Lagrangian Stochastic Model Research Articles

A stochastic approach for the simulation of particle resuspension from rough substrates: Model and numerical implementation

This paper presents a Lagrangian stochastic model to simulate colloid resuspension from rough surfaces. For that purpose, the extension of a recent proposition is discussed as well as the details of its numerical implementation. The basis of this model is a dynamical approach which reproduces explicitly the different steps involved in a three-stage scenario of particle resuspension where particles are set in motion (first stage); roll/slide along a rough surface due to varying force moments (second stage); and can be detached when a large-scale asperity is hit (third stage). The model treats separately hydrodynamic forces (drag force), adhesion forces (mainly due to interface chemical effects) and surface roughness through a two-level description (small-scale asperities and large-scale asperities) within a unified approach that combines the effects of fluid mechanics, interface chemistry and material properties. A description of the key points of the model brings forward the important role played by the number of small-scale asperities in contact with each particle; the pivot point around which particles can roll; the streamwise kinetic energy acquired as particles roll/slide along the surface; the probability to hit a large-scale asperity and the prediction of the actual detachment. Specific methods to simulate the trajectories of these stochastic processes are detailed and validated in a step-by-step manner with a specific emphasis put on the interplay between adhesion forces and particle dynamics. Finally, once each step has been separately assessed, the complete model is evaluated by comparing predictions to a realistic resuspension test-case for airborne colloids, showing that good and consistent numerical predictions are obtained with reasonable time steps.

Simulating Dispersion in the Evening-Transition Boundary Layer

We investigate dispersion in the evening-transition boundary layer using large-eddy simulation (LES). In the LES, a particle model traces pollutant paths using a combination of the resolved flow velocities and a random displacement model to represent subgrid-scale motions. The LES is forced with both a sudden switch-off of the surface heat flux and also a more gradual observed evolution. The LES shows ‘lofting’ of plumes from near-surface releases in the pre-transition convective boundary layer; it also shows the subsequent ‘trapping’ of releases in the post-transition near-surface stable boundary layer and residual layer above. Given the paucity of observations for pollution dispersion in evening transitions, the LES proves a useful reference. We then use the LES to test and improve a one-dimensional Lagrangian Stochastic Model (LSM) such as is often used in practical dispersion studies. The LSM used here includes both time-varying and skewed turbulence statistics. It is forced with the vertical velocity variance, skewness and dissipation from the LES for particle releases at various heights and times in the evening transition. The LSM plume spreads are significantly larger than those from the LES in the post-transition stable boundary-layer trapping regime. The forcing from the LES was thus insufficient to constrain the plume evolution, and inclusion of the significant stratification effects was required. In the so-called modified LSM, a correction to the vertical velocity variance was included to represent the effect of stable stratification and the consequent presence of wave-like motions. The modified LSM shows improved trapping of particles in the post-transition stable boundary layer.

Comparison of PDFs, closure schemes and turbulence parameterisations in Lagrangian stochastic models

Six one–dimensional models, based on the Ito–type stochastic equation, are presented and compared. Four of these take into account up to the fourth order moment of vertical velocity fluctuations, and two up to the third order moment. Four models make use of a bi–Gaussian probability density function (PDF) and the other two are based on a Gram–Charlier series expansion truncated to the third or fourth order. All the models were run with a parameterisation of input turbulence (i.e. w2, w3, and τ profiles). Concerning the fourth order moment w4, two different parameterisations were considered. Comparisons are made between ground–level concentrations, plume height and plume width observed in the Willis and Deardorff water tank experiments and those predicted by the different models here considered. The goal of this study was to find the models that give greater confidence in their applicability in dispersion studies and to verify the importance of considering the fourth order moment. The main conclusions are: simulation results largely depend on the turbulence parameterisation chosen; the Gram–Charlier PDF gives the best agreement with observations; some combinations of models and turbulence parameterisations perform well in simulating the shape of the ground–level concentration (g.1.c.) trend but fail in correctly simulating the form of the plume (plume height and vertical width); in the case of the Gram–Charlier PDF, the fourth order model reproduced the vertical plume width better than the third order one, whereas the two schemes yielded similar g.1.c. distributions.

