## 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 (*x*_{max}/*u*

*σ*

_{υ}due to meandering (

*σ*

^{M}

_{υ}

*σ*

^{M}

_{υ}

*σ*

_{υ}

^{2}

*σ*

^{T}

_{υ}

^{2}

^{1/2}

*σ*

^{T}

_{υ}

*u*

*θ,*and

*σ*

^{M}

_{υ}

*i*=

*σ*

_{u,υ}/

*u*

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

*u*

^{−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*

*w*

*R*(

*τ*) were obtained from calculated power spectra using the relationshipwhere

*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 *R*_{u}(*τ*), *R*_{υ}(*τ*), and *R*_{w}(*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 *R*_{w}(*τ*) exhibits an exponential form, *R*_{u}(*τ*) and *R*_{υ}(*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.

*σ*

_{w}, our observations support a relationship of the form

*σ*

_{w}

*u*

*zL*

^{−1}

*u*∗ denotes the friction velocity, and

*L*is the Monin–Obukhov length. A scatterplot, using (3), of observed versus calculated

*σ*

_{w}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

*R*

^{2}= 0.95 and little scatter along the solid line indicate a fairly good agreement between recorded and computed data. Comparison with the expressiongiven by Hanna (1982), with

*h*being the height of the stable boundary layer, showed a slight underestimation of

*σ*

_{w}for values higher than 0.2 m s

^{−1}(not shown). Apparently,

*σ*

_{w}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

*σ*

_{w}of the form

*σ*

_{w}

*u*

*zL*

^{−1}

## Model description

### Horizontal dispersion

*t*+ Δ

*t*

_{h}are given by

*u*

*υ*

*x*and

*y*directions,

*u*′ and

*υ*′ are the velocity fluctuations,

*σ*

_{u}and

*σ*

_{υ}are the standard deviations of the velocity components,

*χ*are random numbers with zero mean and a standard deviation equal to 1,

*ρ*

_{u}and

*ρ*

_{υ}are the intercorrelation parameters, and Δ

*t*

_{h}is a random time step for which the horizontal velocity fluctuations remain constant.

*T*

_{Lu}and

*T*

_{Lυ}are known, the mean time interval

*t*

_{h}

*ρ*

_{u,υ}< 0, a negative loop is caused in

*R*

_{u,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

*ρ*

_{u,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

*ρ*

_{u,υ}< 0.9 is chosen and for small-scale turbulence diffusion processes if

*ρ*

_{u,υ}≥ 0.9 is taken. In Fig. 3, the effect of different values for

*ρ*

_{u,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

*ρ*

_{u,υ}= 0.9 (top panel) and the same but for

*ρ*

_{u,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

*ρ*

_{u,υ}< 0.9.

*ρ*

_{u,υ}should also be a function of wind speed. Because

*ρ*

_{u,υ}is a dimensionless variable, the standard deviation of the wind direction fluctuations

*σ*

_{θ}was taken for parameterization instead of the wind speed. Observations made by Leahey et al. (1994) support a relationship of the formwith

*σ*

_{θ}in degrees. In Fig. 4, the relationship according to Leahey et al. (1994) between

*σ*

_{θ}and wind speed is drawn. There is almost no change in

*σ*

_{θ}for wind speeds higher than 3 m s

^{−1}; for wind speeds below this value, a substantial increase of

*σ*

_{θ}is the case. If

*σ*

_{θ}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

*ρ*

_{u,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,

*ρ*

_{u,υ}should be limited by a value of about −0.9, because (10) becomes physically unrealistic for

*ρ*

_{u,υ}< −1.0. Hence, a proper expression for

*ρ*

_{u,υ}as a function of

*σ*

_{θ}should give a value of

*ρ*

_{u,υ}= 0.9 for

*σ*

_{θ}< 7.0° and a value of

*ρ*

_{u,υ}= −0.9 for the largest observed

*σ*

_{θ}. A possible shape of a function for

*ρ*

_{u,υ}satisfying these criteria could take the formwhere

*A*and

*B*are constants to be determined empirically, and

*σ*

_{θ}here is in units of radians. For the Lagrangian timescales

*T*

_{Lu,υ}in (10), formulations suggested by Hanna (1982) are used in the model:The height of the stable boundary layer

*h*in (14) and (15) can be approximated withwhere

*f*= 0.0001 s

^{−1}is the Coriolis parameter, and the turbulent part of the observed standard deviations are given byEquations (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

*ρ*

_{u,υ}needs to be varied if the model is used for small-scale turbulence and meandering flows.

