a. The Gaussian plume and puff models

Polyphemus hosts both a Gaussian plume model and a Gaussian puff model. Arya (1999) reviews the main assumptions for the relevance of the two models. A Gaussian plume model assumes a Gaussian distribution of mean concentration in the horizontal (crosswind) and vertical directions, in steady-state and homogeneous meteorological conditions. The dispersion in the downwind direction is supposed to be negligible compared to the transport by the mean wind in the plume model. A Gaussian puff model discretizes the plume into a series of puffs, each of them having a Gaussian shape in all three directions. The puff model can also take into account different meteorological data for each puff, therefore allowing a more accurate simulation of nonstationary and nonhomogeneous conditions. In Polyphemus, the plume and puff models share the same parameterizations. Thus, in stationary and homogeneous conditions—as assumed in the Prairie Grass and Kincaid experiments—the Gaussian puff model gives the same solution as the Gaussian plume model, if averaged over a long enough period. Henceforth, only the Gaussian plume model is therefore used (it was verified that the Gaussian puff model gave the same results).

For a stationary plume, the concentration C is given by the Gaussian plume formula:

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where Q is the source emission rate, given in mass per second, u is the mean wind speed, and σ_y and σ_z are the Gaussian plume standard deviations in the horizontal (crosswind) and vertical directions. The coordinate y refers to the crosswind horizontal direction, and y_s is the source coordinate in that direction. The coordinate z refers to the vertical coordinate and z_p is the plume height above ground—sum of the source height and the plume rise (see section 2c). The second term of the last factor represents the ground reflection; similarly, additional terms can take into account the reflections on the elevated inversion layer. Also, when the plume is mixed enough in the boundary layer (σ_z > 1.5z_i), the formula is modified to consider that the plume is vertically homogeneous. In addition, other factors can be applied to this formula to take loss processes into account, such as dry deposition, scavenging, and radioactive or biological decay. In Polyphemus, full gaseous chemistry may also be handled by the Gaussian puff model.

b. Dispersion parameterization schemes

The expression of the total plume standard deviation is the sum of the spread due to turbulence (σ_yturb, σ_zturb), the additional spread due to plume rise (σ_ypr, σ_zpr), and the initial spread due to the diameter d_s of the source

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For the sake of clarity, σ_yturb and σ_zturb are named in this section σ_y and σ_z. These standard deviations are estimated through empirical schemes, as functions of the downwind distance to the source and the stability, based on a few dispersion field experiments. Here, three parameterizations are described. Two of them are based on a discrete description of the atmospheric boundary layer: the Briggs formulas and the Doury formulas. The third one is based on similarity theory, and involves parameters such as the wind velocity fluctuations, the Monin–Obukhov length, the mixing height, and the friction velocity.

1) Briggs formulas

The Briggs formulas are based on the Pasquill–Turner stability classes (Turner 1969) and fitted on the Prairie Grass experiment. This parameterization is born from an attempt to synthesize several widely used schemes by interpolating them for open country and for urban areas, for which they are particularly recommended. The full formulas can be found in Arya (1999), for instance. The general form is given by

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with x as the downwind distance from source, and α, β, and γ coefficients depending on the Pasquill stability class. There are six stability classes corresponding to different states of the atmosphere (Pasquill 1961), from A (extremely unstable) to F (stable).

2) Doury formulas

An alternative parameterization is described in Doury (1976). This parameterization has been developed for the specific application of radionuclide dispersion, and fitted on a wider experimental field than the Prairie Grass field. The formulas use only two stability situations, corresponding to “normal” and “low” dispersion, determined by the vertical temperature gradient. By default, it is assumed that low dispersion occurs during nighttime with low wind speed (u ≤ 3 m s⁻¹). The standard deviations are given in both cases in the general form

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where t is the transfer time since release time. In the case of a steady-state plume, t = x/u, where x is the distance from the source and u is the wind speed. The formulas for the normal situation can be found in Demaël and Carissimo (2008).

