• Brusasca, G., G. Tinarelli, and D. Anfossi, 1992: Particle model simulation of diffusion in low wind speed stable conditions. Atmos. Environ.,26A, 707–723.

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  • Chiba, O., 1978: Stability dependence of the vertical wind velocity skewness in the atmospheric surface layer. J. Meteor. Soc. Japan,56, 140–142.

  • Cirillo, M. C., and A. A. Poli, 1992: An intercomparison of semiempirical diffusion models under low wind speed, stable conditions. Atmos. Environ.,26A, 765–774.

  • De Haan, P., 1999: On the use of density kernels for concentration estimations within particle and puff dispersion models. Atmos. Environ.,33, 2007–2021.

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  • Franzese, P., A. K. Luhar, and M. S. Borgas, 1999: An efficient Lagrangian stochastic model of vertical dispersion in the convective boundary layer. Atmos. Environ.,33, 2337–2345.

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  • Sharan, M., and A. K. Yadav, 1998: Simulation of experiments under light wind, stable conditions by a variable K-theory model. Atmos. Environ.,32, 3481–3492.

  • Stull, R. B., 1989: An Introduction to Boundary Layer Meteorology. Kluwer Academic, 666 pp.

  • Thomson, D. J., 1987: Criteria for the selection of stochastic models of particle trajectories in turbulent flows. J. Fluid Mech.,180, 529–556.

  • Wang, L.-P., and D. E. Stock, 1992: Stochastic trajectory models for turbulent diffusion: Monte-Carlo process versus Markov chains. Atmos. Environ.,26A, 1599–1607.

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  • View in gallery

    Eulerian autocorrelation functions averaged over all stability classes with u < 1 m s−1 for the u, υ, and w components of the wind vector. The standard deviation is indicated with the shaded area

  • View in gallery

    Observed and calculated σw with (3)

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    Ground-level concentrations (μg m−3) obtained for different values of the intercorrelation parameter ρu,υ (top: ρu,υ = 0.9; bottom:ρu,υ = −0.5)

  • View in gallery

    Plot of σθ as a function of wind speed as proposed by Leahey et al. (1994)

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    Vertical concentration profile obtained for the well-mixed test of the model after an integration time of 1 h. The concentration is normalized by the initial concentration

  • View in gallery

    Observed (solid line) and calculated (dotted line) normalized concentrations on the 100-, 200-, and 400-m arcs for the INEL experiment (Sagendorf and Dickson 1974) for different test numbers

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A New Method to Estimate Diffusion in Stable, Low-Wind Conditions

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  • 1 Institute for Internal Combustion Engines and Thermodynamics, Graz University of Technology, Graz, Austria
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Abstract

Sonic anemometer observations were made 10 m above ground level for a period of 1 yr. From these data, Eulerian autocorrelation functions were computed for the horizontal and vertical wind velocity fluctuations for low wind speeds. Although the autocorrelation function for the vertical velocity component exhibited the well-known exponential form, the function for the horizontal components of the wind vector showed a negative loop for all stability classes at low wind speeds. This result might be an effect of low-frequency meandering of the flow. Observations of the standard deviations of the vertical wind component confirmed the proportionality with the friction velocity, though with a slightly lower constant of proportionality than has been found by other authors. A Lagrangian dispersion model (LDM) with random time steps and a negative intercorrelation parameter ρu,υ for the horizontal wind components was used to take the first of the above-mentioned findings into account. In a simple test case, it could be shown that using a negative tail in the autocorrelation function for the horizontal wind fluctuations in an LDM results in larger plume spreads as if the usual exponential law were used. This model characteristic is in agreement with enhanced dispersion in low-wind situations as found by different authors earlier. Because the model reduces to the Langevin equation for ρu,υ = 0.9, it has the advantage that it can be used for all wind speeds by simply adjusting the intercorrelation parameter. Last, the model was tested against field experiment data gathered by the Idaho National Engineering Laboratory during stable, low-wind conditions. The results with the new method for these experiments are very promising in comparison with methods used by other authors earlier.

