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EDDY DIFFUSIVITY PARAMETERIZATIONS AND IMPROVE FEATURES IN AN ANALYTICAL MULTYLAYER DISPERSION MODEL T. Tirabassi1, F. Baffo1 and D.M. Moreira2 Abstract. The advection-diffusion equation has been widely applied in operational atmospheric dispersion models to predict ground-level concentrations due to low and tall stacks emissions. Analytical solutions of equations are of fundamental importance in understanding and describing physical phenomena, since they are able to take into account all the parameters of a problem, and investigate their influence. Unfortunately, no general solution is known for equations describing the transport and dispersion of air pollution. There are some specific solutions based on particular assumptions concerning the turbulent state of the atmosphere, the best-known being the so-called Gaussian solution, where wind and dispersion parameters coefficients are constant with height. We present an improvement of an analytical model proposed by Vilhena et al. (1998). The model is based on a discretization of the Planetary Boundary Layer (PBL) in N sub-layers; in each sub-layer the advectiondiffusion equation is solved by the Laplace transform technique, considering an average value for eddy diffusivity and wind speed. Resumo. A equação de difusão-advecção tem sido amplamente aplicada em modelos de dispersão atmosférica para predizer concentrações ao nível do solo devido às emissões de fontes altas e baixas. Soluções analíticas destas equações são de fundamental importância no entendimento e descrição de fenômenos físicos, desde que elas são capazes de levar em conta todos os parâmetros de um problema e investigar sua influência. Infelizmente, nenhuma solução geral é conhecida para as equações que descrevem o transporte e dispersão de poluentes atmosféricos. Existem algumas soluções específicas baseadas em hipóteses particulares devido ao estado da atmosfera, bem como a conhecida solução Gaussiana, onde o vento e os parâmetros de dispersão são constantes com a altura. Apresentamos neste trabalho uma melhoria no modelo analítico proposto por Vilhena et al. (1998). O modelo é baseado na discretização da Camada Limite Planetária (CLP) em N sub-camadas; em cada subcamada a equação de difusão-advecção é resolvida pela técnica da Transformada de Laplace, considerando um valor médio para o coeficiente de difusão e velocidade do vento. Key words: air pollution modelling, solutions of advection-diffusion equation, eddy coefficients. ______________________________________________________ 1 – Institute ISAC of CNR, Via Gobetti 101, 40129 Bologna, Italy Phone: +39 051 6399601 Fax: +39 051 6399658 E.mail: t.tirabassi@ isac.cnr.it 2 – ULBRA, Av. Miguel Tostes, 101, Canoas, Brazil Phone: 51 477-9285 Fax: 51 477-1313 Email: davidson@ulbra.tche.br INTRODUCTION The Eulerian approach for modelling the statistical properties of concentrations of contaminants in a turbulent flow like the Planetary Boundary Layer (PBL) is widely used in the field of air pollution studies. One of the most commonly used dispersion equation closures is based on the gradient transport hypothesis which, in analogy to molecular diffusion, assumes that turbulence causes a net movement of material down the gradient of material concentration, at a rate that is proportional to the magnitude of the gradient (Pasquill and Smith, 1983). The simplicity of the approach (the so-called K-theory of turbulent diffusion) has led to the widespread use of this theory as the mathematical basis for simulating atmospheric dispersion. However, K-closure has its own limits. In contrast to molecular diffusion, turbulent diffusion is scale-dependent. This means that the diffusion rate of a cloud of material generally depends on the cloud dimensions and intensity of turbulence. As the cloud grows, larger eddies are incorporated in the expansion process, so that a progressively larger fraction of turbulent kinetic energy is available for the cloud expansion. However, eddies much larger than the cloud itself are relatively unimportant in its expansion. Thus, the gradient-transfer theory works well when the dimension of the dispersed material is much larger than the size of the turbulent eddies involved in the diffusion