A General Analytical Solution of the Advection–Diffusion Equation for Fickian Closure

  • D. Buske
  • M. T. Vilhena
  • C. F. Segatto
  • R. S. Quadros


In the last few years there has been increased research interest in searching for analytical solutions for the advection–diffusion equation (ADE). By analytical we mean that no approximation is done along the derivation of the solution. There exists a significant literature regarding this theme. For illustration we mention the works of (Rounds 1955; Smith 1957; Scriven, Fisher 1975; Demuth 1978; van Ulden 1978; Nieuwstadt, de Haan 1981; Tagliazucca et al. 1985; Tirabassi 1989; Tirabassi, Rizza 1994; Sharan et al. 1996; Lin, Hildemann 1997; Tirabassi 2003). We note that in these works all solutions are valid for very specialized problems having specific wind and eddy diffusivities vertical profiles. Further, also in the literature there is the ADMM (Advection Diffusion Multilayer Method) approach which solves the two-dimensional ADE with variable wind profile and eddy diffusivity coefficient (Moreira et al. 2006). The main idea relies on the discretization of the Atmospheric Boundary Layer (ABL) in a multilayer domain, assuming in each layer that the eddy diffusivity and wind profile take averaged values. The resulting advection–diffusion equation in each layer is then solved by the Laplace transformation technique. For more details about this methodology see the review work done by (Moreira et al. 2006). We are also aware of the recent work of (Costa et al. 2006), dubbed as GIADMT method (Generalized Integral Advection Diffusion Multilayer Technique), which presented a general solution for the time-dependent three-dimensional ADE, again assuming the stepwise approximation for the eddy diffusivity coefficient and wind profile and proceeding further in similar way according the previous work. To avoid this approximation, in this work we report an analytical general solution for this problem, assuming that the eddy diffusivity coefficient and wind profile are arbitrary functions having a continuous dependence on the vertical and longitudinal variables. Without losing generality we specialize the application in micrometeorology, specially for the problem of simulation of contaminant releasing in the ABL.


