contaminated by numerical errors. A ‘test-adjustment’ scheme treating the temporal integration, and a ‘buffer’ layer treating the surface reflection of the LS particles eliminate numerical errors. Forward LS simulations or the flux footprint estimates were quite insensitive to these errors.

Flux and concentration footprint modelling: State of the art

Since early 1990s, the development of footprint models has been rapid with presently four different approaches being available: (i) analytical models, (ii) Lagrangian stochastic particle dispersion models, (iii) large-eddy simulations, and (iv) closure models. Parameterizations of some of these approaches have been developed, simplifying the original algorithms for use in practical applications. The paper provides a review of the footprint modelling. It also discusses our present understanding of the theoretical background, the most successful modelling approaches, as well as the usage and benefits of the footprint concept as it relates to flux measurements. There has recently been a trend emerging in modelling the behavior of the footprint functions using a less idealized, more realistic description of inhomogeneities, vegetation structure and topography, ultimately for reactive compounds. The estimation of footprints for application in the real world, complete with a multitude of interesting gaseous and particulate substances, remains a complex problem.

Determining ammonia emissions from a cattle feedlot with an inverse dispersion technique

An inverse-dispersion technique is used to calculate ammonia (NH 3) gas emissions from a cattle feedlot. The technique relies on a simple backward Lagrangian stochastic (bLS) dispersion model to relate atmospheric NH 3 concentration to the emission rate Q bLS. Because the wind and the source configuration are complicated, the optimal implementation of the technique is unclear. Two categorically different measurement locations (for concentration and winds) are considered: within the feedlot and downwind. The in-feedlot location proved superior, giving a nearly continuous Q bLS timeseries. We found average emissions of 0.15 kg NH 3 animal −1 day −1 in both 2004 and 2005, representing a loss of 63% (2004) or 65% (2005) of the dietary nitrogen in the animal feed. Downwind measurement locations were less useful for several reasons: a narrow range of useable wind directions; ambiguity in the choice of wind statistics to use in the calculations; low NH 3 concentrations; and downwind deposition of NH 3. When addressing a large source (like a feedlot) that modifies the ambient wind flow, we recommend in-source measurements for use in inverse-dispersion applications.

A Large-Eddy Simulation and Lagrangian Stochastic Study of Heavy Particle Dispersion in the Convective Boundary Layer

Large-eddy simulation and Lagrangian stochastic dispersion models were used to study heavy particle dispersion in the convective boundary layer (CBL). The effects of various geostrophic winds, particle diameters, and subgrid-scale (SGS) turbulence were investigated. Results showed an obvious depression in the vertical dispersion of heavy particles in the CBL and major vertical stratification in the distribution of particle concentrations, relative to the passive dispersion. Stronger geostrophic winds tended to increase the dispersion of heavy particles in the lower CBL. The SGS turbulence, particularly near the surface, markedly influenced the dispersion of heavy particles in the CBL. For reference, simulations using passive particles were also conducted; these simulation results agreed well with results from previous convective tank experiments and numerical simulations.

Analysis of tracer particle migration in inhomogeneous turbulence

Previous studies [J.M. Maclnnes, F.V. Bracco, Stochastic particle dispersion modeling and the tracer particle limit, Physics of Fluids A 4 (1992) 2809–2824; X.Q. Chen, Heavy particle dispersion in inhomogeneous, anisotropic, turbulence flows, International Journal of Multiphase Flow 26 (2000) 635–661; T.L. Bocksell, E. Loth, Random walk models for particle diffusion in free-shear flows, AIAA Journal 29 (2001) 1086–1096] have shown that the commonly applied stochastic separated flow (SSF) model predicts unphysical results when dealing with the dispersion of tracer particles in inhomogeneous flows. This problem is explored, with regards to the discontinuous random walk model, by considering an idealized flow with constant mean velocity with two regions of constant turbulent kinetic energy. Using the probability density functions (PDFs) for the turbulent velocities it is shown that there is a higher probability of particles traveling into the low kinetic energy region than there are traveling to the region of high kinetic energy, thus resulting in a net migration of particles to the region of low kinetic energy. Corrections that apply a correction velocity and/or adjust the fluctuating velocity based on the local value of the turbulent kinetic energy are analyzed and tested.