3) Similarity theory

If enough meteorological data are available, σ_y and σ_z can be estimated using the standard deviations of wind velocity fluctuations in crosswind horizontal direction σ_υ and in vertical direction σ_w. Following Irwin (1979), dispersion coefficients are investigated in the form

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where t is the time in seconds, and F_y and F_z are functions of a set of parameters that specify the characteristics of the atmospheric boundary layer. These functions are determined from experimental data. Various expressions of F_y and F_z have been proposed (e.g., Irwin 1979; Weil 1988). Appendix A details the formulations used in Polyphemus for F_y, F_z, σ_υ, and σ_w. A modified formulation from Hanna and Paine (1989) is also available for the case of elevated sources (above 100-m height). Two different formulations are also given for the standard deviations above the boundary layer.

c. Plume rise schemes

When emissions are hotter than the ambient air, or have a significant ejection speed, the plume generally reaches a certain height before behaving like the surrounding air: this difference in heights can be significant (up to a few hundred meters) and is called “plume rise” (see Fig. 1). Estimates of the plume rise can be provided, based on the source initial height, diameter, ejection temperature, and ejection velocity. In all formulas described below, it is assumed that the plume instantaneously reaches its final height at the emission time, then travels downwind. More accurate formulations allow one to determine the plume rise as a function of the downwind distance, but are not investigated here.

If the plume rise is Δh, the plume final height is given by z_p = z_s + Δh, with z_s as the source height. The initial spread due to plume rise is added to the total standard deviations at the emission time according to Eq. (2). It is estimated following Irwin (1979) for the crosswind spread and Hanna and Paine (1989) for the vertical spread that

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Three different formulas to compute plume rise are described in this section. While the Briggs–Hybrid Plume Dispersion Model (HPDM) formulas depend on the stability of the atmosphere, the two other formulas depend only on the source dynamics and buoyancy and are based on the source heat rate.

1) Briggs–HPDM plume rise

The plume rise computation is based on the Briggs formulas—detailed in Seinfeld and Pandis (1998)—assuming a buoyancy-driven plume rise [Eqs. (8)–(12)]. In the formulas, Δh is the plume rise, u is the wind velocity, w_* is the convective velocity, and u_* is the friction velocity. In addition, s_p is the Briggs static stability parameter and F_b is the initial buoyancy flux parameter. They are defined as

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with g as the gravity acceleration, dθ/dz as the vertical gradient of potential temperature, T as the ambient temperature, υ_s as the vertical velocity of the emission, d_s as the source diameter, and T_s as the source temperature.

Stable cases

The formulas for stable cases come from Seinfeld and Pandis (1998) and are also used in the HPDM (Hanna and Paine 1989). The final plume rise is Δh = min(Δh₁, Δh₂), where

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Unstable and neutral cases

The following formulas come from Seinfeld and Pandis (1998) and are originally from Briggs:

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In the above formulas, ambient turbulence is supposed to be negligible, relative to the plume internal turbulence. However, in many unstable and neutral cases, the plume eventually reaches a point where ambient turbulence is large enough to stop the plume vertical progress. In Briggs (1971), this so-called breakup height is determined as the point where both the plume and the ambient eddy dissipation rates are equal. It is reported in Hanna (1984) and used here for neutral cases, whereas the unstable breakup formula comes from Hanna and Paine (1989). When there is the choice between a breakup formula and a Briggs formula, the one giving the minimal plume rise is chosen:

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2) Holland formula

This plume rise formula is the same for all stabilities. It comes originally from Holland (1953) and was modified by Stümke (1963):

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3) Concawe formula

In Brummage (1968), a comparison study was made between several plume rise formulas, including the Holland–Stümke formula, which was among the best. An attempt was made to synthesize the multiple formulations into a relationship in the form Δh = K(Q_h^α/u^β), where Q_h is the source heat rate and K, α, and β are empirical constants. It resulted in the Concawe formulas:

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Here, the ejection heat rate is computed using the source diameter, its ejection velocity, and the temperature difference between the plume and ambient air:

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This is an approximation, based on the assumption that the calorific capacity of the ejected gas is not too different from that of air.

a. Comparison setup

The evolution of the Gaussian standard deviations (without a possible additional spread due to the diameter and/or plume rise) can be plotted for the parameterizations described in the previous section, to evaluate the spread. It must be noted that the Briggs formulas give σ_y and σ_z as a function of the downwind distance x from the source, whereas the Doury and similarity theory parameterizations are expressed according to the travel time t. These two parameters are related through t = x/u, with u as the wind speed. The evolution of the standard deviations against t is given in Fig. 3. The Briggs formulas are computed for the six Pasquill stability classes, taking a wind speed value representative of the stability class: u = 2 m s⁻¹ for A and F classes, u = 2.5 m s⁻¹ for B, u = 4 m s⁻¹ for C, u = 5.5 m s⁻¹ for D, and u = 3 m s⁻¹ for E. To compute the standard deviations with similarity theory, some other meteorological data are needed, and the following values are taken: the friction velocity u_* = 0.3 m s⁻¹ and the boundary layer height h = 500 m. The Monin–Obukhov length L is computed according to the relationship between L and the roughness length z₀ given by Golder (1972), taking the roughness value for Prairie Grass z₀ = 0.01 m. This leads to the value for unstable cases (corresponding to the B class) L = −5.86 m, and for stable cases (E class) L = 11.5 m. The convective velocity is computed with the formula w_*³ = −u_*³h/L. The plume height is taken at z = 1.5 m.

