Short communication
Analytic modeling of two-dimensional transient atmospheric pollutant dispersion by double GITT and Laplace Transform techniques

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Abstract

An analytical solution for the transient two-dimensional atmospheric pollutant dispersion problem is presented. The approach used in this problem utilizes the double GITT (Generalized Integral Transform Technique), the Laplace Transform and the matrix diagonalization. Furthermore, mathematical filters are used due to the existence of non-homogeneous boundary conditions. The results we obtained are compared with experimental data for short range downwind dispersion utilizing two well-known experimental dispersion datasets (Copenhagen and Prairie Grass). It is shown that the present analytical approach give good results for downwind concentration except for receptors very close to the release points were any Eulerian approach based on K-closure is known to fail.

Introduction

The use of partial differential equations to formulate problems that represent physical phenomena is well known, especially in modeling heat and mass transfer problems and particularly in the air-pollution context (Nieuwstadt and van Dop, 1982, Seinfeld, 1986, Blackadar, 1997). Analytical solutions of systems of partial differential equations that describe atmospheric transport and dispersion of air pollution are available since the 1920s (Roberts, 1923). They are of fundamental importance in understanding and describing physical phenomena because they explicitly take into account all parameters of a problem. One analytical technique that has been recently used is the Generalized Integral Transform Technique (GITT) (Cotta and Mikhailov, 1997) associated with Laplace Transform and matrix diagonalization.

The GITT may be described as an integral transformation associated with a series expansion used to solve partial differential equations. In this work, we employ one of the recent advances of this technique: the analytical solution of the transformed problem by the Laplace Transform and matrix diagonalization (Wortmann et al., 2005b).

One of the main characteristics of the GITT technique is its ability to control the error of the results. Moreover, it is worth noting that this is an analytic technique, i.e., no approximation is made in deriving the solution. In this context, it is possible to get a certain confidence that the analytical solution is an exact solution except for round-off error.

Its first application in the Planetary-Boundary Layer (PBL) air-pollution context was presented by Wortmann et al. (2005a), in the case of a stationary two-dimensional advection–diffusion equation.

They also employed the Laplace Transform to solve analytically the resulting transformed ordinary differential equations' system. In that work the authors did not considered the vertical advection and longitudinal diffusion coefficients while the other coefficients were parameterized in terms of the vertical height.

The application of the GITT in the case of non-homogeneous boundary conditions requires the use of mathematical filters (Cotta and Mikhailov, 1997) to transform the problem into a non-homogeneous case with homogeneous boundary conditions. Here we have considered a filter expression obtained as solution of a diffusive–advective steady-state two-dimensional problem with constant coefficients and the same boundary conditions as the main problem.

Recently, Moreira et al. (2006) have proposed a hybrid technique for the unsteady two-dimensional advection–diffusion equation utilizing the GITT method with the Laplace inversion obtained numerically through the Gaussian quadrature scheme. In that work the longitudinal diffusion coefficient was not considered.

Sharan and Modani (2005) described a steady-state mathematical model by solving analytically the three-dimensional advection–diffusion equation using methods of eigenfunction expansion and Fourier Transforms. In that work the vertical wind velocity is neglected, and the horizontal wind velocity and the eddy diffusion coefficients are assumed constant. Sharan and Modani (2006) also described an analytical solution for the steady two-dimensional advection–diffusion equation, in which the vertical eddy diffusion coefficient is parameterized in terms of downwind distance from the source, for applications in low-wind conditions.

In order to overcome the mathematical (and physical) difficulties that arise in the analytical treatment of the advection–diffusion equation, we propose a mathematical model that includes all diffusion and advection terms of the two-dimensional transient advection–diffusion equation and all their, respectively, variable coefficients. The solution is obtained using double GITT, Laplace Transform and matrix diagonalization.

This is a first step toward the more general problem of studying the analytical solution of the fully three-dimensional diffusion equation for the atmospheric pollutants, with coefficients (winds and eddy diffusivities) varying in space and time, as obtained, for example, from a grid-meteorological model. The coupling between meteorological and dispersion models, together with the methodologies used to solve the original set of equations, is a very delicate task, and represents the state of the art in air-pollution modeling. All methodologies so far employed are based on numerical algorithms (Grell et al., 2000, Grell et al., 2005).

This work is organized as follows: in Section 2, we introduce the mathematical model; in Section 3, we describe the mathematical filter; in Section 4, we discuss the application of double GITT to the two-dimensional advection–diffusion equation; Section 5 is devoted to the boundary layer parameterization; in Section 6 we report the steadiness tests of the proposed solutions and the comparison with an experimental dataset; in Section 7, we discuss the main results and the conclusions that can be drawn after this work.

Section snippets

The mathematical model

Let us consider the crosswind-integrated transient advective–diffusive problem represented by the equationC(x,z,t)t+u(x,z)C(x,z,t)x+w(x,z)C(x,z,t)z=x(Kx(x,z)C(x,z,t)x)+z(Kz(x,z)C(x,z,t)z)0zh;0x<;0t<,where C(x,z,t) is the contaminant concentration, u(x,z) and w(x,z) are the longitudinal and vertical wind velocity, respectively, Kx(x,z) and Kz(x,z) are the eddy diffusion coefficients and h is the atmospheric boundary layer height. The initial and boundary conditions are

The mathematical filter

Eq. (1), provided with boundary conditions expressed by Eqs. (1a), (1b), (1c), (1d), represents a classical mathematical problem with non-homogeneous boundary conditions due to the presence of u(x,z) in the denominator. On the other hand, by keeping all advective/diffusive coefficients and letting them vary in space, we are describing the most general diffusion phenomena without introducing any approximations into the governing equations.

A mathematical problem with non-homogeneous boundary

The application of the double GITT

Since the original non-homogeneous problem expressed by Eqs. (1) has been turned into a problem with homogeneous boundary conditions described by Eq. (18), it is straightforward now to apply the GITT. This is done by first considering the vertical direction.

It is important to point out that the application of CITT in the problem (2) is equivalent to the GITT application of the problem (18). Thus, both procedures have the same auxiliary problem (3a), (3b), eigenfunctions (4a), eigenvalues (4b),

Boundary layer parameterization

In this work, we have used two parameterizations for eddy diffusivity coefficient. The first parameterization is derived from Taylor theory (Taylor, 1921) as proposed by Degrazia (Degrazia et al., 1997) isKz(x,z)wh=0.22(zh)1/3(1zh)1/3[1exp(4zh)0.0003exp(8zh)]

The turbulent vertical exchange coefficients expressed by Eq. (26) represent a non-local closure explicitly describing the fact that the energy-containing eddies are scaled by the convective velocity scale and by the height of the PBL.

Experimental data and results

We have first evaluated the stability of the proposed algorithm, then we have compared the ground-level concentration with a measured concentration taken from literature.

Conclusions

In this work an analytical approach for the two-dimensional transient pollutant dispersion problem has been proposed. This method utilizes a double GITT along the space variables and a Laplace Transform in time. In this way, since the procedure is purely analytical, a full control of the error in the solution is guaranteed.

From the physical point of view, we have retained all the coefficients (advective/diffusive) in the two-dimensional transient dispersion equation. This allows to model the

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