M.T. Vilhena

A General Analytical Solution of the Advection–Diffusion Equation for Fickian Closure

In the last few years there has been increased research interest in searching for analytical solutions for the advection–diffusion equation (ADE). By analytical we mean that no approximation is done along the derivation of the solution. There exists a significant literature regarding this theme. For illustration we mention the works of (Rounds 1955; Smith 1957; Scriven, Fisher 1975; Demuth 1978; van Ulden 1978; Nieuwstadt, de Haan 1981; Tagliazucca et al. 1985; Tirabassi 1989; Tirabassi, Rizza 1994; Sharan et al. 1996; Lin, Hildemann 1997; Tirabassi 2003). We note that in these works all solutions are valid for very specialized problems having specific wind and eddy diffusivities vertical profiles. Further, also in the literature there is the ADMM (Advection Diffusion Multilayer Method) approach which solves the two-dimensional ADE with variable wind profile and eddy diffusivity coefficient (Moreira et al. 2006). 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 transformation technique. For more details about this methodology see the review work done by (Moreira et al. 2006). We are also aware of the recent work of (Costa et al. 2006), dubbed as GIADMT method (Generalized Integral Advection Diffusion Multilayer Technique), which presented a general solution for the time-dependent three-dimensional ADE, again assuming the stepwise approximation for the eddy diffusivity coefficient and wind profile and proceeding further in similar way according the previous work. To avoid this approximation, in this work we report an analytical general solution for this problem, assuming that the eddy diffusivity coefficient and wind profile are arbitrary functions having a continuous dependence on the vertical and longitudinal variables. Without losing generality we specialize the application in micrometeorology, specially for the problem of simulation of contaminant releasing in the ABL.

A Closed-Form Formulation for Pollutant Dispersion in the Atmosphere

Transport and diffusion models of air pollution are based either on simple techniques, such as the Gaussian approach, or on more complex algorithms, such as the K-theory differential equation. The Gaussian equation is an easy and fast method, which, however, cannot properly simulate complex nonhomogeneous conditions. The K-theory can accept virtually any complex meteorological input, but generally requires numerical integration, which is computationally expensive and is often affected by large numerical advection errors. Conversely, Gaussian models are fast, simple, do not require complex meteorological input, and describe the diffusive transport in an Eulerian framework, making easy use of the Eulerian nature of measurements.

An Analytical Solution for the Transient Two-Dimensional Advection–Diffusion Equation with Non-Fickian Closure in Cartesian Geometry by the Generalized Integral Transform Technique

Analytical solutions of equations are of fundamental importance in understanding and describing physical phenomena, since they are able to take into account all the parameters of a problem and investigate their influence. In a recent work, [Bus07] reported an analytical solution for the stationary two-dimensional advection–diffusion equation with Fickian closure by the Generalized Integral Laplace Transform Technique (GILTT). The main idea of this method consists of: construction of an auxiliary Sturm–Liouville problem, expansion of the contaminant concentration in a series in terms of the obtained eigenfunctions, replacement of the expansion in the original equation, and finally after taking moments, resulting a set of ordinary differential equations which are then solved analytically by the Laplace transform technique.