An Approach for Measuring Methane Emissions from Whole Farms

Estimates of enteric methane (CH4) emissions from ruminants are typically measured by confining animals in large chambers, using head hoods or masks, or by a ratiometric technique involving sampling respired air of the animal. These techniques are not appropriate to evaluate large-scale farm emissions and the variability between farms that may be partly attributed to different farm management. This study describes the application of an inverse-dispersion technique to calculate farm emissions in a controlled tracer-release experiment. Our study was conducted at a commercial dairy farm in southern Alberta, Canada (total of 321 cattle, including 152 lactating dairy cows). Sulfur hexafluoride (SF6) and CH4 were released from 10 outlet locations (barn and open pens) using mass-flow controllers. A Lagrangian stochastic (LS) dispersion model was then used to infer farm emissions from downwind gas concentrations. Concentrations of SF6 and CH4 were measured by gas chromatography analysis and open path lasers, respectively. Wind statistics were measured with a three-dimensional sonic anemometer. Comparing the inferred emissions with the known release rate showed we recovered 86% of the released CH4 and 100% of the released SF6. The location of the concentration observations downwind of the farm was critically important to the success of this technique.

Evaluation of a turbulent flow and dispersion model in a typical street canyon in York, UK

It is likely that the use of computational fluid dynamics (CFD) modelling approaches to represent the dispersion of pollutants within urban areas will become increasingly common in the future. If such models are to be used within a decision-making context, their evaluation for a range of urban geometries should be undertaken. In this work a k–ε flow model coupled with a Lagrangian stochastic dispersion model are used to model the flow and dispersion of a traffic-related pollutant in a complex urban street canyon within the city of York, UK. The results are compared with measurements of mean wind, turbulence and carbon monoxide concentrations from a field experiment. For comparison, measured concentrations were normalised by the background wind speed and an estimate of the traffic emissions based on traffic flows. The normalisation was found to be inapplicable at background wind speeds below 2 m s −1 and therefore comparisons are restricted to higher wind speeds. The mean flow is found to be well predicted by the model and helps to interpret complex flow structures in the street, which contained vertical recirculation, corner vortices, channelling and convergence. The modelled worst case concentrations are generally higher than the measurements which could potentially be due in part to the reduced vertical velocities in the model and lower levels of predicted turbulence for certain background wind conditions when compared to the field data. Sensitivity analysis suggests that the modelled concentrations are fairly robust with respect to changes in grid resolution and upwind velocity profiles. The variability of the normalised measured concentrations was high, which is thought to be due to variability within the traffic emissions due to the presence of different traffic regimes, as well as large-scale turbulence within the background flow.

Footprint characteristics of scalar concentration in the convective boundary layer

Footprint characteristics for passive scalar concentration in the convective boundary layer (CBL) are investigated. A backward Lagrangian stochastic (LS) dispersion model and a large eddy simulation (LES) model are used in the investigation. Typical characteristics of the CBL and their responses to the surface heterogeneity are resolved from the LES. Then the turbulence fields are used to drive the backward LS dispersion. To remedy the spoiled description of the turbulence near the surface, Monin-Obukhov similarity is applied to the lowest LES level and the surface for the modeling of the backward LS dispersion. Simulation results show that the footprint within approximately 1 km upwind predominates in the total contribution. But influence from farther distances also exists and is even slightly greater than that from closer locations. Surface heterogeneity may change the footprint pattern to a certain degree. A comparison to three analytical models provides a validation of the footprint simulations, which shows the possible influence of along-wind turbulence and the large eddies in the CBL, as well as the surface heterogeneity.

Actuation of the Kagome Double-Layer Grid. Part 2: Effect of imperfections on the measured and predicted actuation stiffness

The actuation stiffness of a set of steel Kagome Double-Layer Grid (KDLG) structures with brazed joints is measured experimentally and compared with predictions by the finite element method. The predicted actuation stiffnesses for the perfect KDLGs much exceed the measured values, and it is argued that the low values of observed actuation stiffness are due to the presence of geometric imperfections introduced during manufacture. In order to assess the significance of geometric defects upon actuation stiffness, finite element calculations are performed on structures with a stochastic dispersion in nodal position from the perfectly periodic arrangement, and on structures with wavy bars. It is found that bar waviness has the dominant effect upon the actuation stiffness. The predicted actuation stiffness for the imperfect structures are in satisfactory agreement with the measured values assuming the same level of imperfection between theory and experiment.