process, i.e. for ground-level releases and large travel times. Strictly speaking, one should introduce a diffusion coefficient function not only of atmospheric stability and release height, but also of travel time or distance from source; however, such time-dependence makes it difficult to treat the diffusion equation in a fixed-coordinate system, when multiple sources must be dealt with simultaneously. Otherwise, one should limit the application of the gradient theory to large travel times (Pasquill and Smith, 1983). A further problem is that the down-gradient transport hypothesis is inconsistent with the observed features of turbulent diffusion in the upper portion of the mixed layer, where countergradient material fluxes are known to occur (Deardoff and Willis, 1975). Despite these well known limits, the K-closure is widely used in several atmospheric conditions, because it describes the diffusive transport in an Eulerian framework where almost all measurements are Eulerian in character. It produces results that agree with experimental data as well as any more complex model, and it is not computationally expensive, as is the case of higher order closures. The reliability of the K-approach strongly depends on the way the eddy diffusivity is determined on the basis of the turbulence structure of the PBL, and on the model’s ability to reproduce experimental diffusion data. A great variety of formulations exist (Ulke, 2000). Bearing the K-theory limitations in mind, the main idea of the approach proposed is to obtain an eddy diffusivity scheme for practical applications in an analytical multilayer dispersion model, as well as a variable vertical discretization in order to represent better transport and diffusion phenomena at the ground, at the top of the boundary layer and near the source. THE MODEL The concentration turbulent fluxes are often assumed to be proportional to the mean concentration gradient. This assumption, along with the equation of continuity, leads to the advection-diffusion equation. For a Cartesian coordinate system, in which the x direction coincides with that one of the average wind, the steady state advection-diffusion equation is written as: U c K x c K y c K z c x x x y y z z (1) where c denotes the average concentration, U the mean wind speed in x direction and Kx , Ky and Kz are the eddy diffusivities. The cross-wind integration of the equation (1) (neglecting the longitudinal diffusion) leads to: U c y x z Kz y c (2) z subject to the boundary conditions of zero flux at the ground and at PBL top, and a source with emission rate Q at height H s : y U c ( 0 ,z)=Q(z-H s ) Kz c in x = 0 (3) y z 0 in z = 0, zi (4) where c y now represents the average cross-wind integrated concentration, and z i is the height of the PBL. Bearing in mind the dependence of the Kz coefficient and wind speed profile U on variable z, following Vilhena et al. (1998), the height z i of a PBL is discretized in N sub-intervals, in such a way that, within each interval, Kz(z) and U(z) assume the average value: zn K z (z)dz zn 1 (5) zn U z (z)dz z n z n 1 zn 1 (6) 1 Kn Un z n z n 1 1 Therefore the solution of problem (2) is reduced to the solution of N problems of the type: Un y cn K z x 2 z 2 y cn zn-1 z zn with (7) for n = 1: N, where cny denotes the concentration at the nth sub-interval. To determine the 2N integration constants, additional (2N-2) conditions, namely continuity of concentration and flux at interface, are considered: y y c n c n 1 n = 1,2,...(N-1) y y c n Kn z (8) K n 1 c n 1 n = 1,2,...(N-1) z (9) Applying the Laplace transform in equation (7) results: 2 z 2 UnS y U y y c n (s,z) c n (s,z)= c n ( 0 ,z) Kn Kn (10) in which c ny ( s, z ) L c ny ( x, z ); x s , which has the well-know solution: c n (s,z) = Ane Rn z Bne Rn z y Q R (z H s ) R (z H s ) (e n e n ) 2 Ra where Rn = UnS Kn and Ra = U n SKn (11) Finally, by applying the interface and boundary conditions a linear system for the integration constants is obtained. Henceforth, the concentration is obtained by inverting numerically the transformed concentration c y using Gaussian quadrature scheme: M c ny ( x, z ) A j . j 1 M c ny ( x, z ) A j . j 1 1 2 Pj . An exp x Pj U n .z Bn exp .z xK n xK n Pj U n (12) Pj U n Pj PjU n . An exp .z B n exp .z xK