Boundary Layer Parameterization for a Non-Gaussian Puff Model

T. Tirabassi Institute FISBAT of CNR, Bologna, Italy

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U. Rizza Institute ISIAtA of CNR, Lecce, Italy

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Abstract

The paper presents two boundary layer parameterizations for a model based on a general technique for solving the K equation using the truncated Gram–Charlier expansion (type A) of the concentration field and a finite set of equations for the corresponding moments. The two parameterizations allow the model to be applied routinely using as input simple ground-level meteorological data acquired by an automatic network. A preliminary performance evaluation is shown in the case of continuous emission from an elevated source in a variable boundary layer (with a time resolution of 10 min).

Corresponding author address: Dr. Tiziano Tirabassi, Institute FISBAT of CNR, Via P. Gobetti 101, I-40129 Bologna, Italy.

tirabassi@area.bo.cnr.it

Abstract

The paper presents two boundary layer parameterizations for a model based on a general technique for solving the K equation using the truncated Gram–Charlier expansion (type A) of the concentration field and a finite set of equations for the corresponding moments. The two parameterizations allow the model to be applied routinely using as input simple ground-level meteorological data acquired by an automatic network. A preliminary performance evaluation is shown in the case of continuous emission from an elevated source in a variable boundary layer (with a time resolution of 10 min).

Corresponding author address: Dr. Tiziano Tirabassi, Institute FISBAT of CNR, Via P. Gobetti 101, I-40129 Bologna, Italy.

tirabassi@area.bo.cnr.it

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

The advection diffusion equation describing the time evolution of concentration 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
i1520-0450-36-8-1031-e1
where 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 Kz and Kh are the eddy diffusivities for vertical and horizontal turbulent transport, respectively. All variables are nondimensional, with the corresponding scale factors being given by H2s/Ks for time, UsH2s/Ks =dHs for a distance along the x axis, Hs for the height and distance along the y axis, Ks for diffusivities, us for wind speed, and Q0/(dH3s) for concentration. Here, K s and us represent the values of the dimensional u and K profiles at the dimensional source Hs.
The initial condition is
i1520-0450-36-8-1031-eq1
and the no-flux boundary conditions applied at the ground level and at the mixing layer height (zi) are
i1520-0450-36-8-1031-eq2
Since C is exponentially small at asymptotic distances from the source on any horizontal plane, we can introduce the moments of its (x, y) distribution:
i1520-0450-36-8-1031-e2
where m and n are nonnegative integers.
Of course, Cm,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):
i1520-0450-36-8-1031-e3
for m + n ≠ 0 and D, the differential operator (∂/∂z)Kz(∂/∂z).
The initial condition is therefore written as
i1520-0450-36-8-1031-eq3
and the boundary conditions become
i1520-0450-36-8-1031-eq4

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).

In the case of one-variate function of the concentration C(x), truncated to the fourth order, if Sk is the skewness and Ku is the kurtosis, we have (Lupini and Tirabassi 1983)
i1520-0450-36-8-1031-e4
where
i1520-0450-36-8-1031-eq4a

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.

We applied a parameterization proposed by Troen and Marth (1986), as presented in Pleim and Chang (1992). During near-neutral and stable conditions (zi/L ≥ −10), we adopted
i1520-0450-36-8-1031-e5
During convective conditions (zi/L < −10), the friction velocity u∗ was replaced by the convective velocity w∗ as scaling velocity to give (Pleim and Chang 1992)
i1520-0450-36-8-1031-e6
where the convective velocity is defined as
wβgwθ0zi1/3
Here, βg is the bouyancy parameter and wθ0 the surface kinematics heat flux (w is vertical velocity and θ the potential temperature). The prime indicates turbulent fluctuation variables.
For the horizontal eddy diffusivity in unstable conditions, we use (Seinfeld 1986)
Khwzi
In neutral stable conditions (Tangerman 1978),
KhKMz
where KMz is the maximum of Kz.
The second parameterization used is based on Freeman’s work (Freeman 1977), in which vertical and horizontal eddy diffusivity coefficients Kz and Kh are derived from the Mellor–Yamada level 2 model (Mellor and Yamada 1974). In particular, we have
KzKH
where KH is the turbulent heat conductivity given by
KHAl2q.
Here, q is the turbulent kinetic energy and
i1520-0450-36-8-1031-eq5

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 KH values. In order to avoid such negative values, we used a modified flux gradient relation, as suggested by Deardoff (1972).