Corresponding author address: Dietmar Oettl, Institute for Internal Combustion Engines and Thermodynamics, Graz University of Technology, Inffeldgasse 25, A-8010 Graz, Austria.

oettl@vkmb.tu-graz.ac.at

Abstract

Sonic anemometer observations were made 10 m above ground level for a period of 1 yr. From these data, Eulerian autocorrelation functions were computed for the horizontal and vertical wind velocity fluctuations for low wind speeds. Although the autocorrelation function for the vertical velocity component exhibited the well-known exponential form, the function for the horizontal components of the wind vector showed a negative loop for all stability classes at low wind speeds. This result might be an effect of low-frequency meandering of the flow. Observations of the standard deviations of the vertical wind component confirmed the proportionality with the friction velocity, though with a slightly lower constant of proportionality than has been found by other authors. A Lagrangian dispersion model (LDM) with random time steps and a negative intercorrelation parameter ρu,υ for the horizontal wind components was used to take the first of the above-mentioned findings into account. In a simple test case, it could be shown that using a negative tail in the autocorrelation function for the horizontal wind fluctuations in an LDM results in larger plume spreads as if the usual exponential law were used. This model characteristic is in agreement with enhanced dispersion in low-wind situations as found by different authors earlier. Because the model reduces to the Langevin equation for ρu,υ = 0.9, it has the advantage that it can be used for all wind speeds by simply adjusting the intercorrelation parameter. Last, the model was tested against field experiment data gathered by the Idaho National Engineering Laboratory during stable, low-wind conditions. The results with the new method for these experiments are very promising in comparison with methods used by other authors earlier.

Corresponding author address: Dietmar Oettl, Institute for Internal Combustion Engines and Thermodynamics, Graz University of Technology, Inffeldgasse 25, A-8010 Graz, Austria.

oettl@vkmb.tu-graz.ac.at

Introduction

Low-frequency meandering of wind occurs whenever the wind speed is below about 2 m s−1. Because surface-layer similarity fails to describe effects when winds are calm (e.g., Stull 1989), it remains a challenging task to develop dispersion models that take meandering flows into consideration. Previous observations and studies have provided evidence for enhanced dispersion in situations with very low velocities (Leahey et al. 1994; Leung and Liu 1996; Sagendorf and Dickson 1974; Kristensen et al. 1981; Hanna 1983; Etling 1990).

Several models have been developed to describe dispersion processes under the conditions described above. Sharan and Yadav (1998) used a model including streamwise diffusion and variable eddy diffusivities. The eddy diffusivities were specified as linear functions of downwind distance. They tested their model against the dispersion data collected by the Idaho National Engineering Laboratory (INEL), which are described in more detail in section 4. They compared the performance of the model using different parameterizations for the eddy diffusivities and varying time intervals of 2 and 60 min. It was pointed out that using a time interval of 2 min and a dependency of the eddy diffusivities on measured standard deviations of wind direction fluctuations gave the best results. The model of Cirillo and Poli (1992) gave almost identical results when compared with the ones of the model of Sharan and Yadav (1998) for the INEL dataset. Hence, only the latter model was considered in the comparison given in section 4. Sagendorf and Dickson (1974) used a Gaussian model and also divided each computation period into 2-min time intervals and summed the results to determine the total concentration. In terms of the physical processes involved, such a model is questionable whenever the chosen time intervals are below the ratio of the maximum travel distance to the average wind speed (xmax/u). The limitations of these kind of models result from the built-in assumptions of a homogenous wind field and restrictions concerning the shape of the source.

Brusasca et al. (1992) used a Lagrangian particle model to take meandering of the flow into account. The idea was based on Gifford’s (1960) fluctuating plume model where the part of the crosswind standard deviation of the wind speed συ due to meandering (σMυ) is given by
σMυσυ2σTυ21/2
where σTυ is the turbulent part of the horizontal velocity fluctuations. The latter was calculated using surface similarity theory. The time period was split into 3-min time intervals. Then randomly picked wind vectors are varied until the hourly values of mean wind speed u, mean wind direction θ, and σMυ were close to observational data. The model was also applied to dispersion data collected by INEL and showed good agreement for two test runs. As pointed out by Brusasca et al. (1992), their model performs worse when the intensity of turbulence i = σu,υ/u (e.g., Pasquill and Smith 1983) is close to 1.0. In such cases, they proposed a method in which the randomly picked wind vectors are ordered according to the size of the wind angle. However, this approach might be a drawback in that statistical stability for different sets of wind vectors is no longer guaranteed.