### Vertical dispersion

*w*

*t*

*t*

_{υ}

*a*

*w,*

*z*

*t*

_{υ}

*C*

_{0}

*z*

^{1/2}

*dW*

*w*

*t*

*z*

*t*

*t*

_{υ}

*z*

*t*

*w*

*t*

*t*

_{υ}

*t*

_{υ}

*w*is the vertical velocity of a particle,

*C*

_{0}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 Δ

*t*

_{υ}, and the time step Δ

*t*

_{υ}is given byThe deterministic acceleration term

*a*(

*w, z*) is assumed to be a function of the vertical velocity:

*a*

*w, z*

*α*

*z*

*w*

^{2}

*β*

*z*

*w*

*γ*

*z*

*α*(

*z*),

*β*(

*z*), and

*γ*(

*z*) are unknown parameters, which are determined from the Fokker–Planck equation:where

*P*

_{E}(

*w, z*) is the Eulerian PDF of the vertical turbulent velocity at a given height

*z.*

*P*

_{E}(

*w, z*) but only requires the first four Eulerian moments of the vertical velocity. The coefficients in (22) can be expressed asIn (24)–(26),

*w*

^{i}

*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) aswhere

*k*is the von Kármán constant. The fourth moment was set aswhich is the Gaussian assumption.

*z*) was taken according to Stull (1989),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.

*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:The roughness length

*z*

_{0}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.

*σ*

_{u}and

*σ*

_{υ}in (8) and (9) were set equal to each other and were calculated using

*σ*

_{u,υ}

*σ*

_{θ}

*u*

*σ*

_{u,υ}/

*u*

*x*= Δ

*y*= 5 m and Δ

*z*= 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 10

^{6}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

*ρ*

_{u,υ}, (12) and (13) were used instead of observed

*σ*

_{θ}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*=

*z*

_{0}and

*z*=

*h.*

*σ*

_{θ}parameterization of Sharan and Yadav (1998), several statistical measures were computed: the relative mean biasthe top-10 relative bias

*c*

_{c10}

*c*

_{o10}

*c*

_{o10}

*c*

_{o}

*c*

_{c}

*c*

_{o}

*c*

_{c}

*c*

_{c}is the computed concentration,

*c*

_{o}is the observed concentration, and

*c*

_{c10}and

*c*

_{o10}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.

*c*

_{n}

*χ*

*u*

*Q*

^{−3}

*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 *ρ*_{u,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 *ρ*_{u,υ} and the wind direction fluctuations *σ*_{θ}, 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 *ρ*_{u,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.Gifford, F. A., 1960: Peak to average concentration ratios according to a fluctuating plume dispersion model.

*Int. J. Air Pollut.,***3,**253–260.Hanna, S. R., 1982: Applications in air pollution modeling.

*Atmospheric Turbulence and Air Pollution Modeling,*F. T. M. Nieuwstadt and H. Van Dop, Eds., Reidel, 275–310.Hanna, S. R., 1983: Lateral turbulence intensity and plume meandering during stable conditions.

*J. Climate Appl. Meteor.,***22,**1424–1431.Hanna, S. R., 1988: Air quality model evaluation and uncertainty.

*J. Air Pollut. Control Assoc.,***38,**406–412.Kristensen, L., N. O. Jensen, and E. L. Peterson, 1981: Lateral dispersion of pollutants in a very stable atmosphere.

*Atmos. Environ.,***15,**837–844.Leahey, D. M., M. C. Hansen, and M. B. Schroeder, 1994: Variations of wind fluctuations observed at 10 m over flat terrain under stable atmospheric conditions.

*J. Appl. Meteor.,***33,**712–720.Leung, D. Y. C., and C. H. Liu, 1996: Improved estimators for the standard deviations of horizontal wind fluctuations.

*Atmos. Environ.,***30,**2457–2461.Panofsky, H. A., and J. A. Dutton, 1984:

*Atmospheric Turbulence.*John Wiley and Sons, 397 pp.Pasquill, F., and F. B. Smith, 1983:

*Atmospheric Diffusion.*3d ed. Ellis Horwood, Halsted Press, John Wiley and Sons, 437 pp.Sagendorf, J. F., and C. R. Dickson, 1974: Diffusion under low windspeed, inversion conditions. NOAA Tech. Memo. ERL ARL-52, 89 pp. [Available from Air Resources Library, 1750 Foote Drive, Idaho Falls, ID 83402.].

Shampine, L. F., S. M. Davenport, and R. E. Huddleston, 1974: Curve fitting by polynomials in one variable. Sandia Laboratories Rep. SLA-74-0270.

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.Wilson, J. D., and B. L. Sawford, 1996: Review of Lagrangian stochastic models for trajectories in the turbulent atmosphere.

*Bound.-Layer Meteor.,***78,**191–210.

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

Reported values of mean wind speed (*ū*), mean wind direction (*θ*), and standard deviation of wind direction (*σ*_{θ}) for the INEL experiment (Sagendorf and Dickson 1974)

Observed and calculated (Nm = new method) concentrations (*μ*g 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)

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

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

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