Abstract
Hankel transform was employed to solve the two-dimensional steady state advection–diffusion equation considering a continuous point source with vertical eddy diffusivity as a power law of vertical height and downwind distance, also, taking wind speed as power law. The analytical model was evaluated and compared with Hanford diffusion experiment in stable conditions and Copenhagen diffusion experiment in unstable and neutral conditions which was done by reducing the general analytical model to a one with linear vertical eddy diffusivity and constant downwind speed profile. Comparison with other analytical models was held. The presented model predictions show a good agreement with observations and lay inside a factor of two with observed data of both Hanford and Copenhagen diffusion experiments.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arya, S. P. (1995). Modeling and parameterization of near source diffusion in weak winds. Journal of Applied Meteorology, 34, 1112–1122.
Arya, S. P. (1999). Air pollution meteorology and dispersion. New York: Oxford University Press.
Degrazia, G. A., Campos Velho, H. F., & Carvalho, J. C. (1997). Nonlocal exchange coefficients for the convective boundary layer derived from spectral properties. Controlling Atmosphere Physics, 1, 57–64.
Demuth, C. (1978). A contribution to the analytical steady solution of the diffusion equation for line sources. Atmospheric Environment, 12, 1255–1258.
Doran, J. C., Abbey, O. B., Buck, J. W., Glover, D. W., & Horst, T. W. (1984). Field validation of exposure assessment models (Vol. 1). Research Triangle Park, NC: Data environmental science research lab.
Doran, J. C., & Horst, T. W. (1985). An evaluation of Gaussian Plume-depletion models with dual-tracer field measurements. Atmospheric Environment, 19, 939–951.
Dyer, A. J. (1974). A review of flux-profile relationships. Boundary-Layer Meteorology, 7, 363–372.
Ermak, D. L. (1977). An analytical model for air pollutant transport and deposition from a point source. Atmospheric Environment, 11, 231–237.
Essa, K. S. M., Mosallem, A. M., & Ibrahim, M. A. E. (2019a). The concentration of pollutants in two dimensional with constant and variable vertical eddy diffusivity. Journal of Multidisciplinary Engineering Science and Technology (JMEST), 6(6), 2458–9403.
Essa, K. S. M., Maha, S., & El-Otaify, M. (2007). Mathematical model for hermitized atmospheric dispersion in low winds with eddy diffusivities linear functions downwind distance. Meteorology and Atmospheric Physics, 96, 265–275.
Essa, K. S. M., Fouad, E. A. (2011). Estimating of crosswind integrated Gaussian and non-Gaussian concentration by using different dispersion schemes. Journal of Basic and Applied Sciences, 5(11):1580–1587
Essa, K. S. M., Ibrahim, M. A. E., Shalaby, A. S., & Mosallem, A. M. (2019b). Analytical solution of two dimensional diffusion equation using Hankel transform. Journal of Scientific and Engineering Research, 6(6), 6–11.
Gradshteyn, I., Jeffrey, A., & Ryzhik, I. (1996). Table of integrals, series, and products. New York: Academic Press.
Gryning, S. E., Holtslag, A. A. M., Irwin, J. S., & Sivertsen, B. (1987). Applied dispersion modeling based on meteorological scaling parameters. Atmospheric Environment, 21, 79–89.
Gryning, S. E., & Lyck, E. (1984). Atmospheric dispersion from elevated sources in an urban area: Comparison between tracer experiments and model calculations. Journal of Climate and Applied Meterology, 23, 651–660.
Hanna, S. R., Briggs, G. A., Rayford, P., & Hosker, J. R. (1982). Handbook on atmospheric diffusion. DOE TIC-11223 (DE 82002045).
Hanna, S. R. (1989). confidence limit for air quality models as estimated by bootstrap and Jacknife resembling methods. Atmospheric Environment, 23, 1385–1395.
Huang, C. H. (1979). A theory of dispersion in turbulent shear flow. Atmospheric Environment, 13, 453–463.
Irwin, J. S. (1979). A theoretical validation of the wind profile power law exponent as a function of surface roughness and stability. Atmospheric Environment, 13, 191–194.
Jacobson, M. (2005). Fundamentals of atmospheric modeling. Cambridge: Cambridge University Press.
Lin, J. S., & Hildemann, L. M. (1996). Analytical solutions of the atmospheric diffusion equationwith multiple sources and height-dependent wind speed and eddy diffusivities. Atmospheric Environment, 30, 239–254.
Llewelyn, R. P. (1983). An analytical model for the transport dispersion and elimination of air pollutants emitted from a point source. Atmospheric Environment, 17, 249–256.
Marrouf, A., Essa, K., El-Otaify, M., Mohamed, A., & Ismail, G. (2015). The influence of eddy diffusivity variation on the atmospheric diffusion equation. Open Journal of Air Pollution, 04(03), 109–118.
Monin, A. S., & Yaglom, A. M. (1971). Statistical fluid mechanics (Vol. 1). Cambridge: MIT.
Mooney, C. J., & Wilson, J. D. (1993). ‘Disagreements between gradient-diffusion and Lagrangian stochastic dispersion models, even for surface near the ground. Boundary-Layer Meteorology, 64, 291–296.
Goyal, P., & Kumar, A. (2011). Mathematical modeling of air pollutants: an application to Indian urban city. INTECH Open Access Publisher.
Pasquill, F., & Smith, F. B. (1983). Atmospheric diffusion (3rd ed., p. 269). NewYork/London: D Van Nostrand/Wiley.
Pasquill, F. (1961). The estimation of the dispersion of windborne material. The Meteorological Magazine, 90(1063), 33–49.
Seinfeld, J. H., & Pandis, S. N. (1998). Atmospheric chemistry and physics (p. 1326). New York: Wiley.
Sharan, M., & Kumar, P. (2009). An analytical model for crosswind integrated concentration released from a continuous source in a finite atmospheric boundary layer. Atmospheric Environment, 43, 2268–2277.
Sharan, M., & Modani, M. (2006). A two-dimensional analytical model for the dispersion of air-pollutants in the atmosphere with a capping inversion. Atmospheric Environment, 40, 3479–3489.
Tirabassi, T. (1989). Analytical air pollution advection and diffusion models. Water, Air, and Soil pollution, 47, 19–24.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Essa, K.S.M., Shalaby, A.S., Ibrahim, M.A.E. et al. Analytical Solutions of the Advection–Diffusion Equation with Variable Vertical Eddy Diffusivity and Wind Speed Using Hankel Transform. Pure Appl. Geophys. 177, 4545–4557 (2020). https://doi.org/10.1007/s00024-020-02496-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00024-020-02496-y