Use of open-path FTIR and inverse dispersion technique to quantify gaseous nitrogen loss from an intensive vegetable production site

An open-path Fourier transform infrared (OP-FTIR) spectroscopic technique in combination with a backward Lagrangian stochastic (bLS) dispersion model (WindTrax) can be used to simultaneously measure gaseous emissions of N2O, NH3, CH4 and CO2. We assessed the capability of this technique for measuring NH3 and N2O emissions following the application of calcium nitrate (Ca(NO3)2), Nitrophoska (NPK) and chicken manure on a celery farm at Boneo, Victoria, during April and May 2013. We found that the OP-FTIR/WindTrax method was able to measure the diurnal variation in NH3 flux from the field site following application of chicken manure with measured emissions ranging from approximately 0.1–9.8 kg NH3–N ha−1 day−1. The OP-FTIR/WindTrax method also detected a diurnal variation in N2O flux of 1.5–6.2 kg N2O–N ha−1 day−1 and N2O flux increased in response to application of the Ca(NO3)2. We concluded that the OP-FTIR/WindTrax technique can quantify gaseous N loss from vegetable production systems.

Footprint methods to separate N2O emission rates from adjacent paddock areas.

Using micrometeorological techniques to measure greenhouse gas emissions from differently treated adjacent plots is a promising avenue to verify the effect of mitigation strategies at the field scale. In pursuing such an approach, it is crucial to accurately characterize the source area of the fluxes measured at each sampling point. Hence, a comprehensive footprint analysis method is required so that emission rates can be obtained for a specific field within a biochemically heterogeneous area. In this study, a footprint analysis method is developed to estimate the emission for an experiment where the flux of N2O is measured from several control and treated plots. The emission rate of an individual plot is estimated using an inverse footprint fraction approach where the footprint fractions are obtained from an analytical footprint model. A numerical solution for obtaining the background flux for such a multiplot measurement system is also provided. Results of the footprint analysis method are assessed, first, by comparing footprint fractions obtained from both an analytical footprint model and a "forward" simulation of a backward Lagrangian stochastic (bLs) model; and second, by comparing the emission rates of a control plot obtained from the footprint analysis method and from the "backward" simulation of the bLs model. It is found that the analytical footprint fractions compare well with the values obtained from the bLs model (correlation coefficient of 0.58 and 0.66 within p value <0.001). An average of 4.3% of the measured fluxes is found to be contributed by sources outside the measured area and, excluding this outside area contribution to the measured flux, footprint corrected emission rates within the defined domain are found to increase by 2.1 to 5.8% of the measured flux. Also, the proposed method of emission rate estimation is found to work well under a wide range of atmospheric stability.

Use of the open-path TDL analyzer to monitor ammonia emissions from winter wheat in the North China Plain

The backward Lagrangian stochastic (BLS) model and open-path tunable diode laser (OPTDL) analyzer were used to monitor ammonia (NH3) emissions from urea applied to winter wheat in the North China Plain. The high-temporal resolution measurements of ammonia concentrations provided an opportunity for estimating the diel patterns of ammonia emissions, as well as valuable information about the factors that influence NH3 emissions. The results showed both large diel variability and daily variability in NH3 volatilization, with NH3 emissions highest during the daytime. The diel pattern of ammonia volatilization depended mainly on the diel variation of wind speed and soil temperature, while the overall pattern of NH3 loss was strongly affected by soil moisture content, soil NH4 +-N concentration, wind speed and soil temperature. At the end of the measurement period, the cumulative NH3 loss was 12.21–16.43 kg N ha−1, calculated based on different time scale average Q BLS. Due to sensitivity of the OPTDL analyzer, the estimated total ammonia loss was still doubtful in this study.

Do small spores disperse further than large spores?

