Analytical solutions of the atmospheric diffusion equation with multiple sources and height-dependent wind speed and eddy diffusivities

Three-dimensional analytical solutions of the atmospheric diffusion equation with multiple sources and height-dependent wind speed and eddy diffusivities are derived in a systematic fashion. For homogeneous Neumann (total reflection), Dirichlet (total adsorption), or mixed boundary conditions, the solutions for a single source are comprised of three components: a source strength, a crosswind dispersion factor, and a vertical dispersion factor. The two dispersion factors together constitute a Greenʹs function—the concentration response due to a unit disturbance (source). When the general point source Greenʹs functions are derived for a bounded domain (inversion effect) with various boundary conditions and arbitrary power-law profiles for wind speed and eddy diffusivities, previously published equations are found to be simplified versions of this more general case. A methodology based on the superposition of Greenʹs functions is proposed, which enables the estimation of ambient concentrations not only from a single source, but also from multiple point, line, or area releases.