In this paper, a new method is presented to estimate dispersion in stable, low-wind conditions based on observations made with a sonic anemometer 10 m above ground level for a period of 1 yr. In the first section, we briefly describe the generation of data and some major results. The derivation of a particle model is dealt with in the next section, and we outline some model results for the INEL dispersion experiment (Sagendorf and Dickson 1974) in the final section.

Turbulence data

The observation site was situated in a suburban area of the city of Graz (Austria) exhibiting mixed land use and low building density. The region is characterized by frequent calm periods (u < 0.8 m s−1) prevailing some 70% of the year. Measurement height was set at 10 m above ground level to allow comparison with similar studies in the literature. A sonic anemometer (METEK, GmbH, USA-1 H) was used, allowing investigations down to very low wind speeds of about 0.1 m s−1. Sampling frequency and storage were one sample per second. For the analysis, time intervals of 1 h were chosen, and only datasets with no missing data were used. After a rotation of the coordinate system so that average crosswind velocity υ = 0 and average vertical velocity w = 0 the data were detrended (e.g., Shampine et al. 1974). Eulerian autocorrelation functions R(τ) were obtained from calculated power spectra using the relationship
i1520-0450-40-2-259-e2
where f(ν) is the spectral energy density for a component of the wind vector divided by the variance of the wind fluctuations, and ν denotes the frequency.

Eulerian autocorrelation functions Ru(τ), Rυ(τ), and Rw(τ) averaged over all stabilities (sample n = 3613) for wind speeds less than 1 m s−1 are displayed in Fig. 1. The range of stability encountered in Fig. 1 is given in Table 1. Although Rw(τ) exhibits an exponential form, Ru(τ) and Rυ(τ) show a negative loop with a maximum at around 600 s. Because the standard deviation (shaded area) was found to be almost the same if only stable cases were taken, the negative tail seems to be unaffected by stability. The negative values found in the autocorrelation functions might be an effect of low-frequency wind meandering. A possible explanation could be that the persistence of large eddies with a vertical axis in low-wind situations is enhanced because of reduced microscale turbulent friction (Etling 1990). Thus, they may cause a negative tail in the Eulerian autocorrelation functions. Clearly, the assumption of an exponential behavior of the horizontal autocorrelation function in dispersion modeling is not a good approach for low-wind situations.

Regarding the standard deviation of the vertical velocity fluctuation σw, our observations support a relationship of the form
σwuzL−1
where u∗ denotes the friction velocity, and L is the Monin–Obukhov length. A scatterplot, using (3), of observed versus calculated σw values (for zL−1 > 0) is shown in Fig. 2. A total of 1782 observations were available for comparison purposes. The solid line is the one-to-one relationship between observed and calculated values. A relatively high coefficient of determination R2 = 0.95 and little scatter along the solid line indicate a fairly good agreement between recorded and computed data. Comparison with the expression
i1520-0450-40-2-259-e4
given by Hanna (1982), with h being the height of the stable boundary layer, showed a slight underestimation of σw for values higher than 0.2 m s−1 (not shown). Apparently, σw is independent of zL−1 in neutral-to-stable conditions. This result was also found by Panofsky and Dutton (1984), who suggested an expression for σw of the form
σwuzL−1
For all the computations presented in this paper, (3) was used for the standard deviation of the vertical wind.

