Air Pollution Steady-State Advection-Diffusion Equation: The General Three-Dimensional Solution


Atmospheric air pollution turbulent fluxes can be assumed to be proportional to the mean concentration gradient. This assumption, along with the equation of continuity, leads to the advection-diffusion equation. Many models simulating air pollution dispersion are based upon the solution (numerical or analytical) of the advection-diffusion equation as- suming turbulence parameterization for realistic physical scenarios. We present the general steady three-dimensional solution of the advection-diffusion equation considering a vertically inhomogeneous atmospheric boundary layer for arbitrary vertical profiles of wind and eddy-diffusion coefficients. Numerical results and comparison with experimental data are shown.

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D. Buske, M. Vilhena, T. Tirabassi and B. Bodmann, "Air Pollution Steady-State Advection-Diffusion Equation: The General Three-Dimensional Solution," Journal of Environmental Protection, Vol. 3 No. 9A, 2012, pp. 1124-1134. doi: 10.4236/jep.2012.329131.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] P. Zanetti, “Air Pollution Modelling,” Comp. Mech. Publications, Southampton, 1990.
[2] J. H. Seinfeld and S. N. Pandis, “Atmospheric Chemistry and Physics,” John Wiley & Sons, New York, 1998.
[3] O. F. T. Roberts, “The Theoretical Scattering of Smoke in a Turbulent Atmosphere,” Proceedings of the Royal Society A, Vol. 104, 1923, pp. 640-654.doi:10.1098/rspa.1923.0132
[4] W. Rounds, “Solutions of the Two-Dimensional Diffusion Equation,” Transactions—American Geophysical Union, Vol. 36, 1955, pp. 395-405.
[5] F. B. Smith, “The Diffusion of Smoke from a Continuous Elevated Point Source into a Turbulent Atmosphere,” Journal of Fluid Mechanics, Vol. 2, No. 1, 1957, pp. 49- 76. doi:10.1017/S0022112057000737
[6] R. A. Scriven and B. A. Fisher, “The Long Range Transport of Airborne Material and Its Removal by Deposition and Washout-II. The Effect of Turbulent Diffusion,” Atmospheric Environment, Vol. 9, No. 1, 1975, pp. 59-69.doi:10.1016/0004-6981(75)90054-2
[7] G. T. Yeh and C. H. Huang, “Three-Dimensional Air Pollutant Modeling in the Lower Atmosphere,” Boundary-Layer Meteorology, Vol. 9, 1975, pp. 381-390.
[8] M. Y. Berlyand, “Contemporary Problems of Atmospheric Diffusion and Pollution of the Atmosphere,” USEPA, Raleigh, 1975.
[9] C. A. Demuth, “Contribution to the Analytical Steady Solution of the Diffusion Equation for Line Sources,” Atmospheric Environment, Vol. 12, No. 5, 1978, pp. 1255-1258. doi:10.1016/0004-6981(78)90399-2
[10] A. P. van Ulden, “Simple Estimates for Vertical Diffusion from Sources near the Ground,” Atmospheric Environment, Vol. 12, No. 11, 1978, pp. 2125-2129.doi:10.1016/0004-6981(78)90167-1
[11] F. T. M. Nieuwstadt, “An Analytical Solution of the Time-Dependent, One-Dimensional Diffusion Equation in the Atmospheric Boundary Layer,” Atmospheric Environment, Vol. 14, No. 12, 1980, pp. 1361-1364. doi:10.1016/0004-6981(80)90154-7
[12] F. T. M. Nieuwstadt and B. J. de Haan, “An Analytical Solution of One-Dimensional Diffusion Equation in a Nonstationary Boundary Layer with an Application to Inversion Rise Fumigation,” Atmospheric Environment, Vol. 15, No. 5, 1981, pp. 845-851.doi:10.1016/0004-6981(81)90289-4
[13] W. Koch, “A Solution of the Two-Dimensional Atmospheric Diffusion Equation with Height-Dependent Diffusion-Coefficient Including Ground-Level Absorption,” Atmospheric Environment, Vol. 23, No. 8, 1989, pp. 1729-1732. doi:10.1016/0004-6981(89)90057-7
[14] C. Chrysikopoulos, L. M. Hildemann and P. V. Roberts, “A Three-Dimensional Atmospheric Dispersion-Deposi- tion Model for Emissions from a Ground Level Area Source,” Atmospheric Environment, Vol. 26A, 1992, pp. 747-757.
