A Closed-Form Formulation for Pollutant Dispersion in the Atmosphere

  • C. P. Costa
  • M. T. Vilhena
  • T. Tirabassi


Transport and diffusion models of air pollution are based either on simple techniques, such as the Gaussian approach, or on more complex algorithms, such as the K-theory differential equation. The Gaussian equation is an easy and fast method, which, however, cannot properly simulate complex nonhomogeneous conditions. The K-theory can accept virtually any complex meteorological input, but generally requires numerical integration, which is computationally expensive and is often affected by large numerical advection errors. Conversely, Gaussian models are fast, simple, do not require complex meteorological input, and describe the diffusive transport in an Eulerian framework, making easy use of the Eulerian nature of measurements.

For these reasons they are still widely used by environmental agencies all over the world for regulatory applications. However, because of its wellknown intrinsic limits, the reliability of a Gaussian model strongly depends on the way the dispersion parameters are determined on the basis of the turbulence structure of the planetary boundary layer (PBL) and the model’s ability to reproduce experimental diffusion data. The Gaussian model has to be completed by empirically determined standard deviations (the “sigmas”), while some commonly measurable turbulent exchange coefficient has to be introduced in the advection–diffusion equation.


Convection Advection Abate 
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  1. [AbVa04]
    Abate, J., Valkó, P.P.: Multi-precision Laplace transform inversion. Internat. J. Numer. Methods Engng., 60, 979-993 (2004).MATHCrossRefGoogle Scholar
  2. [Ar95]
    Arya, P.: Modeling and parameterization of near-source difusion in weak winds. J. Appl. Meteorology, 34, 1112-1122 (1995).CrossRefGoogle Scholar
  3. [CoEA06]
    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. Atmospheric Environment, 40, 5659-5669 (2006).CrossRefGoogle Scholar
  4. [De72]
    Deardoff, J.W.: Theoretical expression for the countergradient heat flux. J. Geophys. Res. Pap., 59, 5900-5904 (1972).CrossRefGoogle Scholar
  5. [DeEA97]
    Degrazia, G.A., Velho, H.F.C., Carvalho, J.C.: Nonlocal exchange coefficients for the convective boundary layer derived from spectral properties. Contributions to Atmosph. Phys., 40, 57-64 (1997).Google Scholar
  6. [Er42]
    Ertel, H.: Der Vertikale Turbulenz-Wärmestrom in der Atmosphäre, Meteor. Z., 59, 250-253 (1942).Google Scholar
  7. [Ha89]
    Hanna, S.R.: Confidence limit for air quality models as estimated by bootstrap and jacknife resampling methods. Atmospheric Environment, 23, 1385-1395 (1989).CrossRefGoogle Scholar
  8. [MoEA06]
    Moreira, D.M., Vilhena, M.T., Tirabassi, T., Costa, C.P., Bodmann, B.: Simulation of pollutant dispersion in the atmosphere by the Laplace transform: the ADMM approach. Water, Air, and Soil Pollution, 177, 411-439 (2006).CrossRefGoogle Scholar
  9. [PaDu88]
    Panofsky, H.A., Dutton, J.A.: Atmospheric Turbulence, Wiley, New York (1988).Google Scholar
  10. [SePa98]
    Seinfeld, J.H., Pandis, S.N.: Atmospheric Chemistry and Physics: From Air Pollution to Climate Change, Wiley, New York (1998).Google Scholar
  11. [StSe66]
    Stroud, A.H., Secrest, D.: Gaussian Quadrature Formulas, Prentice-Hall, Englewood Cliffs, NJ (1966).Google Scholar
  12. [VaVe01]
    van Dop, H., Verver, G.: Countergradient transport revisited. J. Atmospheric Sci., 58, 2240-2247 (2001).CrossRefGoogle Scholar
  13. [WyWe91]
    Wyngaard, J.C., Weil, J.C.: Transport asymmetry in skewed turbulence. Phys. Fluids, A3, 155-162 (1991).Google Scholar
  14. [GrLy84]
    Gryning, S.E., Lyck, E.: Atmospheric dispersion from elevated sources in an urban area: comparison between tracer experiments and model caculations. J. Climate Appl. Meteorology, 23, 651-660 (1984).CrossRefGoogle Scholar

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© Birkhäuser Boston 2010

Authors and Affiliations

  1. 1.Universidade Federal de PelotasPelotasBrazil
  2. 2.Universidade Federal do Rio Grande do SulPorto AlegreBrazil
  3. 3.Istituto di Scienze dell’Atmosfera e del ClimaBolognaItaly

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