In species that disperse by airborne propagules an inverse relationship is often assumed between propagule size and dispersal distance. However, for microscopic spores the evidence for the relationship remains ambiguous. Lagrangian stochastic dispersion models that have been successful in predicting seed dispersal appear to predict similar dispersal for all spore sizes up to -40 microm diameter. However, these models have assumed that spore size affects only the downwards drift of particles due to gravitation and have largely omitted the highly size-sensitive deposition process to surfaces such as forest canopy. On the other hand, they have assumed that spores are certain to deposit when the air parcel carrying them touches the ground. Here, we supplement a Lagrangian stochastic dispersion model with a mechanistic deposition model parameterized by empirical deposition data for 1-10 microm spores. The inclusion of realistic deposition improved the ability of the model to predict empirical data on the dispersal of a wood-decay fungus (aerodynamic spore size 3.8 microm). Our model predicts that the dispersal of 1-10 microm spores is in fact highly sensitive to spore size, with 97-98% of 1 microm spores but only 12-58% of 10-microm spores dispersing beyond 2 km in the simulated range of wind and canopy conditions. Further, excluding the assumption of certain deposition at the ground greatly increased the expected dispersal distances throughout the studied spore size range. Our results suggest that by evolutionary adjustment of spore size, release height and timing of release, fungi and other organisms with microscopic spores can change the expected distribution of dispersal locations markedly. The complex interplay of wind and canopy conditions in determining deposition resulted in some counterintuitive predictions, such as that spores disperse furthest under intermediate wind, providing intriguing hypotheses to be tested empirically in future studies.

An evaluation of ambient ammonia concentrations over southern Ontario simulated with different dry deposition schemes within STILT-Chem v0.8

Abstract. A bidirectional air–surface exchange scheme for atmospheric ammonia was incorporated into the Stochastic Time-Inverted Lagrangian Transport air quality model (STILT-Chem v0.8). STILT-Chem v0.8 was then applied to simulate atmospheric ammonia concentrations at 53 measurement sites in the province of Ontario, Canada for a six-month period from 1 June to 30 November 2006. In addition to the bidirectional scheme, two unidirectional dry deposition schemes were tested. Comparisons of modeled ammonia concentrations against observations show that all three schemes can reasonably predict observations. For sites with low observed ammonia concentrations, the bidirectional scheme clearly overestimated ammonia concentrations during crop-growing season. Although all three schemes tended to underestimate ammonia concentrations after mid-October and for sites with elevated observed concentrations, mainly due to underestimated NH3 emission inventory after mid-October and/or underestimated emission potentials for those sites, the bidirectional scheme performed better because of its introduction of compensation points into the flux calculation parameterization. In addition to uncertainties in the emission inventory, the results of additional sensitivity tests suggest that uncertainties in the input values of emission potentials in the bidirectional scheme greatly affect the accuracy of modeled ammonia concentrations. The use of much larger emission potentials in the bidirectional scheme and larger anthropogenic NH3 emission after mid-October than provided in the model emissions files is needed for accurate prediction of elevated ammonia concentrations at intensive agricultural locations.

Modelling Small-Scale Drifting Snow with a Lagrangian Stochastic Model Based on Large-Eddy Simulations

Observations of drifting snow on small scales have shown that, in spite of nearly steady winds, the snow mass flux can strongly fluctuate in time and space. Most drifting snow models, however, are not able to describe drifting snow accurately over short time periods or on small spatial scales as they rely on mean flow fields and assume equilibrium saltation. In an attempt to gain understanding of the temporal and spatial variability of drifting snow on small scales, we propose to use a model combination of flow fields from large-eddy simulations (LES) and a Lagrangian stochastic model to calculate snow particle trajectories and so infer snow mass fluxes. Model results show that, if particle aerodynamic entrainment is driven by the shear stress retrieved from the LES, we can obtain a snow mass flux varying in space and time. The obtained fluctuating snow mass flux is qualitatively compared to field and wind-tunnel measurements. The comparison shows that the model results capture the intermittent behaviour of observed drifting snow mass flux yet differences between modelled turbulent structures and those likely to be found in the field complicate quantitative comparisons. Results of a model experiment show that the surface shear-stress distribution and its influence on aerodynamic entrainment appear to be key factors in explaining the intermittency of drifting snow.

Comparative study to evaluate three ground-based optical remote sensing techniques under field conditions by a gas tracer experiment

Today, ground-based optical remote sensing (ORS) has become an intensively used method for quantifying pollutant or greenhouse gas emissions from point or area sources, and for the validation of airborne or satellite remote sensing data. In this study, we present the results of a release experiment using acetylene (C2H2) as a tracer gas, where three ORS techniques were simultaneously tested for two main purposes: (1) the detection of emission sources and (2) the quantification of release rates. Therefore, passive and active open-path Fourier transform infrared spectroscopy (OP-FTIR) and open-path tunable diode laser absorption spectroscopy (TDLAS) were applied and evaluated. The concentration results of the active ORS methods are compared to those estimated by a Lagrangian stochastic atmospheric dispersion model. Our results reveal that passive OP-FTIR is a valuable tool for the rapid detection and imaging of emission sources and the spatial tracer gas distribution; while with active OP-FTIR and TDLAS, C2H2 concentrations in the sub-ppm range could be quantified that correlated well with the concentration data obtained by our modeling approach.

