Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T04:05:46.040Z Has data issue: false hasContentIssue false
  • English
  • Français

THREE-DIMENSIONAL ANALYTICAL SOLUTION OF THE ADVECTION-DIFFUSION EQUATION FOR AIR POLLUTION DISPERSION

Part of: Harmonic analysis in one variable Boundary value problems

Published online by Cambridge University Press:  26 April 2022

M. FARHANE Open the ORCID record for M. FARHANE [Opens in a new window] ,
O. ALEHYANE Open the ORCID record for O. ALEHYANE [Opens in a new window]  and
O. SOUHAR Open the ORCID record for O. SOUHAR [Opens in a new window]
M. FARHANE*
Affiliation:
Faculty of Sciences El Jadida, LIMA Laboratory, Department of Mathematics, Chouaib Doukkali University, El Jadida, Morocco; e-mail: souhar.o@ucd.ac.ma
O. ALEHYANE
Affiliation:
Faculty of Sciences El Jadida, MF Laboratory, Department of Mathematics, Chouaib Doukkali University, El Jadida, Morocco; e-mail: alehyane.o@ucd.ac.ma
O. SOUHAR
Affiliation:
Faculty of Sciences El Jadida, LIMA Laboratory, Department of Mathematics, Chouaib Doukkali University, El Jadida, Morocco; e-mail: souhar.o@ucd.ac.ma
*
e-mail: farhane.m@ucd.ac.ma
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a new analytical solution of a three-dimensional atmospheric pollutant dispersion. The main idea is to subdivide vertically the planetary boundary layer into sub-layers, where the wind speed and eddy diffusivity assume average values for each sub-layer. Basically, the model is assessed and validated using data obtained from the Copenhagen diffusion and Prairie Grass experiments. Our findings show that there is a good agreement between the predicted and observed crosswind-integrated concentrations. Moreover, the calculated statistical indices are within the range of acceptable model performance.