The horizontal diffusion coefficients are
i1520-0450-36-8-1031-e12
where
i1520-0450-36-8-1031-eq5a
and KM is the turbulent viscosity defined by
i1520-0450-36-8-1031-e14
Here, following Mellor and Yamada (1982), l1 = 0.92l, l2 = 0.74l, Λ1 = 16.6l, Λ2 = 10.1l, and l is the master length scale.
All diffusion coefficients have been expressed in terms of vertical derivatives of the mean values of boundary layer variables and the turbulent kinetic energy. The relation of the kinetic energy to the mean field quantities is found by solving the algebraic equation (Freeman 1977), where s is the horizontal wind shear,
i1520-0450-36-8-1031-e15
in which additional dependence on q is contained in KH and KM. This equation becomes a quadratic equation, the coefficients of which depend on the mean quantities of the boundary layer through the gradient Richardson number Ri (Freeman 1977). Here, Ri has been evaluated following the relation
i1520-0450-36-8-1031-e16
where α is from Yamada (1975) and the flux Richardson numberRf is evaluated with an iterative process by the two formulas
i1520-0450-36-8-1031-e17
where B1 = 16.6 and the expression for the stability function M is that given in Mellor and Yamada (1974).
The length scale l may be computed following Blackadar (1962):
i1520-0450-36-8-1031-e19
where k is the von Kármán constant and λ is an asymptotic value for l defined as (Moeng and Wyngaard 1989)
i1520-0450-36-8-1031-e20
Here, αb 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: λ1q1λ2q2 · · · until |(λnλn−1)/λn| ≤ 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 SF6 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 (Cy) 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.

Moreover, Table 5 presents some statistical indices, defined as the normalized mean square error (nmse), correlation coefficient (r), factor of 2 (fa2), fractional bias (fb), and fractional standard deviation (fd):
i1520-0450-36-8-1031-eq6
where the subscripts 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 SF6 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 Kz profiles predicted by the two approaches. From Figs. 1 and 2, one can see that the two parameterizations predict about the same values of Kz 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.

REFERENCES

  • Blackadar, A. K., 1962: The vertical distribution of wind and turbulent exchange in a neutral atmosphere. J. Geophys. Res.,67, 3103–3109.

  • Deardoff, J. W., 1972: Theoretical expression for the countergradient vertical heat flux. J. Geophys. Res.,77, 5900–5904.

  • Freeman, B. E., 1977: Tensor diffusivity of a trace constituent in a stratified boundary layer. J. Atmos. Sci.,34, 124–136.

  • Gryning, S. E., and E. Lyck, 1984: Atmospheric dispersion from elevated sources in an urban area: Comparison between tracer experiments and model calculations. J. Climate Appl. Meteor.,23, 651–660.

  • ——, A. A. M. Holtslag, J. S. Irwin, and B. Siversten, 1987: Applied dispersion modelling based on meteorological scaling parameters. Atmos. Environ.,21, 79–89.

  • Kendall, M., and A. Stuart, 1977: The Advanced Theory of Statistic. Vol. 1. Charles Griffin, 472 pp.

  • Lupini, R., and T. Tirabassi, 1983: Solution of the advection-diffusion equation by the moments method. Atmos. Environ.,17, 965–971.

  • Mellor, G., and T. Yamada, 1974: A hierarchy of turbulence closure models for planetary boundary layers. J. Atmos. Sci.,31, 1791–1806.

  • ——, and ——, 1982: Development of a turbulent closure model for geophysical fluid problems. Rev. Geophys. Space Phys.,20, 851–875.

  • Moeng, C. H., and J. C. Wingaard, 1989: Evaluation of turbulent transport and dissipation closures in second-order modeling. J. Atmos. Sci.,46, 2311–2330.

  • Olesen, H. R., 1995: Datasets and protocol for model validation. Int. J. Environ. Pollut.,5, 693–701.

  • Pleim, J., and J. S. Chang, 1992: A non local closure model for vertical mixing in the convective boundary layer. Atmos. Environ.,26A, 965–981.

  • Seinfeld, J. H., 1986: Atmospheric Chemistry and Physics of Air Pollution. John Wiley & Sons, 738 pp.

