Comment on: “Corrections to the Mathematical Formulation of a Backwards Lagrangian Particle Dispersion Model” by Gibson and Sailor (2012: Boundary-Layer Meteorology 145, 399–406)

We discuss the results of Gibson and Sailor (Boundary-Layer Meteorol 145:399–406, 2012) who suggest several corrections to the mathematical formulation of the Lagrangian particle dispersion model of Rotach et al. (Q J R Meteorol Soc 122:367–389, 1996). While most of the suggested corrections had already been implemented in the 1990s, one suggested correction raises a valid point, but results in a violation of the well-mixed criterion. Here we improve their idea and test the impact on model results using a well-mixed test and a comparison with wind-tunnel experimental data. The new approach results in similar dispersion patterns as the original approach, while the approach suggested by Gibson and Sailor leads to erroneously reduced concentrations near the ground in convective and especially forced convective conditions.     

Footprints in Homogeneously and Heterogeneously Driven Boundary Layers Derived from a Lagrangian Stochastic Particle Model Embedded into Large-Eddy Simulation

A Lagrangian stochastic (LS) model, which is embedded into a parallelised large-eddy simulation (LES) model, is used for dispersion and footprint evaluations. For the first time an online coupling between LES and LS models is applied. The new model reproduces concentration patterns, which were obtained in prior studies, provided that subgrid-scale turbulence is included in the LS model. Comparisons with prior studies show that the model evaluates footprints successfully. Streamwise dispersion leads to footprint maxima that are situated less far upstream than previously reported. Negative flux footprints are detected in the convective boundary layer (CBL). The wide range of applicability of the model is shown by applying it under neutral and stable stratification. It is pointed out that the turning of the wind direction with height leads to a considerable dependency of source areas on height. First results of an application to a heterogeneously heated CBL are presented, which emphasize that footprints are severely affected by the inhomogeneity.     

On the Formulation of Lagrangian Stochastic Models of Scalar Dispersion Within Plant Canopies

Lagrangian stochastic models, quadratic in velocity and satisfying the well-mixed condition for two-dimensional Gaussian turbulence, are used to make predictions of scalar dispersion within a model plant canopy. The non-uniqueness associated with satisfaction of the well-mixed condition is shown to be non-trivial (i.e. different models produce different predictions for scalar dispersion). The best agreement between measured and predicted mean concentrations of scalars is shown to be obtained with a small sub-class of optimal models. This sub-class of optimal models includes Thomson's model (J. Fluid Mech. 180, 529–556, 1987), the simplest model that satisfies the well-mixed condition for Gaussian turbulence, but does not include two other models identified recently as being in optimal agreement with the measured spread of tracers in a neutral boundary layer. It is therefore demonstrated that such models are not universal, i.e. applicable to a wide range of flows without readjustment of model parameters. Predictions for scalar dispersion in the model plant canopy are also obtained using the model of Flesch and Wilson (Boundary-Layer Meteorol. 61, 349–374, 1992). It is shown that, when used with a Gaussian velocity distribution or a maximum-missing-information velocity distribution, which accounts for the measured skewness and kurtosis of velocity statistics, the agreement between predictions obtained using the model of Flesch and Wilson and measurements is as good as that obtained using Thomson's model.     

Lagrangian Stochastic Models For Turbulent Dispersion In The Atmospheric Boundary Layer

A one-particle three-dimensional stochastic Lagrangian model fortransport of particles in a horizontally-homogeneous atmosphericsurface layer with arbitrary one-point probability density functionof Eulerian velocity fluctuations is suggested. A uniquely definedLagrangian stochastic model in the class of well-mixed models isconstructed from physically plausible assumptions. These assumptionsare: (i) in the neutrally stratified horizontally homogeneous surface layer, the vertical motion is mainly controlled by eddies whose size is of order of the current height; and (ii), the streamwise drift term is independent of the crosswind velocity. Numerical simulations for neutral stratification have shown a good agreement of our model with the well-known Thomson's model, with Flesch and Wilson's model, and with experimental measurements as well. However there is a discrepancy of these results with the results obtained by Reynolds' model.     