Keywords

Atmospheric dispersionFourier transformSturm–Liouville eigenvalue problem

MSC classification

Primary: 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Secondary: 34B24: Sturm-Liouville theory
Type
Research Article
Information
The ANZIAM Journal , Volume 64 , Issue 1 , January 2022 , pp. 40 - 53
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barad, M. L. (Editor), Project prairie grass: a field program in diffusion, Vols. I and II of Geophysical Research Paper, 59, AFCRL-TR-58-235 (ASTIA Document No. AF-152572) (Air Force Cambridge Research Laboratories, Bedford, 1958). https://www.harmo.org/jsirwin/PrairieGrassDiscussion.html.10.21236/AD0152573CrossRefGoogle Scholar
Boyce, W. E., Diprima, R. C. and Meade, D. B., Elementary differential equations and boundary value problems, 2nd edn (Wiley, Hoboken, NJ, 2017). https://books.google.co.ma/books?id=iE2fAQAACAAJ.Google Scholar
Brown, M., Arya, S. and Snyder, W., “Plume descriptors derived from a non-Gaussian concentration model”, Atmos. Environ. 31 (1997) 183–189; doi:10.1016/1352-2310(96)00487-6.CrossRefGoogle Scholar
Chang, J. and Hanna, S., “Air quality model performance evaluation”, Meteorol. Atmos. Phys. 87 (2004) 167–196; doi:10.1007/s00703-003-0070-7.CrossRefGoogle Scholar
Chrysikopoulos, C., Hildemann, L. and Roberts, P., “A three-dimensional steady-state atmospheric dispersion-deposition model for emissions from a ground-level area source”, Atmos. Environ. Part A 26 (1992) 747–757; doi:10.1016/0960-1686(92)90234-C.CrossRefGoogle Scholar
De Visscher, A., Air dispersion modeling: foundations and applications, 1st edn (John Wiley & Sons, Hoboken, NJ, 2013); doi:10.1002/9781118723098.CrossRefGoogle Scholar
Deaves, D. and Harris, R., A mathematical model of the structure of strong winds, Volume 76 of CIRA Report (Construction Industry Research and Information Association, London, 1978). https://www.worldcat.org/title/mathematical-model-of-the-structure-of-strong-winds/oclc/51357818.Google Scholar
Degrazia, G., Velho, H. C. and Carvalho, J., “Nonlocal exchange coefficients for the convective boundary layer derived from spectral properties”, Contrib. Atmos. Phys. 71 (1997) 57–64; https://www.osti.gov/etdeweb/biblio/477925.Google Scholar
Ema’a, E., Ben-Bolie, G., Patrice, E., Zarma, A. and Pierre, O. A., “A three-dimensional analytical solution for the study of air pollutant dispersion in a finite layer”, Bound.-Layer Meteorol. 155 (2015) 289–300; doi:10.1007/s10546-014-9997-0.Google Scholar
Gryning, S., Holtslag, A. A. M., Irwin, J. S. and Sivertsen, B., “Applied dispersion modeling based on meteorological scaling parameters”, Atmos. Environ. 21 (1987) 79–89; doi:10.1016/0004-6981(87)90273-3.CrossRefGoogle Scholar
Gryning, S. and Lyck, E., “Atmospheric dispersion from elevated sources in an urban area: comparison between tracer experiments and model calculations”, J. Appl. Meteorol. Clim. 23 (1984) 651–660. https://journals.ametsoc.org/view/journals/apme/23/4/1520-0450_1984_023_0651_adfesi_2_0_co_2.xml.2.0.CO;2>CrossRefGoogle Scholar
Guerrero, J. P., Pimentel, L. and Skaggs, T., “Analytical solution for the advection-dispersion transport equation in layered media”, Int. J. Heat Mass Transf. 56 (2013) 274–282; doi: 10.1016/j.ijheatmasstransfer.2012.09.011.CrossRefGoogle Scholar
Hanna, S. R., Briggs, G. A., Rayford, P. and Hosker, R. P. Jr., “Handbook on atmospheric diffusion”, Atmos. Environ. 17 (2013) 673–675; doi:10.2172/5591108.Google Scholar
Huang, C., “A theory of dispersion in turbulent shear flow”, Atmos. Environ. 13 (1979) 453–463; doi:10.1016/0004-6981(79)90139-2.CrossRefGoogle Scholar
Laaouaoucha, D., Farhane, M., Essaouini, M. and Souhar, O., “Analytical model for the two-dimensional advection-diffusion equation with the logarithmic wind profile in unstable conditions”, Int. J. Environ. Sci. Technol. (Tehran) (2021); doi:10.1007/s13762-021-03554-1.CrossRefGoogle Scholar
Lin, J.-S. and Hildemann, L. M., “Analytical solutions of the atmospheric diffusion equation with multiple sources and height-dependent wind speed and eddy diffusivities”, Atmos. Environ. 30 (1996) 239–254; doi:10.1016/1352-2310(95)00287-9.CrossRefGoogle Scholar
Mikhailov, M. D. and Ozisik, M. N., Unified analysis and solution of heat and mass diffusion, 1st edn (Dover Publications, New York, 1984).Google Scholar
Mooney, C. J. and Wilson, J. D., “Disagreements between gradient-diffusion and Lagrangian stochastic dispersion models, even for surface near the ground”, Bound.-Layer Meteorol. 64 (1993) 291–296; doi:10.1007/BF00708967.CrossRefGoogle Scholar
Moreira, D., Tirabassi, T., Vilhenah, M. and Goulart, A. G., “A multi-layer model for pollutant dispersion with dry deposition to the ground”, Atmos. Environ. 44 (2010) 1859–1865; doi:10.1016/j.atmosenv.2010.02.025.CrossRefGoogle Scholar
Nieuwstadt, F., “An analytical solution of the time-dependent, one-dimensional diffusion equation in the atmospheric boundary layer”, Atmos. Environ. 14 (1980) 1361–1364; doi:10.1016/0004-6981(80)90154-7.CrossRefGoogle Scholar
Obukhov, A. M., “Turbulence in an atmosphere with a non-uniform temperature”, Bound.-Layer Meteorol. 2 (1971) 7–29; doi:10.1007/BF00718085.CrossRefGoogle Scholar
Pasquill, F. and Smith, F. B., Atmospheric diffusion, 1st edn, In: Ellis Horwood Series in Environmental Science (Halsted Press, Chichester, UK, 1983).Google Scholar
Seinfeld, J. and Pandis, S., Atmospheric chemistry and physics: from air pollution to climate change, Volume 45 of Environment: Science and Policy for Sustainable Development (American Association for the Advancement of Sciences (AAAS), William T. Golden Center for Science and Engineering, Washington, DC, 1998) 13–26; ISBN: 978-1-118-94740-1.Google Scholar
Wu, X., Nethery, R. C., Sabath, M. B., Braun, D. and Dominic, F., “Air pollution and COVID-19 mortality in the united states: strengths and limitations of an ecological regression analysis”, Sci. Adv. 6 (2020) 1–6; doi:10.1126/sciadv.abd4049.CrossRefGoogle ScholarPubMed