Model description

Horizontal dispersion

Usually Lagrangian particle models make use of an exponential form of the autocorrelation function. To match the form of the autocorrelation function for the horizontal wind component with a negative tail discussed above, an algorithm is introduced based on the paper of Wang and Stock (1992). Therefore, the new positions for a particle at time t + Δth are given by
i1520-0450-40-2-259-e6
u and υ are the mean components of the wind vector in the x and y directions, u′ and υ′ are the velocity fluctuations, σu and συ are the standard deviations of the velocity components, χ are random numbers with zero mean and a standard deviation equal to 1, ρu and ρυ are the intercorrelation parameters, and Δth is a random time step for which the horizontal velocity fluctuations remain constant.
Once the Lagrangian timescales TLu and T are known, the mean time interval Δth can be calculated with (10) if, as in this study, a time step probability density function (PDF) with uniform distribution (11) is used (Wang and Stock 1992):
i1520-0450-40-2-259-e10
For ρu,υ < 0, a negative loop is caused in Ru,υ(τ) as was shown by Wang and Stock (1992), which is desirable for an application of the model for meandering flows. Further, they pointed out that, for ρu,υ ≥ 0.9, (8) and (9) reduce to the Langevin equation, which is often used in dispersion models. The great advantage of the model of Wang and Stock (1992) is that it can be used for meandering flows if ρu,υ < 0.9 is chosen and for small-scale turbulence diffusion processes if ρu,υ ≥ 0.9 is taken. In Fig. 3, the effect of different values for ρu,υ on the plume spread is shown. The results were obtained for a point source at a height of 1.5 m above ground level (indicated by a cross in Fig. 3), a wind speed of 0.5 m s−1 at a height of 2 m above ground level, and stable stratification. A total of one million particles were traced, and concentrations were obtained by counting them in cells with dimensions 5 m × 5 m and 0.2 m in height. The ground level concentration at a height of 0.5 m for ρu,υ = 0.9 (top panel) and the same but for ρu,υ = −0.5 (bottom panel) are shown. It can be seen in Fig. 3 that the effect of meandering flows, namely, the enhanced plume spread, can be simulated by setting ρu,υ < 0.9.
As already discussed in section 1, meandering of the flow depends mainly on wind speed. Therefore, ρu,υ should also be a function of wind speed. Because ρu,υ is a dimensionless variable, the standard deviation of the wind direction fluctuations σθ was taken for parameterization instead of the wind speed. Observations made by Leahey et al. (1994) support a relationship of the form
i1520-0450-40-2-259-e12
with σθ in degrees. In Fig. 4, the relationship according to Leahey et al. (1994) between σθ and wind speed is drawn. There is almost no change in σθ for wind speeds higher than 3 m s−1; for wind speeds below this value, a substantial increase of σθ is the case. If σθ is taken as a measure for meandering of a flow, it can be deduced that below a value of 3 m s−1 meandering is very likely to occur. This result would mean that a proper choice of ρu,υ would be a value of 0.9 for wind speeds higher than 3 m s−1, for which the autocorrelation function becomes exponential. On the other hand, for very low wind speeds, ρu,υ should be limited by a value of about −0.9, because (10) becomes physically unrealistic for ρu,υ < −1.0. Hence, a proper expression for ρu,υ as a function of σθ should give a value of ρu,υ = 0.9 for σθ < 7.0° and a value of ρu,υ = −0.9 for the largest observed σθ. A possible shape of a function for ρu,υ satisfying these criteria could take the form
i1520-0450-40-2-259-e13
where A and B are constants to be determined empirically, and σθ here is in units of radians. For the Lagrangian timescales TLu,υ in (10), formulations suggested by Hanna (1982) are used in the model:
i1520-0450-40-2-259-e14
The height of the stable boundary layer h in (14) and (15) can be approximated with
i1520-0450-40-2-259-e16
where f = 0.0001 s−1 is the Coriolis parameter, and the turbulent part of the observed standard deviations are given by
i1520-0450-40-2-259-e17
Equations (14)–(18) were derived for small-scale turbulence only, so some comments are necessary as to why these relations are used in this paper. To the authors’ knowledge there are no existing expressions for the Lagrangian timescales that are valid for the conditions considered in this study. Further, (16) was used successfully by Brusasca et al. (1992) for low-wind, stable conditions. Using (14) and (15) has the advantage that only the intercorrelation parameter ρu,υ needs to be varied if the model is used for small-scale turbulence and meandering flows.

