An Analytical Formulation for Pollutant Dispersion Simulation in the Atmospheric Boundary Layer

In this work we present the solution of the two-dimensional advection-diffusion equation by the GILTT method. The GILTT approach uses, in the series expansion, eigenfunctions given in terms of cosine functions. Here, a different expansion for the solution of the advection-diffusion equation will be explored. In other words, a Sturm-Liouville problem carrying more information of the original problem is considered, given by Bessel functions. Numerical simulations and comparisons with experimental data are presented.


Introduction
The advection-diffusion equation has long been used to describe the dispersion of contaminants in the atmosphere [1]. Efforts have been made over the years to obtain analytical solutions of this equation in order to modeling air pollution. According to [2,3], these solutions are valid for very specialized practical situations, and in majority with restrictions on wind and eddy diffusivities vertical profiles [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. To solve the advection-diffusion equation for more realistic physical scenario appeared in the literature the ADMM (Advection Diffusion Multilayer Method) approach [19,20], valid for any eddy diffusivity and wind profile depending on the height. The main idea relies on the discretization of the Atmospheric Boundary Layer (ABL) in a multilayer domain, assuming in each layer that the eddy diffusivity and wind profile take averaged values. The resulting advection-diffusion equation in each layer is then solved by the Laplace Transform technique. A more general methodology, which skips the multilayer discretisation of the height z appearing in the ADMM approach, is known in the literature as GILTT (Generalized Integral Laplace Transform Technique) approach [2,3,21]. The main idea of this methodology relies on the expansion of the pollutant concentration in series of eigenfunctions attained from an auxiliary Sturm-Liouville problem, replacement of this equation in the advection-diffusion equation and taking moments. The procedure results a matrix ordinary differential equation which is solved analytically by the Laplace Transform technique. Similar solutions were proposed by [22,23].
To reach our objective, we begin presenting the solution of the two-dimensional advection-diffusion equation in Cartesian geometry by the GILTT approach [2], considering that the eddy diffusivity and the vertical wind profile depend on the z variable. Traditionally, the GILTT approach uses as basis eigenfunctions given in terms of cosine functions. Here, another Sturm-Liouville problem will be considered, carrying more information of the original problem. In this case, the eigenfunctions are given by Bessel functions. Once we construct the general solution, numerical simulations and future perspectives of this methodology are presented.

The Advection-Diffusion Equation and the GILTT Method
For a Cartesian coordinate system the advection-diffusion equation, using first order closure of turbulence, is written like [24]: where c denotes the average concentration of a passive (g/m 3 ), u , v , w are the mean wind (m/s) components along the axis x, y and z, respectively and S is the source term. K x , K y , K z are the Cartesian components of eddy diffusivity (m 2 /s) in the x, y and z directions, respectively. In the first order closure all the information on the turbulence complexity is contained in the eddy diffusivities. Problem (1) is solved analytically by the 3D-GILTT method [3,21,25]. Here, for comparison with experimental data we will assume for the advection-diffusion Equation (1): stationary conditions, crosswind integrated concentrations and that the advection is much higher than the diffusion in the x-direction. After the simplifications, let us consider the problem: which has Bessel functions of first specie and order zero as solution  are the positive roots of the Bessel function of first specie and order one, 1 J . Problem (5) carries more information from the original problem than the previous one.
To determine the unknown coefficient   n c x we replace Equation (3) in Equation (1). Applying the integral operator , we come out with the result: which in matrix form reads like: where is the diagonal matrix with elements , D is the diagonal matrix of eigenvalues n of the matrix F, X is the matrix of the respective eigenfunctions and X −1 it is the inverse.
Therefore, the solution for the concentration given by Equation (3) is now well determined once the vector   n c x is known and given by Equation (9).
The solution of the problem (2) using in the series expansion (3) eigenfunctions given in terms of cosine and Bessel functions will be called here as GILTTC and GILTTB, respectively.