Measuring Greenhouse-Gas Emissions from a Synthetic Tracer Source

This paper describes a controlled tracer-release experiment in which methane was released from a synthetic source at known rates. An open-path Fourier transform infrared (OP-FTIR) spectroscopy system was used to measure line-averaged methane concentrations downwind of the source. A Lagrangian stochastic (LS) dispersion model was employed to infer emission rates from downwind gas concentrations. The main purpose of our study was to investigate the ability of our open-path FTIR system combined with the LS dispersion model to accurately measure greenhouse gas emissions. In our study, the average ratio of the estimated emissions to actual release rates QLS/Q for CH4 was about 0.86 (σQLS/Q = 0.2, n = 6) and 0.84 (σQLS/Q = 0.22, n = 3) after data fi ltering for a 15- and a 30- min period, respectively. Although there is a limited amount of data in this experiment, the results demonstrate the potential of the measurement system for accurate quantifi cation of greenhouse gas emissions.

Interpreting three-dimensional spore concentration measurements and escape fraction in a crop canopy using a coupled Eulerian–Lagrangian stochastic model

Plant disease epidemics caused by pathogenic spores are common threat to agricultural crops. Pathogenic spores are often produced and released inside plant canopies but are transported out of the canopy region by turbulent motions and advected longitudinally over long distances so as to infect other fields. Hence, the fraction of spores that “escape” the canopy sets the effective source strength that determines how fast and far plant diseases can spread. To explore the governing variables of this escape and spatial spread of spores, extensive spore release and recapture experiments were conducted in a maize field and interpreted using a coupled Eulerian–Lagrangian stochastic model (LSM). Spores were released from point sources located at three prototypical depths inside the canopy. Concentration measurements were obtained inside and above the canopy with a 3-dimensional grid of spore collectors. The experimental measurements of mean spore concentration were then used to evaluate an LSM for spore dispersion. The drift and dispersion terms of the LSM were predicted employing a conventional second-order closure model of turbulence within plant canopies thereby allowing this combined Eulerian–Lagrangian formulation to be applied to a broad set of agricultural crops. The dispersion model includes spore deposition on and rebound from canopy elements. The combination of experimental and numerical simulations was then used to quantify the fraction of spores that escape the canopy. Effects of release height and friction velocity on the escape fraction of spores were explored with the LSM.

Transported Joint Probability Density Function Modeling of Turbulent Dilute Spray Flows

A transported joint probability density function (PDF) model for turbulent spray flows is presented, where a one-point one-time statistical description of the gas-phase mixture fraction and the gas velocity is used. This approach requires the closure of the molecular mixing, which is achieved through use of the extended interaction-by-exchange-with-the-mean (IEM) model and a simplified Langevin model for the closure of the gas velocity both of which are extended through additional terms accounting for spray evaporation. These equations require the solution of the turbulent time scales and the mean pressure field through a Eulerian description. The numerical approach includes a Lagrangian Monte Carlo method for the solution of modeled joint PDF equation with a Eulerian finite-volume algorithm to determine the turbulent time scale and the mean pressure field. For the dispersed liquid phase, Lagrangian equations are used to describe the droplet heating, evaporation, and motion in the framework of a discrete droplet model. The convective droplet evaporation model is employed, and the infinite conductivity model with consideration of non-equilibrium effects based on the Langmuir-Knudsen law is used. The droplet turbulent dispersion is modeled with two different Lagrangian stochastic models. The resulting spray evolution equations are solved by a Lagrangian discrete droplet method using the point source approximation for a dilute spray. The numerical results are compared with experimental data of Gounder et al. [1], where the experimental set B of the acetone spray flows SP2 and SP6 are simulated. Comparison of numerical and experimental results includes droplet size, liquid volume flux as well as the mean and fluctuating velocities. Generally, good agreement is achieved, although the radial droplet dispersion is somewhat under-predicted by the computations. The droplet fluctuating velocities show sensitivity to the different dispersion models.