[15] M. Sharan, M. P. Singh and A. K. Yadav, “A Mathematical Model for the Atmospheric Dispersion in Low Winds with Eddy Diffusivities as Linear Function of Downwind Distance,” Atmospheric Environment, Vol. 30, No. 7, 1996, pp. 1137-1145. doi:10.1016/1352-2310(95)00368-1
[16] J. S. Lin and L. M. Hildemann, “A Generalised Mathematical Scheme to Analytically Solve the Atmospheric Diffusion Equation with Dry Deposition,” Atmospheric Environment, Vol. 31, No. 1, 1997, pp. 59-71. doi:10.1016/S1352-2310(96)00148-3
[17] M. T. Vilhena, U. Rizza, G. A. Degrazia, C. Mangia, D. M. Moreira and T. Tirabassi, “An Analytical Air Po- llution Model: Development and Evaluation,” Contri- butions to Atmospheric Physics, Vol. 71, No. 3, 1998, pp. 315-320.
[18] C. P. Costa, M. T. Vilhena, D. M. Moreira and T. Tirabassi, “Semi-Analytical Solution of the Steady Three- Dimensional Advection-Diffusion Equation in the Planetary Boundary Layer,” Atmospheric Environment, Vol. 40, No. 29, 2006, pp. 5659-5669.doi:10.1016/j.atmosenv.2006.04.054
[19] D. M. Moreira, M. T. Vilhena, T. Tirabassi, C. Costa and B. Bodmann, “Simulation of Pollutant Dispersion in Atmosphere by the Laplace Transform: The ADMM Approach,” Water, Air and Soil Pollution, Vol. 177, No. 1-4, 2006, pp. 411-439. doi:10.1007/s11270-006-9182-2
[20] M. Vilhena, C. Costa, D. Moreira and T. Tirabassi, “A Semi-Analytical Solution for the Three-Dimensional Advection Diffusion Equation Considering Non-Local Turbulence Closure,” Atmospheric Research, Vol. 90, No. 1, 2008, pp. 63-69. doi:10.1016/j.atmosres.2008.04.002
[21] T. Tirabassi, “Operational Advanced Air Pollution Modelling,” Pure and Applied Geophysics, Vol. 160, No. 1-2, 2003, pp. 5-16. doi:10.1007/s00024-003-8762-y
[22] S. Wortmann, M. T. Vilhena, D. M. Moreira and D. Buske, “A New Analytical Approach to Simulate the Pollutant Dispersion in the PBL,” Atmospheric Environ- ment, Vol. 39, No. 12, 2005, pp. 2171-2178. doi:10.1016/j.atmosenv.2005.01.003
[23] D. M. Moreira, M. T. Vilhena, T. Tirabassi, D. Buske and R. Cotta, “Near Source Atmospheric Pollutant Dispersion Using the New GILTT Method,” Atmospheric Environment, Vol. 39, No. 34, 2005, pp. 6290-6295. doi:10.1016/j.atmosenv.2005.07.008
[24] D. M. Moreira, M. T. Vilhena, D. Buske and T. Tirabassi, “The GILTT Solution of the Advection-Diffusion Equa- tion for an Inhomogeneous and Nonstationary PBL,” Atmospheric Environment, Vol. 40, No. 17, 2006, pp. 3186- 3194. doi:10.1016/j.atmosenv.2006.01.035
[25] D. M. Moreira, M. T. Vilhena, D. Buske and T. Tirabassi, “The State-of-Art of the GILTT Method to Simulate Pollutant Dispersion in the Atmosphere,” Atmospheric Research, Vol. 92, No. 1, 2009, pp. 1-17. doi:10.1016/j.atmosres.2008.07.004
[26] R. Courant and D. Hilbert, “Methods of Mathematical Physics,” John Wiley & Sons, 1989.
[27] A. K. Blackadar, “Turbulence and Diffusion in the Atmosphere: Lectures in Environmental Sciences,” Springer- Verlag, 1997.
[28] R. H. Torres, “Spaces of Sequences, Sampling Theorem, and Functions of Exponential Type,” Studia Mathematica, Vol. 100, No. 1, 1991, pp. 51-74.
[29] B. Bodmann, M. T. Vilhena, L. S. Ferreira and J. B. Bardaji, “An Analytical Solver for the Multigroup Two Domensional Neutron-Diffusion Equation by Integral Transform Techniques,” Il Nuovo Cimento C, Vol. 33, 2010, pp. 199-206.
