A General Lagrangian Approach to Simulate Pollutant Dispersion in Atmosphere for Low-wind Condition

In this work we present a semi-analytical Lagrangian particle model to simu late the pollutant dispersion during low wind speed conditions. The model is based on a methodology, which solves the Langevin equation through the assumption that coefficient of the integrating factor is a complex function. The method leads to a non-linear stochastic integral equation, which is solved by the Method of Successive Approximations or Picard's Iterat ive Method. Taking into account the isomorphis m between the co mp lex and real p lane by writing down the low wind fo rmulat ion in polar form, the procedure allow to determine a formu la for the lo w wind direction. Furthermo re, an exp ression analogous to the Eulerian autocorrelation function suggested by Frenkiel(1) appears in the real co mponent solution. The model results present an improvement in relat ion to the other models and are shown to agree very well with the field tracer data collected during stable conditions at Idaho National Engineering Laboratory (INEL).


Introduction
Recent years have been seen the flowering of the work of searching analytical solution for the Langevin equation with the main purpose of simulat ing pollutant dispersion in the atmosphere. The mean ing of analyticity relies on the fact that no approximation is made in the derivatives or domain discretizat ion along the solution derivation. In this direction, appeared in the literature the works of Carvalho et al. [2][3], which solves the Langevin equation by the following steps: linearization of the Langevin equation and solution of the resultant stochastic integral equation by the Picard iterative scheme. Th is procedure leads to an analytical solution in each iterative step.
Carvalho and Vilhena [4] solved by this methodology the Langevin equation for low wind speed condition. In order to model the pollutant dispersion during meandering effect in the so lut ion , t he autho rs made the assu mpt ion th at the coefficient of the integrating factor of the first order linear differential equation is a co mplex function, the imag inary component models the low wind condition. Furthermore, the au th o rs cons id ered on ly the real co mp o n ent o f the integrating factor. At this point, it is relevant to mention that by this procedure, the Frenkiel [1] autocorrelation function naturally appears in the solution.
In this work we obtain a more general model, unlike the work of Carvalho and Vilhena [4], considering the real and imaginary parts of the co mplex function before performing the mu ltip lication of the integrating factor, exp ressed by the Eu ler formu la, inside and outside of the integral solution. Taking into account the isomorphism between the co mplex and real plane by writing down the lo w wind formu lation in polar form, the procedure allow to determine a formu la for the low wind direct ion. Furthermo re, an expression analogous to the Eulerian autocorrelation function suggested by Frenkiel [1] appears in the real co mponent solution. Finally, it is necessary to mention that when the non-dimensional quantity that controls the meandering oscillation frequency goes to zero this solution reduces to the solutions encountered by Carvalho et al. [2][3] for windy condition. The low wind speed data collected during stable conditions at Idaho National Engineering Laboratory (INEL) [5] has been used to evaluate the new model. The paper is outlined as follows: in section two the model is presented, in section three the modelling results are discussed and in section four the conclusions.

The Low Wind Model
The approach consists in the linearization of the Langevin equation as stochastic differential equation: which has the well known solution in terms of the integrating factor: In order to embody the low wind speed condition in the Langevin equation, it is assumed that U and ) (t f are complex functions written as: where u and v are the real and imaginary parts of U , respectively, and p and q are the real and imag inary parts of ) (t f , respectively. Therefore, the exponentials appearing in Equation (2) reads like: Applying the Euler formu la, Equation (2) Multiplying the Equation (6) Considering , we can write the Equation (7) as In order to determine the wind d irection we cast equation where by co mparison with Equation (3) Bearing in mind the isomorphis m between the comp lex and real planes, the low wind expression given by Equation (9) is described in the complex p lane. This procedure allows as determining the lo w wind direction, using polar form. For this end, we rewrite Equation (9) where θ is the low wind direct ion relat ive the x-axis Note that the real component of the Equation (9) where T is the time scale for a fu lly developed turbulence and m is a non-dimensional quantity that controls the meandering oscillation frequency. At this point, it is important to mention that when m goes to zero the Equation (9) reduces to the solution for windy conditions, which is written in terms of the exponential form of the autocorrelation function.
The Equation (9) is a non-linear stochastic integral equation, which must be solved iteratively. The method applied to solve the Equation (9) is the Method of Successive Approximations or Picard's Iteration Method [7], assuming that the initial guess for the iterative approximation is determined fro m a Gaussian distribution. For applications, the values for the parameters m and T have been calculated according to Carvalho

Modelling Results
The data utilized to evaluate the model performance are composed by a series of field experiments conducted under stable conditions in low winds over flat terrain. The tracer data were collected at Idaho National Engineering Laboratory (INEL) and the results are published in a U.S. National Oceanic and At mospheric Admin istration (NOAA) report [5].
For the simulat ions, the turbulent flow is assumed inhomogeneous only in the vertical and the transport is realized by the longitudinal co mponent of the mean wind velocity. The horizontal do main was determined according to sampler distances and the vertical domain was set equal to the observed PBL height. The t ime step was maintained Observed wind speeds were used to calculate the coefficients for the exponential wind profiles. The Monin-Obukhov length L and the friction velocity * u were appro ximated through the numerical best fit between the observed wind speeds and calculated wind profile suggested by Businger et al. [10]. To calculate h (the stable PBL height), the relat ion [11], where c f is the Coriolis parameter.

Conclusions
In this paper was presented a more general model to simu late the pollutant dispersion in meandering low wind conditions. The model was obtained by solving the Langevin equation through the integrating factor with its coefficient being a complex function. The method leads to a stochastic integral equation whose solution was obtained through the Method of Successive Approximat ions or Picard's Iteration Method. Taking into account the isomorphism between the complex and real p lane by writing down the low wind formulat ion in polar form, it was possible to determine a formula for the low wind direct ion. An expression analogous to the Eu lerian autocorrelat ion function for meandering conditions appeared in the real co mponent solution. The proposed method can be used to simulate de contaminant dispersion in meandering or non-meandering situations. The model was evaluated through the comparison with experimental data. The results obtained by the new model agree very well with the experimental data, indicating that it represents the dispersion process correctly in low wind speed conditions.