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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The Advection-Diffusion Equation is solved for a constant pollutant emission from a point-like source placed inside an unstable Atmospheric Boundary Layer. The solution is obtained adopting the novel analytical approach: Generalized Integral Laplace Transform Technique. The concentration solution of the equation is expressed through an infinite series expansion. After setting a realistic scenario through the wind and diffusivity parameterizations, the Ground Level Concentration (GLC) is determined, and an explicit approximate expression is provided for it, allowing an analytically simple expression for the position and value of the maximum. Remarks arise regarding the ability to express value and position of the GLC as explicit functions of the parameters defining the Atmospheric Boundary Layer scenario and the source height.

Irreversible consequences of air pollution in the Atmospheric Boundary Layer (ABL) and instances of environmental accidents or even catastrophes demand increasing real time environmental monitoring and control as a routine instrument. In order to evaluate such scenarios one needs efficient procedures, which yield immediate results, for instance evaluating the ground level concentration of pollutants, and especially the maximum concentration and its position. Numerical simulation approaches may in fact still be too slow to provide a map of concentrations in real time, when immediate decisions are necessary. However, analytical solutions for theoretical models are independent of a specific situation and function by parameter estimation. The computational evaluation of numerical data of the concentration field or for a set of positions is an instant task. In view of this, the current work presents a derivation of compact phenomenological formula extracted from the analytical GILTT (Generalized Integral Laplace Transform Technique) [

The analytical solution of the Advection-Diffusion Equation (ADE) has been performed following different approaches based on Gaussian and non-Gaussian solutions. Gaussian solutions represent a rather easy operative tool to handle. Non-Gaussian analytical solutions represent a more realistic approach to represent atmospheric diffusion. However, solutions using non-Gaussian approaches are much harder to achieve, and are often restricted only to rather simple parameterization profiles. A short review in analytically solving the ADE is provided.

A two-dimensional (2-D) steady-state solution of the ADE is shown by [_{z}

Another 2-D solution has been worked out by Smith [_{z}_{z}_{z}

Scriven and Fisher [_{z}_{z}

Demuth [

Nieuwstadt [

Koch [

In the work [

Due to the limitedness of generality and to the increasing development of Large Eddy Simulation (LES) models, analytical approaches to solve the ADE have been largely ignored. In this paper, a complete and coherent analytical solution of the ADE is presented. The solution is based on the GILTT method [

The two dimensional steady-state ADE for an emitting point-like source in a stationary ABL reads:

Where, along the

The horizontal wind _{z}_{s}_{s}

The GILTT technique provides a solution for _{i}_{i}_{N}

The choice of the turbulent parameterization is set to account for the dynamic processes occurring in the ABL. In the following, we restrict our discussion to simple vertical profiles of wind and eddy diffusivity still a reasonably realistic, but more specifically for an unstable regime. For an extension including stable regimens we refer to a future work. The choice of the vertical profile for the wind _{1} is the mean wind velocity at the height _{1}, while _{1}(0.01^{−1}; these values are quite consistent with the whole range of unstable regimes [

The vertical diffusivity parameterization is chosen according to reference [_{*} is the convective scaling parameter related to the Monin-Obukhov length _{MO}_{*} as:
_{MO}_{*}_{0} is the roughness (10^{−5}

The chosen profiles ensure simple functions whilst maintaining rather realistic horizontal wind _{z}

From the solution of the ADE, the Ground Level Concentration (GLC) is obtained after setting

If we consider the definition of

It would be redundant to compare the GILTT results with experimental data as outcomes have already been extensively reported in the literature [_{MGLC}_{M}_{M}_{i}

Based on these facts, and bearing in mind the Gaussian solution and the GLC obtained with power low profile of wind and eddy diffusivity, the dimensionless GLC defined in

Due to the negative values assumed by the Monin-Obukhov length, it will be defined in the following calculations as the positive dimensionless parameter _{MO}_{MO} / h_{s}_{s}/h

_{s}_{1} and the convection scaling parameter _{*} (it compares in _{MO}_{*} by the relationship (7).

From the explicit approximation for _{GLC}

Finally, putting _{M}_{MGLC}_{M}

Two considerations are important here. Firstly, the expression for the position _{M}_{M}_{s}_{s} >

_{s}_{s}_{sl}_{s}_{s}_{M0}_{s}_{s}

In _{s}_{s}_{M0}_{M0}

_{s}_{s}_{s} ≤_{GLC}

_{MGLC}_{M}_{M}_{s}_{MO}_{s}_{M}

A final remark should be made in regard to

Note that for the three-dimensional case this is no longer true. It is evident that diffusive parameters do not play a part and it confirms that turbulence has the only effect that determines the distance where maximum GLC occurs. The results shown above can be generalized (see

The results presented in this paper show the possibility of expressing the GLC from an emitting point-like source in a steady convective ABL by a compact analytical expression. The function was determined analyzing the behavior of the series expansion provided by the GILTT solution, the predictive power of which has been extensively demonstrated in the literature when applied to several experimental data sets. Despite the simplifications due to restricting only to unstable ABL regimes, the analysis allows a high level of understanding of the form of the ground level concentration.

The main progress worth emphasizing is the following: for a function given in

From the operative point of view,

GILTT (Generalized Integral Laplace Transform Technique) Ground Level Concentration (GLC) _{s}_{MO}

GLC _{s}_{s}_{s}_{MO}_{MO}

GLC _{s}_{s}_{s}_{MO}_{MO}

Position of the maximum GLC _{s}_{s} / h

Value of the maximum GLC _{s}_{s} / h

Ground level concentration _{s}_{s} / h

The authors thank Brazilian CNPq, Italian CNR and ENVIREN for the partial financial support of this work.