b. Evaluation of the standard deviations

Figure 3a shows the evolution of the horizontal standard deviations up to about 3 h. The values for σ_y at t = 9000 s range from 100 m (similarity theory) to 3000 m (Doury) in stable cases. In unstable cases, there is less variability, since σ_y ranges from 1200 to 3000 m. The standard deviation computed with similarity theory is the smallest in stable and neutral cases, but it compares to the Briggs results in unstable cases. The Doury standard deviations are comparable to the others in unstable cases, and give the largest values in the case of stable situations. For larger travel times, it would grow above all other parameterizations. Also, if the wind speed is very low (about 0.5 m s⁻¹), the downwind distance remains relatively small even for large travel times, therefore the Doury and similarity theory parameterizations give larger standard deviations at a given distance than Briggs’s. The spread for the vertical standard deviation is also very large (Fig. 3b). In stable cases σ_z values range from about 20 m (similarity theory) to 400 m (Doury). In unstable cases, there are larger differences, with σ_z ranging from 100 m (similarity theory) to 3500 m (Briggs, class A), so over 30 times larger. Of all parameterizations, only the Briggs formulas for the unstable classes A, B, and C show standard deviations above 500 m. Similarity theory is the parameterization giving the smallest vertical standard deviation for all stabilities.

These preliminary results highlight a large variability in the standard deviations, depending on the stability and the parameterization. Thus, a substantial variability can also be expected in the output results. Besides, because of the differences in estimations of σ_z, the plume might penetrate the inversion layer with some parameterizations and not with others, which would increase the variability. It should also be pointed out that the stability diagnosis depends on the standard deviation formula used. For instance, in the Prairie Grass experiments, the Pasquill–Turner diagnosis used with the Briggs formulas considers 30% of the experiments to be unstable, 20% stable, and 50% neutral, whereas the diagnosis based on Monin–Obukhov length (Table A1, described below)—used for similarity theory—gives about 48% of unstable cases and 52% of stable cases. This might compensate some of the differences in the standard deviation estimations: for instance, similarity theory in unstable cases and Briggs’s formulas in neutral cases give similar estimations.

a. Model evaluation criteria

The statistical evaluation method used for all models in this comparison is described in Chang and Hanna (2004). It is also the method used in the Model Validation Kit (Olesen and Chang 2005), which has been developed and tested during a series of workshops and conferences on harmonization within atmospheric dispersion modeling for regulatory purposes. In many field experiments, the sensors are placed in arcs at various distances downwind of the source, and the maximum predicted and observed concentrations for each arc (“maximum arcwise concentrations”) are compared. Therefore, all other concentration values on the arc are not used in the comparison. The maximum observed value is determined on time-averaged concentration measurements (the averaging time for Prairie Grass experiments is 10 min). The comparison study is carried out using maximum arcwise concentrations normalized by the emission rate Q—so-called centerline values. The results are discussed using scatter diagrams as well as statistical performance measures such as the fractional bias (FB), the normalized mean square error (NMSE), the fraction of predictions within a factor 2 of observations (FAC2), and the correlation coefficient (Corr). The formulas to compute these statistics are given in appendix B. It should be noted that these formulas give negative values of FB for model overprediction and positive values if the model tends to underpredict concentrations. A perfect model would have Corr and FAC2 equal to 1.0 and FB and NMSE equal to 0.0. As there is no such thing as a perfect model, it is useful to determine typical performance measures for acceptable models. Those criteria have been summarized in Chang and Hanna (2004) and in Hanna et al. (2004) based on the evaluation of many models with many field datasets. A model is deemed acceptable if the mean bias is within ±30% of the mean (−0.3 ≤ FB ≤ 0.3), if the random scatter is about a factor of 2–3 of the mean (NMSE ≤ 4), and if the fraction of predictions within a factor 2 of the observations is about 50% (FAC2 ≥ 0.5).