On the moments approximation method for constructing a Lagrangian Stochastic model

The exact Eulerian velocity probability density function (pdf) of a turbulent field is generally unknown, and one normally has available only partial information in the form of low order moments. We compare two alternative Lagrangian Stochastic (LS) approaches formed from this partial information, (i) the moments approximation approach (Kaplan and Dinar, 1993); and (ii) the well-mixed model (Thomson, 1987) that corresponds to the maximum missing information pdf formed from the available information. We show that the moments approximation model does not in general satisfy the well-mixed constraint, and can give an inferior prediction of dispersion.     

Modelling of concentration fluctuations in canopy turbulence

The knowledge of the concentration probability density function (pdf) is of importance in a number of practical applications, and a Lagrangian stochastic (LS) pdf model has been developed to predict statistics and concentration pdf generated by continuous releases of non-reactive and reactive substances in canopy generated turbulence. Turbulent dispersion is modelled using a LS model including the effects of wind shear and along-wind turbulence. The dissipation of concentration fluctuations associated with turbulence and molecular diffusivity is simulated by an Interaction by Exchange with the Conditional Mean (IECM) micromixing model. A general procedure to obtain the micromixing time scale needed in the IECM model useful in non-homogeneous conditions and for single and multiple scalar sources has been developed. An efficient algorithm based on a nested grid approach with particle splitting, merging techniques and time averaging has been used, thus allowing the calculation for cases of practical interest. The model has been tested against wind-tunnel experiments of single line and multiple line releases in a canopy layer. The approach accounted for chemical reactions in a straightforward manner with no closure assumptions, but here the validation is limited to non-reacting scalars.     

Trajectory Curvature As A Selection Criterion For valid Lagrangian Stochastic Dispersion Models

Among well-mixed multi-dimensional Lagrangian stochastic (LS) dispersion models, we observe that those in poorest agreement with observations produce spiralling trajectories, with an associated reduction in dispersion. We therefore investigate statistics of increments d ' to the orientation '= arctan(W'/U') of the Lagrangian velocity-fluctuation vector – as a possible means to distinguish the better LS models within the well-mixed class. Zero-spin models, having d' = 0, are found to provide best agreement with observations. It is not clear however, whether imposition of the zero-spin property selects (in conjunction with the well-mixed condition) a unique model.     

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.     

A Homogeneous Langevin Equation Model, Part ii: Simulation of Dispersion in the Convective Boundary Layer

We present a Lagrangian stochastic model of vertical dispersion in the convective boundary layer (CBL). This model is based on a generalized Langevin equation that uses the simplifying assumption that the skewed vertical velocity probability distribution is spatially homogeneous. This approach has been shown to account for two key properties of CBL turbulence associated with large-scale coherent turbulent structures: skewed vertical velocity distributions and long velocity correlation time. A 'linear-skewed' form of the generalized Langevin equation is used, which has a linear (in velocity) deterministic acceleration and a skewed random acceleration. 'Reflection' boundary conditions for selecting a new velocity for a particle that encounters a boundary were investigated, including alternatives to the standard assumption that the magnitudes of the particle incident and reflected velocities are positively correlated. Model simulations were tested using cases for which exact, analytic statistical properties of particle velocity and position are known, i.e., well-mixed spatial and velocity distributions. Simulations of laboratory experiments of CBL dispersion show that (1) the homogeneous linear-skewed Langevin equation model (as well as an alternative 'nonlinear-Gaussian' Langevin equation model) can simulate the important aspects of dispersion in the CBL, and (2) a negatively-correlated-speed reflection boundary condition simulates the observed dispersion of material near the surface in the CBL significantly better than alternative reflection boundary conditions. The homogeneous linear-skewed Langevin equation model has the advantage that it is computationally more efficient than the homogeneous nonlinear-Gaussian Langevin equation model, and considerably more efficient than inhomogeneous Langevin equation models.     