  • Tangerman, G., 1978: Numerical simulations of air pollutant dispersion in a stratified planetary boundary layer. Atmos. Environ.,12, 1365–1369.

  • Troen, I., and L. Marth, 1986: A simple model of the atmospheric boundary layer; sensitivity to surface evaporation. Bound.-Layer Meteor.,37, 129–148.

  • Yamada, T., 1975: The critical Richardson number and the ratio of eddy transport coefficient obtained from a turbulence closure model. J. Atmos. Sci.,32, 926–933.

Fig. 1.
Fig. 1.

Hourly average eddy diffusivity coefficients as functions of height (normalized by source height), evaluated with the Pleim and Chang, and Freeman parameterizations during run 1.

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

Fig. 2.
Fig. 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

Fig. 3.
Fig. 3.

Scatterplot of observed (Co) vs predicted (Cp) 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

Fig. 4.
Fig. 4.

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

Table 1.

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

Table 1.
Table 2.

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

Table 2.
Table 3.

Boundary layer height for the different runs.

Table 3.
Table 4.

Observed (Co) and predicted (Cp) crosswind-integrated concentrations normalized with emission rate (10−4 s m−2) at differentdistances from the source (m) for the Pleim and Chang (1) and Freeman (2) parameterizations.

Table 4.
Table 5.

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

Table 5.
Save
  • Blackadar, A. K., 1962: The vertical distribution of wind and turbulent exchange in a neutral atmosphere. J. Geophys. Res.,67, 3103–3109.

  • Deardoff, J. W., 1972: Theoretical expression for the countergradient vertical heat flux. J. Geophys. Res.,77, 5900–5904.

  • Freeman, B. E., 1977: Tensor diffusivity of a trace constituent in a stratified boundary layer. J. Atmos. Sci.,34, 124–136.

  • Gryning, S. E., and E. Lyck, 1984: Atmospheric dispersion from elevated sources in an urban area: Comparison between tracer experiments and model calculations. J. Climate Appl. Meteor.,23, 651–660.

  • ——, A. A. M. Holtslag, J. S. Irwin, and B. Siversten, 1987: Applied dispersion modelling based on meteorological scaling parameters. Atmos. Environ.,21, 79–89.

  • Kendall, M., and A. Stuart, 1977: The Advanced Theory of Statistic. Vol. 1. Charles Griffin, 472 pp.

  • Lupini, R., and T. Tirabassi, 1983: Solution of the advection-diffusion equation by the moments method. Atmos. Environ.,17, 965–971.

  • Mellor, G., and T. Yamada, 1974: A hierarchy of turbulence closure models for planetary boundary layers. J. Atmos. Sci.,31, 1791–1806.

  • ——, and ——, 1982: Development of a turbulent closure model for geophysical fluid problems. Rev. Geophys. Space Phys.,20, 851–875.

  • Moeng, C. H., and J. C. Wingaard, 1989: Evaluation of turbulent transport and dissipation closures in second-order modeling. J. Atmos. Sci.,46, 2311–2330.

  • Olesen, H. R., 1995: Datasets and protocol for model validation. Int. J. Environ. Pollut.,5, 693–701.

  • Pleim, J., and J. S. Chang, 1992: A non local closure model for vertical mixing in the convective boundary layer. Atmos. Environ.,26A, 965–981.

  • Seinfeld, J. H., 1986: Atmospheric Chemistry and Physics of Air Pollution. John Wiley & Sons, 738 pp.

  • Tangerman, G., 1978: Numerical simulations of air pollutant dispersion in a stratified planetary boundary layer. Atmos. Environ.,12, 1365–1369.

  • Troen, I., and L. Marth, 1986: A simple model of the atmospheric boundary layer; sensitivity to surface evaporation. Bound.-Layer Meteor.,37, 129–148.

  • Yamada, T., 1975: The critical Richardson number and the ratio of eddy transport coefficient obtained from a turbulence closure model. J. Atmos. Sci.,32, 926–933.

  • Fig. 1.

    Hourly average eddy diffusivity coefficients as functions of height (normalized by source height), evaluated with the Pleim and Chang, and Freeman parameterizations during run 1.

  • Fig. 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.

  • Fig. 3.

    Scatterplot of observed (Co) vs predicted (Cp) 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.

  • Fig. 4.

    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.

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