Diffusion Equation Atmospheric Boundary Layer Eddy Diffusivity Wind Profile Pollutant Dispersion 
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  1. [AbVa04]
    Abate, J., Valkó, P.P.: Multi-precision Laplace transform inversion. Int. J. for Num. Methods in Engineering, 60, 979–993 (2004). MATHCrossRefGoogle Scholar
  2. [Ary95]
    Arya, S. Pal: Modeling and parameterization of near-source diffusion in weak winds. J. Appl. Meteor., 34, 1112–1122 (1995). CrossRefGoogle Scholar
  3. [Bla97]
    Blackadar, A.K.: Turbulence and Diffusion in the Atmosphere: Lectures in Environmental Sciences, Springer-Verlag, 185 pp. (1997). MATHGoogle Scholar
  4. [Bod10]
    Bodmann, B., Vilhena, M.T., Ferreira, L.S., Bardaji, J.B.: An analytical solver for the multi-group two-dimensional neutron-diffusion equation by integral transform techniques. Il Nuovo Cimento C, 33, 199–206 (2010). Google Scholar
  5. [Bus10]
    Buske, D., Vilhena, M.T., Moreira, D.M., Tirabassi, T.: An Analytical Solution for the Transient Two-Dimensional Advection-Diffusion Equation with Non-Fickian Closure in Cartesian Geometry by Integral Transform Technique. Integral Methods in Science and Engineering: Computational Methods, Birkhauser, Boston, 33–40 (2010). Google Scholar
  6. [Cos06]
    Costa, C.P., Vilhena, M.T., Moreira, D.M., Tirabassi, T.: Semi-analytical solution of the steady three-dimensional advection-diffusion equation in the planetary boundary layer. Atmos. Environ., 40, n. 29, 5659–5669 (2006). CrossRefGoogle Scholar
  7. [CoBa07]
    Cotta, R.M., Barros, F.P.J.: Integral transforms for three-dimensional steady turbulent dispersion in rivers and channels. Applied Mathematical Modelling, 31, 2719–2732 (2007). MATHCrossRefGoogle Scholar
  8. [CoHi89]
    Courant, R., Hilbert, D.: Methods of Mathematical Physics, John Wiley & Sons, New York (1989). CrossRefGoogle Scholar
  9. [Deg02]
    Degrazia, G.A., Moreira, D.M., Campos, C.R.J., Carvalho, J.C., Vilhena, M.T.: Comparison between an integral and algebraic formulation for the eddy diffusivity using the Copenhagen experimental dataset. Il Nuovo Cimento, 25C, 207–218 (2002). Google Scholar
  10. [Dem78]
    Demuth, C.: A contribution to the analytical steady solution of the diffusion equation for line sources. Atmos. Environ., 12, 1255–1258 (1978). CrossRefGoogle Scholar
  11. [GrLy84]
    Gryning, S.E., Lyck, E.: Atmospheric dispersion from elevated source in an urban area: comparison between tracer experiments and model calculations. J. Appl. Meteor., 23, 651–654 (1984). CrossRefGoogle Scholar
  12. [Han89]
    Hanna, S.R.: Confidence limit for air quality models as estimated by bootstrap and jacknife resampling methods. Atm. Env., 23, 1385–1395 (1989). CrossRefGoogle Scholar
  13. [LiHi97]
    Lin, J.S., Hildemann, L.M.: A generalised mathematical scheme to analytically solve the atmospheric diffusion equation with dry deposition. Atm. Env., 31, 59–71 (1997). MATHCrossRefGoogle Scholar
  14. [Mor05]
    Moreira, D.M., Vilhena, M.T., Tirabassi, T., Buske, D., Cotta, R.M.: Near source atmospheric pollutant dispersion using the new GILTT method. Atmos. Environ., 39, n. 34, 6290–6295 (2005). CrossRefGoogle Scholar
  15. [Mor06]
    Moreira, D.M., Vilhena, M.T., Tirabassi, T., Costa, C., Bodmann, B.: Simulation of pollutant dispersion in atmosphere by the Laplace transform: the ADMM approach. Water, Air and Soil Pollution, 177, 411–439 (2006). CrossRefGoogle Scholar
  16. [Mor09a]
    Moreira, D.M., Vilhena, M.T., Buske, D., Tirabassi, T.: The state-of-art of the GILTT method to simulate pollutant dispersion in the atmosphere. Atmospheric Research, 92, 1–17 (2009). CrossRefGoogle Scholar
  17. [Mor09b]
    Moreira, D.M., Vilhena, M.T., Buske, D.: On the GILTT Formulation for Pollutant Dispersion Simulation in the Atmospheric Boundary Layer. Air Pollution and Turbulence: Modeling and Applications, CRC Press, Boca Raton – Flórida (USA), 179–202 (2009). CrossRefGoogle Scholar
  18. [Mor10]
    Moreira, D.M., Vilhena, M.T., Tirabassi, T., Buske, D., Costa, C.P.: Comparison between analytical models to simulate pollutant dispersion in the atmosphere. Int. J. Env. and Waste Management, 6, 327–344 (2010). CrossRefGoogle Scholar
  19. [NiHa81]
    Nieuwstadt, F.T.M., de Haan, B.J.: An analytical solution of one-dimensional diffusion equation in a nonstationary boundary layer with an application to inversion rise fumigation. Atmos. Environ., 15, 845–851 (1981). CrossRefGoogle Scholar
  20. [PaDu88]
    Panofsky, A.H., Dutton, J.A.: Atmospheric Turbulence, John Wiley & Sons, New York (1988). Google Scholar
  21. [ScFi75]
    Scriven, R.A., Fisher, B.A.: The long range transport of airborne material and its removal by deposition and washout-II. The effect of turbulent diffusion. Atmos. Environ., 9, 59–69 (1975). CrossRefGoogle Scholar
  22. [Rou55]
    Rounds, W.: Solutions of the two-dimensional diffusion equation. Trans. Am. Geophys. Union, 36, 395–405 (1955). MathSciNetGoogle Scholar
  23. [Sha96]
    Sharan, M., Singh, M.P., Yadav, A.K.: A mathematical model for the atmospheric dispersion in low winds with eddy diffusivities as linear functions of downwind distance. Atmos. Environ., 30, n. 7, 1137–1145 (1996). CrossRefGoogle Scholar
  24. [Smi57]
    Smith, F.B.: The diffusion of smoke from a continuous elevated poinr source into a turbulent atmosphere. J. Fluid Mech., 2, 49–76 (1957). MathSciNetMATHCrossRefGoogle Scholar
  25. [StSe66]
    Stroud, A. H., Secrest, D.: Gaussian Quadrature Formulas, Prentice Hall Inc., Englewood Cliffs, N.J. (1966). MATHGoogle Scholar
  26. [VaAb04]
    Valkó, P.P., Abate, J.: Comparison of sequence accelerators for the Gaver method of numerical Laplace transform inversion. Computers and Mathematics with Application, 48, 629–636 (2004). MATHCrossRefGoogle Scholar
  27. [Tag85]
    Tagliazucca, M., Nanni, T., Tirabassi, T.: An analytical dispersion model for sources in the surface layer. Nuovo Cimento, 8C, 771–781 (1985). Google Scholar
  28. [Tir89]
    Tirabassi, T.: Analytical air pollution and diffusion models. Water, Air and Soil Pollution, 47, 19–24 (1989). CrossRefGoogle Scholar
  29. [TiRi94]
    Tirabassi, T., Rizza U.: Applied dispersion modelling for ground-level concentrations from elevated sources. Atmos. Environ., 28, 611–615 (1994). CrossRefGoogle Scholar
  30. [Tir03]
    Tirabassi, T.: Operational advanced air pollution modeling. PAGEOPH, 160, n. 1–2, 5–16 (2003). CrossRefGoogle Scholar
  31. [Van78]
    van Ulden, A.P.: Simple estimates for vertical diffusion from sources near the ground. Atmos. Environ., 12, 2125–2129 (1978). CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • D. Buske
    • 1
  • M. T. Vilhena
    • 2
  • C. F. Segatto
    • 2
  • R. S. Quadros
    • 1
  1. 1.Universidade Federal de PelotasPelotasBrazil
  2. 2.Universidade Federal do Rio Grande do SulPorto AlegreBrazil

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