Vertical dispersion

In order that the model fulfills the well-mixed condition (Thomson 1987), an algorithm suggested by Franzese et al. (1999) was used. The new vertical coordinate for a particle is obtained by the following equations:
wttυaw,ztυC0z1/2dWwt
and
zttυztwttυtυ
where w is the vertical velocity of a particle, C0 is a universal constant set at a value of 2.1 (see, e.g., Wilson and Sawford 1996), ε(z) is the ensemble-average rate of dissipation of turbulent kinetic energy, dW are the increments of a Wiener process with zero mean and variance Δtυ, and the time step Δtυ is given by
i1520-0450-40-2-259-e21
The deterministic acceleration term a(w, z) is assumed to be a function of the vertical velocity:
aw, zαzw2βzwγz
here α(z), β(z), and γ(z) are unknown parameters, which are determined from the Fokker–Planck equation:
i1520-0450-40-2-259-e23
where PE(w, z) is the Eulerian PDF of the vertical turbulent velocity at a given height z.
By assuming a quadratic functional form for the acceleration, the model of Franzese et al. (1999) does not need any information about the form of PE(w, z) but only requires the first four Eulerian moments of the vertical velocity. The coefficients in (22) can be expressed as
i1520-0450-40-2-259-e24
In (24)–(26), wi (i = 1, 2, 3, 4) denote the highest Eulerian moments of the vertical velocity. The first moment is the mean of the vertical velocity, which is set equal to zero, and the second moment is the variance and was derived from (3). The third moment was taken according to Chiba (1978) as
i1520-0450-40-2-259-e27
where k is the von Kármán constant. The fourth moment was set as
i1520-0450-40-2-259-e28
which is the Gaussian assumption.
The ensemble-average rate of dissipation of turbulent kinetic energy ε(z) was taken according to Stull (1989),
i1520-0450-40-2-259-e29
which is valid for the stable surface layer and is independent of height.

A simple test case was set up to test whether the model does satisfy the well-mixed condition. For that purpose, a total of one million particles were equally distributed in a box with dimensions of 50 m × 50 m in the horizontal directions and 20 m in height. A logarithmic wind profile with a wind speed of 0.5 m s−1 at a height of 2 m and a stable stratification were taken for the simulation. The vertical profile of the concentrations in the center of the box was evaluated after 1 h. At the upper and lower boundary of the box, perfect reflection of the particles was implemented. At the lateral boundaries of the box, cyclic boundary conditions were used, that is, a particle leaving the box at one side enters it at the opposite side again. The result of the test case is shown in Fig. 5. The deviations from the initial concentration within the box were found to be less than 5%, which is, in our opinion, satisfactory for usual applications of the model.

Application of the model

To test the accuracy of the model, dispersion data from INEL (Sagendorf and Dickson 1974) derived under stable conditions with light winds (<2 m s−1) over flat, even terrain, were taken for comparison purposes. As a tracer gas, sulfur hexafluoride, released 1.5 m above ground at a rate of 0.032 mg s−1, was used. Receptors were placed along three arcs with radii r = 100, 200, and 400 m, respectively, and the source located in the center. On each arc, 60 receptors were equally distributed. Samples were taken 0.76 m above ground. Of the total of 14 tests conducted, results from 10 of them have been used in this study for comparison purposes.

Wind and temperature data were recorded at a 61-m tower at 2, 4, 8, 16, 32, and 61 m above the ground. Because the additional surface-layer parameters u∗ and L were needed to run the model, an approximation was made similar to that used in Brusasca et al. (1992), where both values were determined through a numerical best fit of (30), given by Businger et al. (1971), to the measured vertical wind profile at the levels at 2, 4, 8, and 16 m:
i1520-0450-40-2-259-e30
The roughness length z0 was pinned down at 0.005 m by Brusasca et al. (1992) and Sharan and Yadav (1998). As earlier pointed out by both these authors, this method has a limitation in that (30) is based on Monin–Obukhov similarity theory, which can be questionable in low-wind situations, because it was not derived for such conditions.
Values of σu and συ in (8) and (9) were set equal to each other and were calculated using
σu,υσθu
which is a good approximation for σu,υ/u < 1.2 (Leung and Liu 1996). The meteorological data used for the simulations are summarized in Table 2. Concentrations were computed by counting the particles in each grid volume with a spacing of Δx = Δy = 5 m and Δz = 0.2 m every time step and then conducting a final averaging over all time steps. The influence of the grid volume size on the computed concentrations is kept low by using 106 particles in each of the simulations [see, e.g., de Haan (1999) for some more discussion on that topic]. For the parameterization of the intercorrelation parameter ρu,υ, (12) and (13) were used instead of observed σθ values during the INEL experiment. In the simulations, the values for A and B in (13) were set at 0.5 and 0.2, respectively. Particles were perfectly reflected at the levels z = z0 and z = h.
To compare the results with those obtained by the segmented plume method of Sagendorf and Dickson (1974) and Brusasca et al. (1992) and the segmented plume method with σθ parameterization of Sharan and Yadav (1998), several statistical measures were computed: the relative mean bias
i1520-0450-40-2-259-e32
the top-10 relative bias
cc10co10co10
the fractional bias
cocccocc
the normalized mean-square error
i1520-0450-40-2-259-e35
the correlation coefficient
i1520-0450-40-2-259-e36
and FAC2, the fraction of data for which
i1520-0450-40-2-259-e37
where cc is the computed concentration, co is the observed concentration, and cc10 and co10 are the mean of the 10 highest calculated and observed concentrations, respectively (Hanna 1988). In regulatory use it is more important to get a correct estimate of the peak concentration rather than the very exact location of occurrence. Therefore, the ratio of calculated to observed peak values has also been worked out and compared with results obtained by the segmented plume method reported in Sagendorf and Dickson (1974) and by the model of Sharan and Yadav (1998) for each test case.