n x xK n Pj U n Q . exp ( z H s ) xK n P j K nU n x Pj U n exp ( z H s) xK n (13) Solution (12) is valid for layers that do not contain the contaminant source. At the same time, solution (13) can be used to evaluate the concentration field in the layer that contains the continuous source. Here, Aj and Pj are the weights and roots of the Gaussian quadrature scheme. In the present study, M=8 was considered, because this value provides the required accuracy with small computational effort. Obviously, the greater the number of layers (N), the more accurate the concentration pattern calculated, although the relative code running time is consequently greater. Moreover, the strata in which the PBL is divided are not constant in thickness. A more detailed description is required of wind and diffusion coefficients in proximity to the ground, where their gradients are high and more strongly influence pollutant dispersion. Therefore layers close to the terrestrial surface are assigned a smaller thickness than those located higher up. PARAMETERIZATION OF THE VERTICAL TURBULENT DIFFUSION COEFFICIENT The literature reports many, greatly varied formulae for the calculation of the vertical turbulent diffusion coefficient (Ulke, 2000). Some of them are presented here and will be used in the following section to assess the relative model performances: Formulae of Degrazia: employed throughout the PBL Unsatable condition L 0 (Degrazia et al. , 1997): 13 z z z K z 0.22 w* hz h 1 3 1 1 exp 4 0.0003 exp 8 zi zi zi (14) Stable condition L 0 (Degrazia et al., 2000; Degrazia et al. , 2001): Kz 0.31 z h u z 1 3.7 z (15) where z is height; zi the thickness of the stable layer; L1 z z i 5 4 , where L is the MoninObukhov length. Similarity formulation: employed only within the Surfer Layer (Panofsky and Dutton, 1988). Kz ku* z h z L (16) The function h is calculated with the formulae of Dyer: z Unstable conditions: z / L 0 h 1 16 L Neutral conditions: z / L 0 h 1 Stable conditions: z / L 0 h 1 5 1 2 z L Formulae of Lamb and Durran: employed throughout the PBL in unstable conditions (Seinfeld and Pandis, 1997). 43 z z k zz w z i 2.5 k 1 15 L zi 14 0 z 0.05 zi z z k zz w z i 0.021 0.408 1.351 zi zi z k zz w z i 0.2 exp 6 10 zi 2 z 4.096 zi 3 z 2.560 zi 4 0.05 z 0.6 zi 0 .6 z 1 .1 zi (17) z 1 .1 zi k zz w z i 0.0013 Formulae of Myrup and Ranzieri: employed throughout the PBL in neutral conditions (Seinfeld and Pandis, 1997). k zz ku z z 0 .1 zi z k zz ku z1.1 zi 0.1 k zz 0 z 1 .1 zi z 1.1 zi (18) Formula of Shir: employed throughout the PBL in neutral conditions (Seinfeld and Pandis, 1997). 8 fz k zz ku z exp u (19) Formulae of Lamb et al. : employed throughout the PBL in neutral conditions (Seinfeld and Pandis, 1997). 2 3 4 2 zf zf zf u 4 2 zf 7.396 *10 6.082 *10 2.532 12.72 15.17 k zz f u u u u k zz 0 for zf 0 0.45 u for zf 0.45 u (20) where f represents the Coriolis coefficient: f 1.46 *10 4 Formula of Businger and Arya: employed throughout the PBL in stable conditions (Seinfeld and Pandis, 1997). Kz 8 fz ku* z exp 0.74 4.7z L u* (21) Formulae of Troen and Mahrt: employed throughout the PBL (Pleim e Chang, 1992). Unstable conditions zi 10 : L z Stable or almost neutral conditions i 10 : L where: z h 1 16 L h 1 5 z L z k zz kw z1 zi z k zz ku z 1 zi 2 (22) h z L (23) 1 2 L0 L0 WIND PARAMETERIZATION The equations used by the model to calculate mean wind are those of similarity (Panofsky and Dutton, 1988): u u ka z z ln m L z0 (24) where, u* is the scale velocity relative to mechanical turbulence, k a the von Karman constant, and m the stability function expressed in Businger relations: z L m 4.7 z L for 1 x2 z 1 x ln m ln 2 arctan x 2 L 2 2 1 L0 2 z with x 1 15 L for 1 L0 14 The similarity expression is utilised of within the surfer layer. Alternatively, the wind speed profile can be described by a power law expressed as follows (Panofsky and Dutton, 1988): uz z u1 z1 n (25) where u z and u1 are the mean wind speeds horizontal to heights z and z1 and n is an exponent that is related to the intensity of turbolence (Irwin, 1979) MODEL EVALUATION AGAINST EXPERIMENTAL DATASETS The new parameterisations of the model have been evaluated using two experimental datasets with different emission and meteorological scenarios. The Copenhagen field campaign (Gryning and Lyck, 1984) took place in the suburbs of Copenhagen in 1978. A SF6 tracer was released without buoyancy from a tower at a height of 115m and collected at ground-level on arcs located 2000, 4000, and 6000 meters from the release point. The site was mainly residential with a roughness length, z 0 , of 0.6m. The meteorological conditions during the dispersion experiments ranged from moderately unstable to convective. The Prairie Grass dataset (Barad, 1958) is composed of dispersion data from a field experiment conducted in open country ( z 0 was 0.008m) during the summer of 1956 in O’Neill, Nebraska. Sulphur dioxide was released from a continuous point source at a height of 0.46m and collected at 5 arcs, 50, 100, 200, 400, 800 meters from the source. Here, we use a part of the values of the crosswind-integrated concentrations as calculated and reported by Van Ulden (1978). The two experiments cover a vast range of atmospheric turbulence. Figures 1-4 show the eddy coefficient vertical profiles for the various turbulence regimes found for the experimental datasets. One of the parameters taken into consideration in the evaluation of the model performance is mean wind. A comparison was made of the results obtained using three versions of the model. In the first, the wind profile was calculated with the similarity formulae within the surface layer, and was assumed constant above it. In the second, a power law was used to represent the wind profile throughout the entire PBL. In the third, the wind was calculated using the similarity within the surface layer and the power law above it. Table 1 presents some performances measures obtained by using the statistical evaluation procedure described by Hanna (1989) and defined in the following way, NMSE (normalized mean square) = (Co C p ) 2 / Co C p COR (correlation)= (Co Co )(C p C p ) / o p FA2 = fraction of Co values within a factor two of corresponding Cp values FB (fractional bias)= Co C p / 0.5 Co C p where the subscripts o and p refer respectively to observed and predicted quantities, and an overbar indicates an average. Table 1. Statistical indices for the tests of different expressions of mean wind. Wind profile NMSE COR FA2 FB Similarity 0.06 0.92 1.00 0.07 Power 0.13 0.91 1.00 0.22 Misto 0.12 0.91 1.00 0.20 Analysing the statistical indices in Table 1, it can be seen that there are no significant differences in the model performances using the different formulae for calculating mean wind, although the best result was obtained with the Similarity formulae (similarity profile in the surface layer and constant wind above) PROFILI CONVETTIVI Degrazia Troen e Mahrt Lamb e Durran 1 0.9 0.8 0.7 z / zi 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.05 0.1 0.15 Kz / w* zi Figure 1. Convective conditions. 0.2 0.25 PROFILI INSTABILI Degrazia Troen e Mahrt Lumb e Durran 1 0.9 0.8 0.7 z / zi 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.05 Figure 2. Unstable condictions. 0.1 0.15 Kz / w* zi 0.2 0.25 0.3 PROFILI NEUTRI Degrazia Similarità Troen e Mahrt Myrup e Ranzieri Shir 1 0.9 0.8 0.7 z / zi 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.05 0.1 Figure 3. Neutral condictions. 0.15 0.2 Kz / u * zi 0.25 0.3 0.35 0.4 PROFILI STABILI Degrazia Similarità Troen e Mahrt Businger ed Arya 1 0.9 0.8 0.7 z / zi 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.01 0.02 Kz / u * zi Figure 4. Stable condictions. 0.03 The values of the statistical indices calculated for the diverse parameterizations of the eddy diffusion coefficients with the data of Copenhagen are reported in Table 2. Tabel 2. Statistical indices for different K z profiles with Copenhagen data set. Kz profile NMSE COR FA2 FB Degrazia et al. 0.06 0.91 1.00 0.07 Similarity 0.08 0.87 1.00 0.07 Troen and Mart 0.09 0.91 1.00 0.16 Myrup and Ranzieri 0.23 0.69 0.78 0.21 Shir 0.23 0.70 0.78 0.22 Lamb et al. 0.33 0.51 0.78 0.23 12 10 Cp Q-1 10-4 (sm-2 ) 8 6 4 Degrazia Similarità Troen e Mahrt Myrup e Ranzieri Shir Lamb et. al 2 0 0 2 4 6 Co Q-1 10-4 (sm-2 ) 8 10 12 Figure 5. Copenhagen dataset. Scatter plot of observed (Co) versus predicted (Cp) crosswindintegrated concentrations normalized with the emission source rate. Points between dashed lines are in a factor of two Figure 5 shows a comparison between