A Velocity–Dissipation Lagrangian Stochastic Model for Turbulent Dispersion in Atmospheric Boundary-Layer and Canopy Flows

An extended Lagrangian stochastic dispersion model that includes time variations of the turbulent kinetic energy dissipation rate is proposed. The instantaneous dissipation rate is described by a log-normal distribution to account for rare and intense bursts of dissipation occurring over short durations. This behaviour of the instantaneous dissipation rate is consistent with field measurements inside a pine forest and with published dissipation rate measurements in the atmospheric surface layer. The extended model is also shown to satisfy the well-mixed condition even for the highly inhomogeneous case of canopy flow. Application of this model to atmospheric boundary-layer and canopy flows reveals two types of motion that cannot be predicted by conventional dispersion models: a strong sweeping motion of particles towards the ground, and strong intermittent ejections of particles from the surface or canopy layer, which allows these particles to escape low-velocity regions to a high-velocity zone in the free air above. This ejective phenomenon increases the probability of marked fluid particles to reach far regions, creating a heavy tail in the mean concentration far from the scalar source.

A Lagrangian stochastic model for estimating the high order statistics of a fluctuating plume in the neutral boundary layer

We use a Lagrangian stochastic micromixing model to predict the concentration fluctuations of a continuous release in a neutral boundary layer. We present the computational algorithm that implements the interaction by exchange with the conditional mean model and we compare the numerical solutions with the experimental values in order to evaluate the reliability of the model. The influence of the source size on the concentration probability density function in the near and far-field is discussed and some shortcomings of the model are pointed out.

Estimates of total ash content from 2006 and 2009 explosion events at Bezymianny volcano with use of a regional atmospheric modeling system

The December 24, 2006, and December 16, 2009, strong explosion events at Bezymianny Volcano (Kamchatka Peninsula) were accompanied by extensive ash-falls in proximal and medium–distal area (<100km) downwind following the direction of prevailing upper tropospheric winds. In the present study, we apply a limited area atmospheric modeling system RAMS6.0 and a lagrangian stochastic model HYPACT1.5 to predict local airflows in a mountain area around the volcano during these events and quantify effects of atmospheric dispersal, gravitational settling, and particle aggregation on the observed ash-fall deposit patterns. It was found that the orography-induced atmospheric disturbances provided first-order influence on ash dispersal regime in the events owing to enhanced turbulence rates in a free troposphere above mountains and low-level airflows generated by mesoscale pressure perturbations. A total mass of ash from these eruptions is inverted based on grain-size sample data and model-calculated Green's function for atmospheric transport with use of a multiple regression approach. We demonstrate that in the absence of precise data on individual and collective settling rates the proposed inversion technique, which explicitly constrains fall velocity spectrum within individual sieve classes and aggregated modes, provides more reliable estimate for total erupted mass compared to procedures employing constant shape factor or prescribed settling rates within the framework of a simple linear regression model.

Orientation dynamics of small, triaxial–ellipsoidal particles in isotropic turbulence

AbstractThe orientation dynamics of small anisotropic tracer particles in turbulent flows is studied using direct numerical simulation (DNS) and results are compared with Lagrangian stochastic models. Generalizing earlier analysis for axisymmetric ellipsoidal particles (Parsa et al., Phys. Rev. Lett., vol. 109, 2012, 134501), we measure the orientation statistics and rotation rates of general, triaxial–ellipsoidal tracer particles using Lagrangian tracking in DNS of isotropic turbulence. Triaxial ellipsoids that are very long in one direction, very thin in another and of intermediate size in the third direction exhibit reduced rotation rates that are similar to those of rods in the ellipsoid’s longest direction, while exhibiting increased rotation rates that are similar to those of axisymmetric discs in the thinnest direction. DNS results differ significantly from the case when the particle orientations are assumed to be statistically independent from the velocity gradient tensor. They are also different from predictions of a Gaussian process for the velocity gradient tensor, which does not provide realistic preferred vorticity–strain-rate tensor alignments. DNS results are also compared with a stochastic model for the velocity gradient tensor based on the recent fluid deformation approximation (RFDA). Unlike the Gaussian model, the stochastic model accurately predicts the reduction in rotation rate in the longest direction of triaxial ellipsoids since this direction aligns with the flow’s vorticity, with its rotation perpendicular to the vorticity being reduced. For disc-like particles, or in directions perpendicular to the longest direction in triaxial particles, the model predicts noticeably smaller rotation rates than those observed in DNS, a behaviour that can be understood based on the probability of vorticity orientation with the most contracting strain-rate eigendirection in the model.