Forward-in-time and Backward-in-time Dispersion in the Convective Boundary Layer: the Concentration Footprint

The turbulence field obtained using a large-eddy simulation model is used to simulate particle dispersion in the convective boundary layer with both forward-in-time and backward-in-time modes. A Lagrangian stochastic model is used to treat subgrid-scale turbulence. Results of forward dispersion match both laboratory experiments and previous numerical studies for different release heights in the convective boundary layer. Results obtained from backward dispersion show obvious asymmetry when directly compared to results from forward dispersion. However, a direct comparison of forward and backward dispersion has no apparent physical meaning and might be misleading. Results of backward dispersion can be interpreted as three-dimensional or generalized concentration footprints, which indicate that sources in the entire boundary layer, not only sources at the surface, may influence a concentration measurement at a point. Footprints at four source heights in the convective boundary layer corresponding to four receptors are derived using forward and backward dispersion methods. The agreement among footprints derived with forward and backward methods illustrates the equivalence between both approaches. The paper shows explicitly that Lagrangian simulations can yield identical footprints using forward and backward methods in horizontally homogeneous turbulence.     

A Lagrangian Stochastic Model for Heavy Particle Dispersion in the Atmospheric Marine Boundary Layer

The dispersion of heavy particles and pollutants is often simulated with Lagrangian stochastic (LS) models. Although these models have been employed successfully over land, the free surface at the air-sea interface complicates the implementation of traditional LS models. We present an adaptation of traditional LS models to the atmospheric marine boundary layer (MBL), where the bottom boundary is represented by a realistic wavy surface that moves and deforms. In addition, the correlation function for the turbulent flow following a particle is extended to the anisotropic, unsteady case. Our new model reproduces behaviour for Lagrangian turbulence in a stratified air flow that departs only slightly from the expected behaviour in isotropic turbulence. When solving for the trajectory of a heavy particle in the air flow, the modelled turbulent forcing on the particle also behaves remarkably well. For example, the spectrum of the turbulence at the particle location follows that of a massless particle for time scales approximately larger than the Stokes’ particle response time. We anticipate that this model will prove especially useful in the context of sea-spray dispersion and its associated momentum, sensible and latent heat, and gas fluxes between spray droplets and the atmosphere.     

A Two-Dimensional Lagrangian Stochastic Dispersion Model for Convective Boundary Layers with Wind Shear

Rotach, Gryning and Tassone constructed a two-dimensional Lagrangian stochastic model to describe the dispersion of passive tracers in turbulent boundary layers with stabilities ranging from ideally-neutral (w_* = 0) to fully-convective (u_* = 0). They found that the value of the Kolmogorov constant, C₀, as determined by optimizing model agreement with the measured spread of passive tracers, was dependent upon stability. Here, it is shown that the non-uniqueness, associated with satisfaction of the well-mixed condition, can be exploited to construct an alternative version of the model of Rotach et al. for which C₀ = 3 is universally applicable over the entire range of stabilities under consideration. This alternative model is shown to be in very close agreement with predictions, obtained in large-eddy simulations, for the dispersion of passive tracers in turbulent boundary layers with stabilities ranging from ideally-neutral to fully-convective.     

On the forward-in-time upstream advection scheme for non-uniform and time-dependent flow

Summary The classical forward-in-time upstream advection scheme for uniform flow field has been extended to include non-uniform and time-dependent advective flow. This generalised scheme is described in one dimension for an advective flow which varies both in time and in space. The classical upstream advection scheme is only first-order accurate both in time and in space if the advective flow is not uniform. Higherorder accuracy in both time and space, however, can be easily obtained in the generalised scheme.This generalised scheme with third-order accuracy is applied to the one-dimensional inviscid Burgers equation (socalled self-advection problem), two-dimensional steady flow, and to a time-split shallow water equation model. The results are compared with those obtained from the Takacs' (1985) scheme and from a standard third-order semi-Lagrangian scheme, and also with those obtained from the fourth-order Lax-Wendroff scheme of Crowley (1968) in the time-split shallow water equation model. It is shown that the generalised scheme performs as well as, but is more efficient than, the standard semi-Lagrangian scheme with same order. It is much more accurate than the Takacs' scheme which has large dissipation errors, especially for the flow with strong deformation. In contrast, the generalised scheme has very weak dissipation and has much better dispersion and shapeconserving properties. Although the fourth-order Lax-Wendroff scheme has higher accuracy and can give more accurate numerical solutions for uniform advective flow or solid rotational flow (Crowley, 1968), it is inferior to the generalised third-order scheme for non-uniform flow with strong deformation or large spatial gradients. This generalised scheme, therefore, has considerable application potential in different numerical models, especially for the models using time-split algorithms.With 8 Figures     