the data calculated by the model and experimental data. The points between the dashed lines are in a factor of two. The values of the statistical indices calculated for the different parameterizations of the eddy diffusion coefficient with the Prairie Grass data are reported in Table 3, while Figure 6 shows the comparison between the data calculated by the model and experimental data 200 180 160 Cp Q-1 10-3 (sm-2 ) 140 120 100 80 60 40 Degrazia Similarità Troen e Mahrt Myrup e Ranzieri Shir Lamb et. al 20 0 0 20 40 60 100 120 80 Co Q-1 10-3 (s m-2) 140 160 180 200 Figure 6. Prairie Grass dataset. Scatter plot of observed (Co) versus predicted (Cp) crosswindintegrated concentrations normalized with the emission source rate. Points between dashed lines are in a factor of two Table 3. Statistical indices for different K z profiles with Prairie Grass data set Kz profile NMSE COR FA2 FB Degrazia et al. 0.07 0.94 0.97 -0.13 Similarity 0.06 0.95 0.97 -0.06 Troen and Mart 0.10 0.93 0.86 -0.15 Myrup and Ranzieri 0.06 0.95 0.97 -0.04 Shir 0.06 0.95 0.97 -0.04 Lamb et al. 0.07 0.95 0.97 0.02 CONCLUSIONS The parameters considered for the evaluation of the model performances are: mean wind and the vertical turbulent diffusion coefficient. Different expressions of the vertical turbulent diffusion coefficient were introduced, together with different expressions for the calculation of mean wind, available in the literature. A comparison was made of the concentrations measured and calculated by the model. This was done through widely used indices for the evaluation of model performances. On the basis of the values of the indices, it is possible to affirm that, in the case of high emissions in moderately unstable to convective meteorological conditions, i.e. similar to those of the Copenhagen campaign, the best performances are obtained using the formulae of Degrazia, Similarity and Troen and Mahrt, for the calculation of the vertical turbulent diffusion coefficient. Conversely, in the case of ground-level emissions in meteorological conditions from convective to stable, i.e. similar to those of the Prairie Grass campaign, the best model performances are obtained using the formulae of Lamb and Durran for unstable cases, those of Businger and Arya for stable cases, and those of Myrup and Ranzieri, Shir and Lamb et al. for neutral cases. However, the performances obtained with the formulae of Degrazia and of Similarity are also good. As far as the wind profile is concerned, there are non significant differences in the model performances using the diverse formulae for the calculation of wind, although slightly better performances are obtained using the Similarity formulae within the surface layer and considering wind above the said layer. In all cases, the values of the statistical indices, both in the sensitivity analysis and assessment of model performances varying the characteristic parameters, in diverse turbulence regimes, turn out to be good, when compared with those of other models available in the literature (Olesen, 1995). Such remarks lead to the conclusion that the proposed approach could be used in an operative model of pollutant dispersion into the atmosphere, as a tool for the evaluation and management of air quality. Ackowledgements The authors thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) and FAPERGS (Fundação de Amparo à Pesquisa do Estado do Rio Grande do Sul) for the partial financial support of this work. REFERENCES Barad, M.L, 1958. Project Prairie Grass. Geophysical Research Paper, No. 59, Vols. I and II, GRD, Bedford, MA. Deardoff, J.W., Willis G.E., 1975. A parameterization of diffusion into the mixed layer. Journal of Applied Meteorology 14, 1451-1458. Degrazia G.A., Anfossi D., Carvalho J.C., Mangia C., Tirabassi T., Campos Velho H.F., 2000. Turbulence parameterisation for PBL dispersion models in all stability conditions. Atmospheric Environment 34, 3575-3583. Degrazia G.A., Moreira, D.M., Vilhena, M.T., 2001. Derivation of an eddy diffusivity depending on source distance for vertically inhomogeneous turbulence in a convective boundary layer. Journal of Applied Meteorology., 40, 12331240. Degrazia, G.A., Rizza U., Mangia C., Tirabassi T., 1997. 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