Observed and Modeled Global Ocean Turbulence Regimes as Deduced from Surface Trajectory Data

Abstract A large-scale tool for systematic analyses of the dispersal and turbulent properties of ocean currents and the subsequent separation of dynamical regimes according to the prevailing trajectories taxonomy in a certain area was proposed by Rupolo. In the present study, this methodology has been extended to the analysis of model trajectories obtained by analytical computations of the particle advection equation using the Lagrangian open-source software package Tracing the Water Masses of the North Atlantic and the Mediterranean (TRACMASS), and intercomparisons have been made between the surface velocity fields from three different configurations of the global Nucleus for European Modelling of the Ocean (NEMO) ocean/sea ice general circulation model. Lagrangian time scales of the observed and synthetic trajectory datasets have been calculated by means of inverse Lagrangian stochastic modeling, and the influence of the model field spatial and temporal resolution on the analyses has been investigated. In global-scale ocean modeling, compromises are frequently made in terms of grid resolution and time averaging of the output fields because high-resolution data require considerable amounts of storage space. Here, the implications of such approximations on the modeled velocity fields and, consequently, on the particle dispersion, have been assessed through validation against observed drifter tracks. This study aims, moreover, to shed some light on the relatively unknown turbulent properties of near-surface ocean dynamics and their representation in numerical models globally and in a number of key regions. These results could be of interest for other studies within the field of turbulent eddy diffusion parameterization in ocean models or ocean circulation studies involving long-term coarse-grid model experiments.

Hydrogen Sulfide Emissions from Sow Farm Lagoons across Climates Zones.

Hydrogen sulfide (HS) emissions were measured periodically over the course of 2 yr at three sow waste lagoons representing humid mesothermal (North Carolina, NC), humid microthermal (Indiana, IN), and semiarid (Oklahoma, OK) climates. Emissions were determined using a backward Lagrangian stochastic model in conjunction with line-sampled HS concentrations and measured turbulence. The median annual sow-specific (area-specific) lagoon emissions at the OK farm were approximately 1.6 g head [hd] d (5880 µg m s), whereas those at the IN and NC sow farms were 0.035 g hd d (130 µg m s), and 0.041 g hd d (260 µg m s), respectively. Hydrogen sulfide emissions generally increased with wind speed. The daily HS emissions from the OK lagoon were greatest during the first half of the year and decreased as the year progressed. Emissions were episodic at the NC and IN lagoons. The generally low emissions at the NC and IN lagoons were probably a result of significant populations of purple sulfur bacteria maintained in the humid mesothermal and humid microthermal climates. Most of the large HS emission events at the NC and IN lagoons appeared to be a result of either precipitation events or liquid pump-out events. The high emissions at the OK lagoon in a semiarid climate were largely a result of high wind speeds enhancing both lagoon and air boundary layer mixing. The climate (air temperature, winds, and precipitation) appeared to influence the HS emissions from lagoons.

A Stochastic Single-Particle Lagrangian Model for the Concentration Fluctuations in a Plume Dispersing Inside an Urban Canopy

A new approach for estimating concentration fluctuations intensity in dense built-up environments using a Lagrangian stochastic (LS) particle model is described. Following past success in modelling the dynamics of concentration variance as a diffusion-advection process, the ensemble-averaged concentration variance is represented by particles that advect and diffuse throughout the computational domain. The calculation of the concentration variance is addressed by assuming an appropriate distribution of effective variance sources for a given mean concentration field. Dissipation is treated by allowing the variance carried by every particle to decay exponentially with a locally-estimated decay time. The approach has the benefit of easily handling complex boundary conditions. It can also be easily and naturally implemented as an extension to an existing LS model, which is used for mean concentration estimations. The method differs from existing two-particle methods that demand knowledge of the structure function of the flow. It is also more computationally efficient than micro-mixing approaches that involve maintaining high population levels of particles in every grid volume. The model is compared with high frequency concentration measurements, taken as part of the JU2003 (Joint Urban 2003) experiment that was carried out in Oklahoma City. Good agreement is observed.