## Introduction

A distorting effect of the variation with height on the mean wind, both in speed and direction, is often observed in the development of puffs or plume smoke. This effect is most prominent in stable stratified conditions. In fact, wind shear creates a variance in the wind direction, while vertical diffusion destroys this variance and tries to reestablish a nonskewed distribution. The interaction between vertical mixing and velocity shear is continuously effective.

In order to take into account the above phenomenon, we developed a model for the dispersion of passive non-Gaussian puffs. The model is based on a general technique for solving the *K* equation using the truncated Gram–Charlier expansion of the concentration field and the finite set of equations for the corresponding moments. Actually, the Gram–Charlier expansion of type A is a classic method for approximating a given distribution with moments of any order, basically consisting of a truncated expansion in terms of Hermite functions, whose coefficients are chosen so as to reproduce the sequence of moments of the distribution up to a given order. In particular, the model is well suited to applications where interest is focused mainly on certain overall properties of the horizontal patterns, rather than on specific values at particular point receptors.

## The model

*C,*due to a release at time

*t*= 0 of a quantity

*Q*of passive material by an elevated source placed at (0, 0, 1), in a horizontally homogeneous atmospheric boundary layer is

*x*is the along-wind coordinate,

*y*the crosswind one, and

*z*the height;

*δ*means delta function; (

*u, υ,*0) is the wind velocity vector; and

*K*

_{z}and

*K*

_{h}are the eddy diffusivities for vertical and horizontal turbulent transport, respectively. All variables are nondimensional, with the corresponding scale factors being given by

*H*

^{2}

_{s}

*K*

_{s}for time,

*U*

_{s}

*H*

^{2}

_{s}

*K*

_{s}=

*dH*

_{s}for a distance along the

*x*axis,

*H*

_{s}for the height and distance along the

*y*axis,

*K*

_{s}for diffusivities,

*u*

_{s}for wind speed, and

*Q*

_{0}/(

*d*

*H*

^{3}

_{s}

*K*

_{ s}and

*u*

_{s}represent the values of the dimensional

*u*and

*K*profiles at the dimensional source

*H*

_{s}.

*z*

_{i}) are

*C*is exponentially small at asymptotic distances from the source on any horizontal plane, we can introduce the moments of its (

*x, y*) distribution:

*m*and

*n*are nonnegative integers.

*C*

_{m,n}are functions of height and of time. Their time evolution is governed by the double sequence of one-dimensional diffusion equations, equivalent to the single three-dimensional Eq. (1):

*m*+

*n*≠ 0 and

*D,*the differential operator (∂/∂

*z*)

*K*

_{z}(∂/∂

*z*).

A classic method for approximating a given distribution with moments of any order is the Gram–Chalier expansion of type A, which is basically constituted by a truncated expansion in terms of Hermite functions, whose coefficients are chosen so as to reproduce the sequence of moments of the function up to a given order (Kendall and Stuart 1977).

*C*(

*x*), truncated to the fourth order, if

*S*

_{k}is the skewness and

*K*

_{u}is the kurtosis, we have (Lupini and Tirabassi 1983)

## Boundary layer parameterization

In order to evaluate the diffusion coefficients in Eq. (3), utilizing as input simple ground-level meteorological data acquired by an automatic network, we tested two different boundary layer parameterizations.

*z*

_{i}/

*L*≥ −10), we adopted

*z*

_{i}/

*L*< −10), the friction velocity

*u*∗ was replaced by the convective velocity

*w*∗ as scaling velocity to give (Pleim and Chang 1992)

*w*

*βg*

*w*′

*θ*

^{′}

_{0}

*z*

_{i}

^{1/3}

*βg*is the bouyancy parameter and

*w*′

*θ*

^{′}

_{0}

*w*is vertical velocity and

*θ*the potential temperature). The prime indicates turbulent fluctuation variables.

*K*

_{h}

*w*

*z*

_{i}

*K*

_{h}

*K*

_{Mz}

*K*

_{Mz}is the maximum of

*K*

_{z}.

*K*

_{z}and

*K*

_{h}are derived from the Mellor–Yamada level 2 model (Mellor and Yamada 1974). In particular, we have

*K*

_{z}

*K*

_{H}

*K*

_{H}is the turbulent heat conductivity given by

*K*

_{H}

*Al*

_{2}

*q.*

*q*is the turbulent kinetic energy and

With strong convection, ∂*θ*/∂*z* can change sign above the surface layer and remain slightly positive over most of the mixed layer. This implies negative *K*_{H} values. In order to avoid such negative values, we used a modified flux gradient relation, as suggested by Deardoff (1972).