b. Experiment and modeling setup

The Prairie Grass experiment has become a standard database on which parameterizations have been fitted, and has been used for many model evaluation studies. The experiment took place in O’Neil, Nebraska, during summer 1956. The site was a flat terrain of short-cut grass. A continuous plume of SO₂ was released, without plume rise, near the ground (at 0.46 m). Measurements were taken on five arcs at 50, 100, 200, 400, and 800 m from the source. A set of 43 of the 68 trials is usually used for model validations, including a wide range of stability conditions during day and night. In our simulations, the stability classes are taken from the database used in Hanna et al. (2004) for Flame Acceleration Simulator (FLACS) evaluation, and the other experimental data are taken from the U.S. EPA database (http://www.epa.gov/scram001/dispersion_prefrec.htm).

c. Comparison with other Gaussian models

Many Gaussian models and a few computational fluid dynamics (CFD) models were evaluated with the Prairie Grass experiment. Three of the most used Gaussian models were compared in CERC (2007): ISCST3 and AERMOD are recommended by the U.S. EPA, and ADMS4 is developed in the United Kingdom and widely used in Europe. ISCST uses discrete categories to describe the atmospheric stability (Pasquill–Guifford parameterization for rural areas and Briggs’s formulas for urban area), whereas AERMOD and ADMS use boundary layer depth and Monin–Obukhov length to describe the state of the atmospheric boundary layer. The CFD models FLACS (Hanna et al. 2004) and Mercure (Demaël and Carissimo 2008) were also evaluated using Prairie Grass data.

Table 1 shows the Prairie Grass statistics for the Polyphemus plume model with several parameterizations, along with the aforementioned Gaussian models. Results for ADMS4, AERMOD, and ISCST3 come from CERC (2007). The statistics for these three models are computed with a subset from the usual set of 43 experiments (used for Polyphemus evaluation), hence the differences in the mean observed centerline values. The statistics presented here are computed for the centerlines of the five arcs. Most values in Table 1 are within the acceptable range defined in section 4a. AERMOD, ISCST3, and Polyphemus with the Briggs formulas show good results, as expected since the parameterizations they use for standard deviations are fitted on the Prairie Grass experiment. On the contrary, the results for Polyphemus with Doury’s formula and ADMS4 are hardly within the range of acceptable model performance for all indicators: ADMS4 underpredicts the mean concentrations by about 30%, and only 29% of the values predicted by the Doury model are within a factor 2 of the observations. Polyphemus with similarity theory shows very good results—there is almost no bias, and the correlation and NMSE are the best of all models presented here. A more detailed analysis of the results with Polyphemus Gaussian models is provided below.

d. Statistics for each arc

The model performance is now analyzed for the five monitoring arcs, so as to compare the behavior of the three standard deviation parameterizations. Figure 4 shows the statistical indicators, described in appendix B, computed on the centerline values for each arc separately for the 43 experiments. The “acceptable range” as defined in section 4a is delimited by the dashed lines. Most indicators show a decreasing performance at farther distances from the source, except the correlation. Whereas Briggs’s and Doury’s parameterizations underpredict the concentrations—this tendency increases with the distance (FB grows higher)—similarity theory overpredicts the concentrations, especially at the farthest arcs (400 and 800 m). This can be related to the small standard deviation values given by this parameterization, pointed out in section 3. At the first two arcs (50 and 100 m), similarity theory gives the best results for all indicators. Finally, the Doury parameterization shows a poor performance in terms of FAC2, as well as for NMSE and FB, for all distances. In comparison to these results, the FLACS results reported in Hanna et al. (2004) underestimated the concentrations at the shortest distances (up to 200 m), and slightly overpredicted the results at the farthest arcs. Demaël and Carissimo (2008) also found the underestimation tendency for the Mercure CFD model at short distances. They attributed it to the Eulerian dispersion, which corresponds to a plume behavior in σ ∝ t (with t the travel time), whereas the plume dispersion at short range should be in σ ∝ t. For C, D, and E stability classes, the overall performance of the Mercure CFD model was shown to be between that of the Briggs and Doury parameterizations.

Figure 5 shows the scatterplots for the three parameterizations. While the behavior of the model with the Briggs parameterization is quite well fitted to the observations, the overprediction tendency at the two further arcs is clearly shown for similarity theory. The scatterplot for the Doury parameterization confirms its poor performance, especially in terms of FAC2: the results for one arc are almost flat, which means that the modeled concentrations at one arc do not vary much with the different experiments. This could come from the representation of the atmosphere stability in only two classes (normal and low diffusion), which may be insufficient in some meteorological situations. Section 3 also showed that the Doury parameterization can give a very high value of σ_y relative to the other two parameterizations at a given distance, especially at low wind speeds.

