A unified analytical solution of the steady-state atmospheric diffusion equation

A unified analytical solution of the steady-state atmospheric diffusion equation for a finite and semi-infinite/infinite media was developed using the classic integral transform technique (CITT) which is based on a systematized method of separation of variable.The solution was obtained considering an arbitrary mean wind velocity depending on the vertical coordinate (z) and a generalized separable functional form for the eddy diffusivities in terms of the longitudinal (x) and vertical coordinates (z).The examples described in this article show that the well known closed-form analytical solutions, available in the literature, for both finite and semi-infinite/infinite media are special cases of the present unified analytical solution. As an example of the strength of the developed methodology, the Copenhagen and Prairie Grass experiments were simulated (finite media with the mean wind speed and the turbulent diffusion coefficient described by different functional forms). The results indicate that the present solutions are in good agreement with those obtained using other analytical procedures, previously published in the literature. It is important to note that the eigenvalue problem is associated directly to the atmospheric diffusion equation making possible the development of the unified analytical solution and also resulting in the improvement of the convergence behavior in the series of the eigenfunction-expansion.

► The three-dimensional steady-state atmospheric diffusion equation was solved analytically using the integral transform.
► The solution includes a generalized functional form for the mean wind speed and the eddy diffusivities.
► The solution for a finite domain was extended to semi-infinite and infinite domains using a unified procedure.
► The associated eigenvalue problem improves the convergence of the solution.
► The unified analytical solution showed to be robust when evaluated against literature results.