Numerical Results
The performance of the discussed solution was evaluated against experimental ground-level concentration using different dispersion experiments available in the literature. Below we briefly discuss the Copenhagen, Prairie-Grass and Hanford dispersion experiments, which allow us to validate the results encountered by the mentioned solutions.
The Copenhagen field campaign took place in the suburbs of Copenhagen in 1978, and is described by [26]. It consisted of tracer released without buoyancy from a tower at a height of 115 m, and collection of tracer sampling units at the ground-level positions at the maximum of three crosswind arcs. The sampling units were positioned at two to six kilometers from the point of release. The site was mainly residential with a roughness length of the 0.6 m. The meteorological conditions during the dispersion experiments ranged from moderately unstable to convective. Table 1 shows a summary of meteorological conditions during the Copenhagen experiments.
In the Prairie-Grass experiment, according [27], the tracer SO 2 was released without buoyancy at a height of 0.46 m, and collected at a height of 1.5 m at five downwind distances (50, 100, 200, 400 and 800 m) at O'Neill,  Nebraska in 1956. The Prairie Grass site was quite flat and much smooth with a roughness length of 0.6 cm.
Here we consider the experimental data appearing in the paper [28]. Table 2 summaries the meteorological conditions during the Prairie-Grass experiments. The Hanford diffusion experiment was conducted in May-June, 1983, on a semi-arid region of south eastern Washington on generally flat terrain. The detailed description of the experiment was provided by [29]. Data were obtained from six dual-tracer releases located at 100, 200, 800, 1600 and 3200 m from the source during moderately stable to near-neutral conditions. The release height of SF 6 was 2 m and average release rate was around 0.3 g/s. The pollutant was collected at a height of 1.5 m. The terrain was considered as an urban terrain with roughness length of 3 cm. The values of ABL parameters are given in Table 3.
The choice of the turbulent parameterization represents a fundamental aspect for pollutant dispersion modeling. In terms of the convective scaling parameters, the while for stable conditions [31]: where z is height; h is the thickness of the ABL; * is the convective velocity scale; ; L is the Monin-Obukhov length and is the friction velocity. * In our simulations, we use the wind speed profile described by a power law, according [32], where z u and 1 u are the mean wind velocity respectively at the heights z and 1 , while z  is an exponent that is related to the intensity of turbulence [33].  Table 4 point out that a good agreement is obtained between experimental data and the GILTT method for both cosine and Bessel basis, regarding the NMSE, FB and FS values relatively near to zero and COR relatively near to 1. At this point, we can affirm that no significant difference between the models was observed for the high source of the Copenhagen experiment. Table 5 shows the performance of the solution for the Prairie-Grass experiment. The statistical indices of the table point out that a reasonable agreement is obtained between experimental data and the GILTT method. It is important to notice that the GILTTB numerically converges faster than GILTTC (while GILTTB needs 100 eigenvalues, GILTTC needs 300 eigenvalues to reach a   . The statistical indices of Table 6 point out that a good agreement is obtained between experimental data and models. Again, the GILTTC need more eigenvalues to reach a similar numerical result obtained with the GILTTB.
In the following are presented in Tables 7-9 the numerical comparisons of the GILTT method results against the experimental data of Copenhagen, Prairie-Grass and Hanford experiments.
Furthermore, Figures 1-3 show the observed and predicted scatter diagram of crosswind ground-level concentrations for the three experiments considered in this work. In the graphics the symbol represents the GILTTC, Table 7. Observed and predicted crosswind-integrated concentrations C/Q (10 −4 sm −2 ) at the Copenhagen experiment.    Table 9. Observed and predicted crosswind-integrated concentrations C/Q (10 −3 sm −2 ) at Hanford experiment.    and lines the GILTTB solution.
In all the tables and figures, we considered the data numerically converged for both bases. In this respect, it is important to note that the model simulates quite well the observed concentration for all the cases. The greatest difference between the models is seen for the Prairie-Grass experiment.

Conclusions
Focusing our attention on the pollution dispersion simulation in atmosphere, we present an analytical solution in series expansion given by the well-known GILTT method to solve the two-dimensional advection-diffusion equation by the GILTT approach. A Sturm-Liouville problem carrying more information of the original problem, given by Bessel functions, was considered, For the problems discussed, we promptly realize the very good results achieved, under statistical point of view, by the GILTT method when compared with the experimental data for both cosine and Bessel basis used. For the case of high source no significant difference was observed between GILTTC and GILTTB. However, for the low source, GILTTB numerically converges faster than GILTTC. We focus our future attention on the direction of the generalization of this solution considering an infinite boundary layer.