[30] C. Mangia, D. M. Moreira, I. Schipa, G. A. Degrazia, T. Tirabassi and U. Rizza, “Evaluation of a New Eddy Diffusivity Parameterisation from Turbulent Eulerian Spectra in Different Stability Conditions,” Atmospheric Environment, Vol. 36, No. 1, 2002, pp. 67-76.doi:10.1016/S1352-2310(01)00469-1
[31] G. A. Degrazia, “Lagrangian Particle Models,” In: P. Zannetti, Ed., Air Quality Modeling: Theories, Methodologies, Computational Techniques and Available Databases and Software, EnviroComp, 2005, pp. 93-162.
[32] G. A. Degrazia, H. F. Campos Velho and J. C. Carvalho, “Nonlocal Exchange Coefficients for the Convective Boundary Layer Derived from Spectral Properties,” Contributions to Atmospheric Physics, Vol. 70, 1997, pp. 57-64.
[33] G. K. Batchelor, “Diffusion in a Field of Homogeneous Turbulence, Eulerian Analysis,” Australian Journal of Scientific Research, Vol. 2, 1949, pp. 437-450.
[34] J. C. Kaimal, J.C. Wyngaard, D. A. Haugen, O. R. Coté, Y. Izumi, S. J. Caughey and C. J. Readings, “Turbulence Structure in the Convective Boundary Layer,” Journal of the Atmospheric Sciences, Vol. 33, No. 11, 1976, pp. 2152-2169.doi:10.1175/1520-0469(1976)033<2152:TSITCB>2.0.CO;2
[35] S. J. Caughey, “Observed Characteristics of the Atmospheric Boundary Layer,” In: F. T. M. Nieuwstadt and H. van Dop, Eds., Atmospheric Turbulence and Air Pollution Modeling, Reidel, Boston, 1982.
[36] J. E. Pleim and J. S. Chang, “A Non-Local Closure Model for Vertical Mixing in the Convective Boundary Layer,” Atmospheric Environment, Vol. 26A, 1992, pp. 965-981.
[37] G. A. Degrazia, M. T. Vilhena and O. L. L. Moraes, “An Algebraic Expression for the Eddy Diffusivities in the Stable Boundary Layer: A Description of Near-Source Diffusion,” Il Nuovo Cimento, Vol. 19C, 1996, pp. 399- 403.
[38] Z. Sorbjan, “Structure of the Atmospheric Boundary Layer,” Prentice Hall, New Jersey, 1989.
[39] F. T. M. Nieuwstadt, “The Turbulent Structure of the Stable Nocturnal Boundary Layer,” Journal of the Atmospheric Sciences, Vol. 41, No. 14, 1984, pp. 2202-2216. doi:10.1175/1520-0469(1984)041<2202:TTSOTS>2.0.CO;2
[40] H. A. Panofsky and J. A. Dutton, “Atmospheric Turbulence,” John Wiley & Sons, New York, 1984.
[41] J. S. Irwin, “A Theoretical Variation of the Wind Profile Power-Low Exponent as a Function of Surface Roughness and Stability,” Atmospheric Environment, Vol. 13, No. 1, 1979, pp. 191-194. doi:10.1016/0004-6981(79)90260-9
[42] S. E. Gryning and E. Lyck, “Atmospheric Dispersion from Elevated Source in an Urban Area: Comparison between Tracer Experiments and Model Calculations,” Journal of Applied Meteorology, Vol. 23, 1984, pp. 651-654.
[43] S. R. Hanna and R. J. Paine, “Hybrid Plume Dispersion Model (HPDM) Development and Evaluation,” Journal of Applied Meteorology, Vol. 28, No. 3, 1989, pp. 206-224. doi:10.1175/1520-0450(1989)028<0206:HPDMDA>2.0.CO;2
[44] H. R. Olesen, “Datasets and Protocol for Model Validation,” International Journal of Environment and Pollution, Vol. 5, 1995, pp. 693-701.
[45] S. R. Hanna, “Confidence Limit for Air Quality Models as Estimated by Bootstrap and Jacknife Resampling Methods,” Atmospheric Environment, Vol. 23, No. 6, 1989, pp. 1385-1395. doi:10.1016/0004-6981(89)90161-3

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