For the peak values (Table 3), the method presented here performed considerably better in most of the cases than did the segmented plume method of Sagendorf and Dickson (1974) and, in some cases, performed slightly better than the method suggested by Brusasca et al. (1992). On the average, the model of Sagendorf and Dickson (1974) gave slightly better results for the ratio of computed peak to observed peak concentrations than the new method presented here (Table 4) did. For RMB and RMB-TT (Table 5), the new method performed somewhat worse than did the model of Brusasca et al. (1992). In Table 6, NMSE, R, FAC2, and FB are summarized. In comparison with the other models, the new method presented here performed best, as indicated by all the statistical measures except FB.

In Fig. 6, the normalized concentrations along the three arcs are illustrated to provide a visual impression. The normalization was carried out according to the procedure described by Sagendorf and Dickson (1974) using
cnχuQ−3
Here χ denotes the concentration (mg m−3) and Q is the release rate (mg s−1).

Conclusions

As observations with a sonic anemometer have shown, the Eulerian autocorrelation function shows a negative loop when winds are below about 2 m s−1. This result is important for modelers who want to estimate diffusion in low-wind conditions. The negative loop might be caused by large eddies whose lifetime is increased because of low microscale turbulent friction. In this study, the shape of the Lagrangian autocorrelation function was assumed to be similar to the Eulerian autocorrelation function above. For Lagrangian particle models, this effect can be taken into consideration by using a negative intercorrelation parameter ρu,υ and random time steps for the velocity fluctuations. The physical interpretation of the PDF for the time-step function is not clear at this stage of the study. Instead a relationship between the intercorrelation parameter ρu,υ and the wind direction fluctuations σθ, which can be expressed as a function of wind speed, was derived.

The new method was tested against field data taken by INEL and was compared with results obtained by other models that were evaluated with these data earlier. Good agreement was found between the observed concentrations and those computed with the new method. Although the new method performed slightly better than the other models in most cases, one has to be careful in saying one model is superior to the others, because of the limited test cases available for comparison purposes. Further, it has to be said that the input parameters u∗ and L were not available for the INEL experiment but could only be approximated roughly. Hence, for future research it is planned to perform tracer tests similar to that of INEL but including measurements of boundary layer quantities such as u∗ and L.

The model of Brusasca et al. (1992) and the new method presented in this paper have the advantage that they can be applied to all kinds of sources, for example, line sources and inhomogeneous wind fields, in contrast to the models of Sagendorf and Dickson (1974) and Sharan and Yadav (1998). The model of Brusasca et al. (1992) needs an ad hoc algorithm for very low wind speeds (<0.5 m s−1), but the new method presented should be applicable over all ranges of wind speed, because it reduces to standard methods for ρu,υ = 0.9. The advantage of the new model over methods that use time splitting is that it does not need preprocessing of data [unlike Brusasca et al. (1992)] or increased storage of 2-min quantities.

Acknowledgments

We thank J. F. Sagendorf for providing the INEL data upon which this study was based. Many thanks also to D. Anfossi for the FORTRAN codes provided. Last, we thank the reviewers for their valuable comments, which helped much in improving this paper. The study was partly funded by the Austrian research fund project 12168-TEC.

REFERENCES

  • Brusasca, G., G. Tinarelli, and D. Anfossi, 1992: Particle model simulation of diffusion in low wind speed stable conditions. Atmos. Environ.,26A, 707–723.

  • Businger, J. A., J. C. Wyngaard, Y. Izumi, and E. F. Bradley, 1971: Flux–profile relationships in the atmospheric surface layer. J. Atmos. Sci.,28, 181–189.