A bulk model of the well-mixed boundary layer

A bulk boundary-layer model is developed to predict surface fluxes and conditions in the well-mixed layer between the surface and the lower troposphere. The model includes the effects of all the dominant processes, including advection, in a dry boundary layer. The numerical model is compared with theoretical predictions for the growth of an internal boundary layer, and it is used to simulate the generation of a sea breeze by the diurnal cycle of radiative heating.     

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.     

Turbulent dispersion of gaseous pollutants: Numerical study of ground level concentration as a function of Eddy diffusivity parametrization on the atmospheric boundary layer

Theoretical and Applied Climatology - A numerical model of the dispersion of gaseous atmospheric pollutants based on the K-theory has been proposed, and the numerical procedure has been validated...     

Non-Gaussian Lagrangian Stochastic Model for Wind Field Simulation in the Surface Layer

Wind field simulation in the surface layer is often used to manage natural resources in terms of air quality,gene flow(through pollen drift),and plant disease transmission(spore dispersion).Although Lagrangian stochastic(LS)models describe stochastic wind behaviors,such models assume that wind velocities follow Gaussian distributions.However,measured surface-layer wind velocities show a strong skewness and kurtosis.This paper presents an improved model,a non-Gaussian LS model,which incorporates controllable non-Gaussian random variables to simulate the targeted non-Gaussian velocity distribution with more accurate skewness and kurtosis.Wind velocity statistics generated by the non-Gaussian model are evaluated by using the field data from the Cooperative Atmospheric Surface Exchange Study,October 1999 experimental dataset and comparing the data with statistics from the original Gaussian model.Results show that the non-Gaussian model improves the wind trajectory simulation by stably producing precise skewness and kurtosis in simulated wind velocities without sacrificing other features of the traditional Gaussian LS model,such as the accuracy in the mean and variance of simulated velocities.This improvement also leads to better accuracy in friction velocity(i.e.,a coupling of three-dimensional velocities).The model can also accommodate various non-Gaussian wind fields and a wide range of skewness–kurtosis combinations.Moreover,improved skewness and kurtosis in the simulated velocity will result in a significantly different dispersion for wind/particle simulations.Thus,the non-Gaussian model is worth applying to wind field simulation in the surface layer.     

Development of a Single-Column Model in RegCM4 and Its Preliminary Application for Evaluating PBL Schemes in Simulating the Dry Convection Boundary Layer

A single-column model (SCM) is developed in the regional climate model RegCM4. The evolution of a dry convection boundary layer (DCBL) is used to evaluate this SCM. Moreover, four planetary boundary layer (PBL) schemes, namely the Holtslag-Boville scheme (HB), Yonsei University scheme (YSU), and two University of Washington schemes (UW01, Grenier-Bretherton-McCaa scheme and UW09, Bretherton-Park scheme), are compared by using the SCM approach. A large-eddy simulation (LES) of the DCBL is performed as a benchmark to examine how well a PBL parameterization scheme reproduces the LES results, and several diagnostic outputs are compared to evaluate the schemes. The results show that the SCM is proper constructed. In general, with the DCBL case, the YSU scheme performs best for reproducing the LES results, which include well-mixed features and vertical sensible heat fluxes; the simulated wind speed, turbulent kinetic energy, entrainment flux, and height of the entrainment zone are all underestimated in the UW09; the UW01 has all those biases of the UW09 but larger, and the simulated potential temperature is not well mixed; the HB is the least skillful scheme, by which the PBL height, entrainment flux, height of the entrainment zone, and the vertical gradients within the mixed layer are all overestimated, and a inversion layer near the top of the surface layer is wrongly simulated.Although more cases and further testing are required, these simulations show encouraging results towards the use of this SCM framework for evaluating the simulated physical processes by the RegCM4.