*K*

_{M}is the turbulent viscosity defined by

*l*

_{1}= 0.92

*l, l*

_{2}= 0.74

*l,*Λ

_{1}= 16.6

*l,*Λ

_{2}= 10.1

*l,*and

*l*is the master length scale.

*s*is the horizontal wind shear,

*q*is contained in

*K*

_{H}and

*K*

_{M}. This equation becomes a quadratic equation, the coefficients of which depend on the mean quantities of the boundary layer through the gradient Richardson number

*R*

_{i}(Freeman 1977). Here,

*R*

_{i}has been evaluated following the relation

*α*is from Yamada (1975) and the flux Richardson number

*R*

_{f}is evaluated with an iterative process by the two formulas

*B*

_{1}= 16.6 and the expression for the stability function

*S̃*

_{M}is that given in Mellor and Yamada (1974).

*l*may be computed following Blackadar (1962):

*k*is the von Kármán constant and

*λ*is an asymptotic value for

*l*defined as (Moeng and Wyngaard 1989)

*α*

_{b}is an empirical constant set to 0.1 by Mellor and Yamada (1974);

*λ*contained an additional dependence on

*q,*so we applied an iterative process to calculate 1:

*λ*

_{1}⇒

*q*

_{1}⇒

*λ*

_{2}⇒

*q*

_{2}· · · until |(

*λ*

_{n}−

*λ*

_{n−1})/

*λ*

_{n}| ≤

*ε.*

## Preliminary validation against experimental data

We evaluated the performance of the puff model with the two boundary layer parameterizations proposed using the Copenhagen dataset (Gryning and Lyck 1984). The Copenhagen dataset is composed of tracer SF_{6} data from dispersion experiments carried out in northern Copenhagen, Denmark. The tracer was released without buoyancy from a tower at a height of 115 m and was collected at ground-level positions in up to three crosswind arcs of tracer sampling units. The sampling units were positioned 2–6 km away from the point of release. We used the values of the crosswind-integrated concentrations (*C*_{y}) normalized with the tracer release rate from Gryning et al. (1987). Tracer releases typically started up 1 h before the tracer sampling and stopped at the end of the sampling period. The site was mainly residential, with a roughness length of 0.6 m. Generally, the distributed dataset contains hourly mean values of concentrations and meteorological data. However, in this model validation, we used data with a greater time resolution kindly made available to us by Gryning. In particular, we used 20-min averaged measured concentrations and 10-min averaged values for meteorological data.

The validation has to be considered a preliminary one. As a matter of fact, we have checked only two boundary layer parameterizations with data referring to continuous emission in variable meteorology (with a time resolution of 10 min) and at receptor points far from the source (2–6 km).

Tables 1, 2, and 3 report the friction velocity, the Monin–Obukhov length, and the boundary layer height (only one value for each run), respectively, used in the simulations.

Figures 1 and 2 show two typical examples of eddy diffusivity profiles obtained with the two parameterizations presented. The eddy diffusivity profiles in the figures were obtained with hourly average meteorological data and thus represent the mean profiles for a complete run. In Fig. 1, the two profiles are equal to the source height, while above this level, the *K* values proposed by Pleim and Chang (1992) are greater. In the second case (Fig. 2), the eddy diffusivity coefficients proposed by Pleim and Chang are always higher. The profile shape is common to all the remaining runs—the exchange coefficients proposed by Pleim and Chang in all cases are greater than those evaluated with the Freeman approach.

In Fig. 3, the calculated concentrations are plotted against the measured ones for the two different parameterizations.

In Table 4, the measured ground-level concentration values are presented, together with the computed ones with the two different parameterizations for each time period of the simulation.

*r*), factor of 2 (fa2), fractional bias (fb), and fractional standard deviation (fd):

*o*and

*p*are for the observed and predicted concentrations, respectively, while

*σ*is the standard deviation.

Table 4 and statistical indices show that the model performs better with the parameterization of Pleim and Chang (1992).

## Conclusions

Two boundary layer parameterizations for a non-Gaussian puff model are presented. The model can be applied routinely using as input simple ground-level meteorological data acquired by an automatic network. In fact, diffusion parameterizations are used based on fundamental parameters to describe characteristics of the atmospheric surface and boundary layer that can be evaluated by ground measurements. Model performances were evaluated using data from the Copenhagen dataset, but with a time resolution greater than that of the data generally distributed (Olesen 1995). In particular, we used 20-min average concentrations of SF_{6} and 10-min average values for meteorological data.