  • Chiba, O., 1978: Stability dependence of the vertical wind velocity skewness in the atmospheric surface layer. J. Meteor. Soc. Japan,56, 140–142.

  • Cirillo, M. C., and A. A. Poli, 1992: An intercomparison of semiempirical diffusion models under low wind speed, stable conditions. Atmos. Environ.,26A, 765–774.

  • De Haan, P., 1999: On the use of density kernels for concentration estimations within particle and puff dispersion models. Atmos. Environ.,33, 2007–2021.

  • Etling, D., 1990: On plume meandering under stable stratification. Atmos. Environ.,24A, 1979–1985.

  • Franzese, P., A. K. Luhar, and M. S. Borgas, 1999: An efficient Lagrangian stochastic model of vertical dispersion in the convective boundary layer. Atmos. Environ.,33, 2337–2345.

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Fig. 1.
Fig. 1.

Eulerian autocorrelation functions averaged over all stability classes with u < 1 m s−1 for the u, υ, and w components of the wind vector. The standard deviation is indicated with the shaded area

Citation: Journal of Applied Meteorology 40, 2; 10.1175/1520-0450(2001)040<0259:ANMTED>2.0.CO;2

Fig. 2.
Fig. 2.

Observed and calculated σw with (3)

Citation: Journal of Applied Meteorology 40, 2; 10.1175/1520-0450(2001)040<0259:ANMTED>2.0.CO;2

Fig. 3.
Fig. 3.

Ground-level concentrations (μg m−3) obtained for different values of the intercorrelation parameter ρu,υ (top: ρu,υ = 0.9; bottom:ρu,υ = −0.5)

Citation: Journal of Applied Meteorology 40, 2; 10.1175/1520-0450(2001)040<0259:ANMTED>2.0.CO;2

Fig. 4.
Fig. 4.

Plot of σθ as a function of wind speed as proposed by Leahey et al. (1994)

Citation: Journal of Applied Meteorology 40, 2; 10.1175/1520-0450(2001)040<0259:ANMTED>2.0.CO;2

Fig. 5.
Fig. 5.

Vertical concentration profile obtained for the well-mixed test of the model after an integration time of 1 h. The concentration is normalized by the initial concentration

Citation: Journal of Applied Meteorology 40, 2; 10.1175/1520-0450(2001)040<0259:ANMTED>2.0.CO;2

Fig. 6.
Fig. 6.

Observed (solid line) and calculated (dotted line) normalized concentrations on the 100-, 200-, and 400-m arcs for the INEL experiment (Sagendorf and Dickson 1974) for different test numbers

Citation: Journal of Applied Meteorology 40, 2; 10.1175/1520-0450(2001)040<0259:ANMTED>2.0.CO;2

Fig. 6.
Fig. 6.

(Continued)

Citation: Journal of Applied Meteorology 40, 2; 10.1175/1520-0450(2001)040<0259:ANMTED>2.0.CO;2

Table 1.

Histogram of the range of stability in terms of z/L encountered in Fig. 1

Table 1.
Table 2.

Reported values of mean wind speed (ū), mean wind direction (θ), and standard deviation of wind direction (σθ) for the INEL experiment (Sagendorf and Dickson 1974)

Table 2.
Table 3.

Observed and calculated (Nm = new method) concentrations (μg m−3) for the three arcs of the test runs. Here S.D. refers to the segmented plume method by Sagendorf and Dickson (1974), and B.T.A. stands for the results obtained by Brusasca et al. (1992)

Table 3.
Table 4.

Ratios of computed peak to observed peak concentrations on each arc averaged over all test runs. Here S.D. refers to the segmented plume method by Sagendorf and Dickson (1974), S.Y. stands for the results obtained by Sharan and Yadav (1998), and Nm stands for the new method presented here

Table 4.
Table 5.

Relative mean bias RMB and top-10 relative bias RMB-TT. Here B.T.A. stands for the results obtained by Brusasca et al. (1992), and Nm stands for the new method presented here

Table 5.
Table 6.

Normalized mean-square error NMSE, correlation coefficient R, percentage of predicted concentration within a factor of 2 FAC2, and fractional bias FB evaluated for peak concentrations (n = 30). Here S.D. refers to the segmented plume method by Sagendorf and Dickson (1974), S.Y. stands for the results obtained by Sharan and Yadav (1998), and Nm stands for the new method presented here

Table 6.
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