Application of the Bessel function to compute the air pollutant with the stratification of the atmospheric

The Bessel differential equation with the Bessel function of solution has been applied. Bessel functions are the canonical solutions of Bessel's differential equation. Bessel's equation arises when finding separable solutions to Laplace's equation in cylindrical or spherical coordinates. Bessel functions are important for many problems of advection–diffusion progress and wave propagation. In this paper, authors present the analytic solutions of the atmospheric advection-diffusion equation with the stratification of the boundary condition. The solution has been found by applied the separation of variable method and Bessel’s equation.


INTRODUCTION
The air pollution modeling often leads to solving the general second order partial differential equations (PDE) [11]. The most commonly equation is steady state atmospheric advection -diffusion equation. The separation of variable method is used to solve the PDE. This method is simpler than the Green function method [8], [10], [9]. The atmospheric advection -diffusion equation is transformed to the Bessel equation, with the solution is Bessel function [1].
In this paper, the authors introduce the applications of Bessel equations to solve atmospheric advection -diffusion. The boundary conditions considering the factors of atmospheric stratification and divided into four main types: Dirichlet (total absorption), Neumann (total reflection), Mixed type I (reflections at the ground, absorption at inversion layer) and Mixed Type II (absorption at the ground, reflections at the inversion layer). This model uses Berliand's profile with the wind speed and diffusion coefficients are described by the power law functions [2], The Berliand's profile is closer to reality than the constants of wind speed and diffusion from Gauss plume model [3], [6]. In other hand, the separation of variable method simpler than the Green function method.

AIR PO LLUTANT MO DEL
The atmospheric advection -diffusion equation can be written as where x, y, z are coordinates in the alongwind, cross wind and vertical directions, C is the concentration of pollutant from the emission source located at the point (xs, ys, zs), U is the wind speed in downwind direction, Ky and Kz are eddy diffusivities in the crosswind and vertical directions respectively, S is the point source's function.
The point source's function can be described as Where Q is the source strength,  is the Dirac delta function [7].
The wind speed U and the eddy diffusivity Kz are depended on the height, which are given as [2]. The boundary conditions can be divided to four case as follows The solution of equation (1) is is the standard deviation in the crosswind direction [4].
By using the separation of variables in the , the solution of the equation (5) is given as Where  is the constant depend on the boundary conditions.
The solution of the equation (6) with the constant A, which depend on the boundary condition is given as The solution of the equation (7) depend on the boundary conditions of the atmospheric advection-diffusion equation. In this paper, the authors present the scheme to solve equation (7) with the Dirichlet boundary condition. The form of the Dirichlet boundary condition of the equation (7) is Setting a non-zero value of  , then transform variables as , the equation (7) becomes The equation (10) is the Bessel equation with the solution is given as And n  given as In other case of boundary condition, the solutions of the advection-diffusion equation can be found with similar schemes. The concentration of pollutant are obtained in follows table

NUMERICAL RESULT
To illustrate three-dimensional dispersion for a point source, the parameters of the model are setting as follows: The point source located at (xs= 10 m, ys = 0 m, z s = 50 m) with the strength Q= 10 mg/s.
The meteorological input parameters are taken from [5], [8]:     The concentration reach to maximum with z closed to s z = 50 m.
The resulting contour profiles in the Oxy plane are plotted in Fig 4. In this study, the analytic solution of equation (14) cannot be found. Therefor, the numerical method is used to approximate the solution. Fig 4 shows the equation (14) always have a solution.