Preliminary model performance evaluation shows that the model performs better with the parameterization of Pleim and Chang (1992). This is due to a difference in the *K*_{z} profiles predicted by the two approaches. From Figs. 1 and 2, one can see that the two parameterizations predict about the same values of *K*_{z} up to the height of the source. In upper level, the vertical exchange coefficients predicted by Pleim and Chang’s approach are greater, and they increase with height, leading the diffusion of the pollution aloft. Indeed, looking at Fig. 4, it is possible to see that near the source, the ground-level concentrations predicted with Pleim and Chang’s approach are greater than those predicted utilizing Freeman’s parameterization. On the contrary, far from the source (when upper-level diffusion becomes important), the values predicted using Freeman’s parameterization are greater. The above considerations, together with that in the Copenhagen experiment, the receptors were located always beyond the maximum ground-level concentration (Olesen 1995) can explain the different results predicted by the presented model with the two parameterizations.

## Acknowledgments

We wish to thank Dr. Sven-Erik Gryning for making his data available to us.

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Hourly average eddy diffusivity coefficients as functions of height (normalized by source height), evaluated with the Pleim and Chang, and Freeman parameterizations during run 3.

Citation: Journal of Applied Meteorology 36, 8; 10.1175/1520-0450(1997)036<1031:BLPFAN>2.0.CO;2

Hourly average eddy diffusivity coefficients as functions of height (normalized by source height), evaluated with the Pleim and Chang, and Freeman parameterizations during run 3.

Citation: Journal of Applied Meteorology 36, 8; 10.1175/1520-0450(1997)036<1031:BLPFAN>2.0.CO;2

Hourly average eddy diffusivity coefficients as functions of height (normalized by source height), evaluated with the Pleim and Chang, and Freeman parameterizations during run 3.

Citation: Journal of Applied Meteorology 36, 8; 10.1175/1520-0450(1997)036<1031:BLPFAN>2.0.CO;2

Scatterplot of observed (*C*_{o}) vs predicted (*C*_{p}) crosswind-integrated concentrations, normalized with the emission source rate, using the Pleim and Chang, and Freeman parameterizations. Points between dashed lines have a factor of 2.

Citation: Journal of Applied Meteorology 36, 8; 10.1175/1520-0450(1997)036<1031:BLPFAN>2.0.CO;2

Scatterplot of observed (*C*_{o}) vs predicted (*C*_{p}) crosswind-integrated concentrations, normalized with the emission source rate, using the Pleim and Chang, and Freeman parameterizations. Points between dashed lines have a factor of 2.

Citation: Journal of Applied Meteorology 36, 8; 10.1175/1520-0450(1997)036<1031:BLPFAN>2.0.CO;2

Scatterplot of observed (*C*_{o}) vs predicted (*C*_{p}) crosswind-integrated concentrations, normalized with the emission source rate, using the Pleim and Chang, and Freeman parameterizations. Points between dashed lines have a factor of 2.

Citation: Journal of Applied Meteorology 36, 8; 10.1175/1520-0450(1997)036<1031:BLPFAN>2.0.CO;2

Variation of crosswind-integrated concentrations normalized with the emission rate, with distance, according to measurements (points) and predicted values using the parameterizations of Plein and Chang, and Freeman forrun 3 and period 3.

Citation: Journal of Applied Meteorology 36, 8; 10.1175/1520-0450(1997)036<1031:BLPFAN>2.0.CO;2

Variation of crosswind-integrated concentrations normalized with the emission rate, with distance, according to measurements (points) and predicted values using the parameterizations of Plein and Chang, and Freeman forrun 3 and period 3.

Citation: Journal of Applied Meteorology 36, 8; 10.1175/1520-0450(1997)036<1031:BLPFAN>2.0.CO;2

Variation of crosswind-integrated concentrations normalized with the emission rate, with distance, according to measurements (points) and predicted values using the parameterizations of Plein and Chang, and Freeman forrun 3 and period 3.

Citation: Journal of Applied Meteorology 36, 8; 10.1175/1520-0450(1997)036<1031:BLPFAN>2.0.CO;2

Friction velocity (m s^{−1}) for the different runs and time steps. Every time step corresponds to 10 min.

Monin–Obukhov length (m) for the different runs and time steps. Every time step corresponds to 10 min.

Boundary layer height for the different runs.

Observed (*C** _{o}*) and predicted (

*C*

*) crosswind-integrated concentrations normalized with emission rate (10*

_{p}^{−4}s m

^{−2}) at differentdistances from the source (m) for the Pleim and Chang (1) and Freeman (2) parameterizations.

Statistical evaluation of model results. Models 1 and 2 use the Pleim and Chang, and Freeman parameterizations, respectively.