a. Experiment and modeling setup

The Kincaid experiment took place at the Kincaid power plant in Illinois in 1980 and 1981. The plant stack is 187 m high with a diameter of 9 m and is surrounded by flat farmland and some lakes. Meteorological data came from an instrumented tower, and the conditions during the experiments ranged from neutral to convective. The tracer material was a buoyant plume of SF₆ gas, released from the stack and measured by monitors deployed on arcs between 0.5 and 50 km from the source at ground level. There were 200 monitors that were shifted day to day to correspond to the wind direction, and the arcs distances also varied with the meteorological situation. The recorded measurements are the arc maximum concentrations averaged over 1 h. The database contains 171 h of measurements. This dataset is often used to validate plume rise parameterizations because of the highly buoyant release that led to substantial plume rise. All data used for the following simulations is given with the Model Validation Kit (http://www.harmo.org/kit/Download.asp). In this dataset, a quality indicator was assigned to each measurement value, denoting how reliable the maximum concentration should be considered (Olesen 1995). Only the most reliable data, of quality 2 and 3, were used here. The statistics are computed on 165 experiments.

It has to be pointed out that in the following study the simulations with the Polyphemus Gaussian model use the observed inversion height when available and, for some cases, the predicted height. On the contrary, other models presented here like ADMS, AERMOD, and most likely Operationelle Meteorologiske Luftkvalitetsmodeller (OML) and HPDM recomputed the inversion height with their own preprocessor. This could lead to significant differences, since this height is of great importance to know whether the plume touches the ground or is trapped above the inversion layer, as explained in section 2. Simulations were carried out with three parameterizations for standard deviations (similarity theory, Briggs’s, and Doury’s) and three parameterizations for plume rise described in section 2. The chimney diameter as well as the initial plume spread due to plume rise was taken into account in the total standard deviations [Eq. (2)]. When similarity theory is used, the alternative HPDM formulas for elevated release detailed in appendix A are used. Relative to the usual similarity theory parameterization, it slightly increases the overestimation but it also improves the correlation coefficient, especially with Briggs–HPDM plume rise. The model evaluation consists, first, in comparing the results with the different parameterizations, with an emphasis on the plume rise description. Second, we compare the results with those of other models.

b. Model evaluation

c. Comparison with other Gaussian models

For this comparison, we use the results obtained with the Concawe parameterization that gives the best results for all standard deviation formulas. In addition, the results with similarity theory and Briggs–HPDM plume rise are also provided since they are good despite the lower correlation. Results for the other models come from CERC (2007), as in the case of Prairie Grass, for quality 2 and 3 (Table 6). As noted in section 5a, we used 165 experiments instead of 171, hence the slight differences in the observed mean and standard deviations. Results with Polyphemus models are between those of AERMOD and ADMS4, except for the correlations that are lower as already observed, and comparable to those of ISCST3.

Using data of quality 3 only, we can also compare with HPDM (United States; Hanna and Paine 1989), OML (Denmark), Simulation of Air Pollution from Emissions above Inhomogeneous Regions (SAFE_AIR; Italy; Canepa et al. 2000), as well as the Lagrangian model Numerical Atmospheric-Dispersion Modelling Environment (NAME; United Kingdom; Webster and Thomson 2002). The statistics for HPDM and OML come originally from Olesen (1995). Table 7 shows the results for all these models. ISCST3, AERMOD, and Polyphemus with the Doury formulas show a significant underestimation of the mean values, hence the high bias and NMSE. All other models slightly underestimate the mean values except SAFE_AIR and Polyphemus with similarity theory and Concawe plume rise, which slightly overpredicts the observations. The correlation results are surprisingly against this trend: for instance, AERMOD has a correlation of 40%, which is comparable to the best results. On the contrary, models that have an otherwise good performance such as SAFE_AIR, OML, and Polyphemus models with Concawe formulas have relatively low correlations. The correlation for Polyphemus with similarity theory and HPDM plume rise is very low (5%) but this configuration is among the best models for all other indicators. This correlation is not surprising since the HPDM model evaluation in Hanna and Paine (1989) also showed low correlations with this experiment. Further improvements were made in HPDM, such as a correction to take into account the plume lofting at the inversion layer during convective conditions, and this seems to have greatly improved the results. Table 7 also shows that the spread in the model outputs is very large: the predicted mean concentrations range from about 20 to 70 ng m⁻³ (g s⁻¹)⁻¹, which means more than a factor of 3 between some models. It is interesting to note that, with the different parameterizations available in Polyphemus for standard deviations and plume rise, we can reproduce the same spread, the model configuration being otherwise unchanged. Only the correlations are still low, no matter the formula used, which tends to prove that it comes from the shared inversion-height modeling (plume penetration and lofting).