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Boundary-Layer Meteorology

, Volume 170, Issue 2, pp 285–304 | Cite as

Mathematical Model Using Fractional Derivatives Applied to the Dispersion of Pollutants in the Planetary Boundary Layer

  • Palmira Santana Acioli
  • Frederico Andrade Xavier
  • Davidson Martins MoreiraEmail author
Research Article
  • 104 Downloads

Abstract

We present an analytical solution of the advection–diffusion equation of integer and fractional order applied to the dispersion of pollutants in the planetary boundary layer. The solution is obtained using the Laplace decomposition method, and the perturbation is obtained by homotopy, considering the Caputo derivative in the fractional case. To obtain the solution, two types of eddy diffusivities are used: in the integer-order equation, the eddy diffusivity is dependent on the longitudinal distance from the source (K\( \propto \)x and K\( \propto \)x2); in the fractional-order equation, the eddy diffusivity is constant. To validate the model, the results are compared with experimental data from the literature (Copenhagen and Prairie Grass). In the Copenhagen experiment, which was conducted under moderately unstable conditions, the best results are obtained under the influence of the memory effect with the eddy diffusivity dependent upon the source distance as K\( \propto \)x (with constant eddy diffusivity in the equation with a fractional derivative). However, in the strongly convective case of the Prairie Grass experiment, the best results are obtained only when the eddy diffusivity depends on the source distance as K\( \propto \)x2.

Keywords

Advection–diffusion equation Decomposition method Planetary boundary layer Pollutant dispersion 

Notes

Acknowledgements

We thank CNPq and FAPESB for financial support.

References

  1. Albani RAS, Duda FP, Pimentel LCG (2015) On the modeling of atmospheric pollutant dispersion during a diurnal cycle: a finite element study. Atmos Environ 118:19–27CrossRefGoogle Scholar
  2. Arya S (1995) Modeling and parameterization of near-source diffusion in weak winds. J Appl Meteorol 34(5):1112–1122CrossRefGoogle Scholar
  3. Barad ML (1958) Project Prairie-Grass: a field program in diffusion. Geophys Res. Air Force Cambridge Research Centre, USA, vol I and II (59)Google Scholar
  4. Debnath L (2003) Recent applications of fractional calculus to science and engineering. Int J Math Math Sci 54:3413–3442CrossRefGoogle Scholar
  5. Degrazia GA, Velho HF, Carvalho JC (1997) Nonlocal Exchange coefficients for the convective boundary layer derived from spectral properties. Contrib Atmos Phys 70(1):57–64Google Scholar
  6. Degrazia GA, Moreira DM, Vilhena MT (2001) Derivation of an eddy diffusivity depending on source distance for a vertically inhomogeneous turbulence in a convective boundary layer. J Appl Meteorol 40:1233–1240CrossRefGoogle Scholar
  7. Essa KSM, Etman SM, Embaby M (2007) New analytical solution of the dispersion equation. Atmos Res 84:337–344CrossRefGoogle Scholar
  8. Ganji DD, Rafei M (2006) Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method. Phys Lett A 356(2):131–137CrossRefGoogle Scholar
  9. Ghorbani A (2009) Beyond Adomian polynomials: he polynomials. Chaos Soliton Frac 39(3):1486–1492CrossRefGoogle Scholar
  10. Goulart AGO, Lazo MJ, Suarez JMJ, Moreira DM (2017) Fractional derivative models for atmospheric dispersion of pollutants. Phys A 477:9–19CrossRefGoogle Scholar
  11. Gryning SE, Larsen SE (1984) Evaluation of a K-model formulated in terms of Monin–Obukhov similarity with the results from the Prairie Grass experiments. In: De Wispelaere C (ed) Air pollution modeling and its application III. Nato challenges of modern society (Energy engineering and advanced power systems), vol 5. Springer, BostonGoogle Scholar
  12. Gryning SE, Lyck E (1984) Atmospheric dispersion from elevated sources in an urban area: comparison between tracer experiments and model calculations. J Clim Appl Meteorol 23(4):651–660CrossRefGoogle Scholar
  13. Gryning SE, Holtslag AMM, Irwin J, Sivertsen B (1987) Applied dispersion modelling based on meteorological scaling parameters. Atmos Environ 21(1):79–89CrossRefGoogle Scholar
  14. Hanna SR (1989) Confidence limit for air quality models as estimated by bootstrap and jacknife resampling methods. Atmos Environ 23(6):1385–1395CrossRefGoogle Scholar
  15. He JH (1999) Homotopy perturbation technique. Comput Methods Appl Mech Eng 178:257–262CrossRefGoogle Scholar
  16. He JH (2006) Recent development of the homotopy perturbation method. Topol Methods Nonlinear Anal 31(2):205–209Google Scholar
  17. He JH (2009) An elementary introduction to the homotopy perturbation method. Comput Math Appl 57(3):410–412CrossRefGoogle Scholar
  18. Khan Y, Wu Q (2011) Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Comput Math Appl 61:1963–1967CrossRefGoogle Scholar
  19. Kumar P, Sharan M (2010) An analytical model for dispersion of pollutants from a continuous source in the atmospheric boundary layer. Proc R Soc A 466:383–406CrossRefGoogle Scholar
  20. Moreira DM, Moret M (2018) A new direction in the atmospheric pollutant dispersion inside of the planetary boundary layer. J Appl Meteorol Clim 57(1):185–192CrossRefGoogle Scholar
  21. Moreira DM, Vilhena MT (2009) Air pollution and turbulence: modeling and applications. CRC Press, Boca Raton, p 354CrossRefGoogle Scholar
  22. Moreira DM, Carvalho JC, Degrazia GA, Vilhena MT, Moraes MR (2002) Dispersion parameterization applied to strong convection: low sources case. Hyb Meth Eng 4(1–2):89–107Google Scholar
  23. Moreira DM, Rizza U, Vilhena MT, Goulart AG (2005a) Semi-analytical model for pollution dispersion in the Planetary Boundary Layer. Atmos Environ 39(14):2689–2697CrossRefGoogle Scholar
  24. Moreira DM, Vilhena MT, Tirabassi T, Buske D, Cotta RM (2005b) Near source atmospheric pollutant dispersion using the new GILTT method. Atmos Environ 39(34):6289–6294CrossRefGoogle Scholar
  25. Moreira DM, Vilhena MT, Buske D, Tirabassi T (2009) The state-of-art of the GILTT method to simulate pollutant dispersion in the atmosphere. Atmos Res 92(1):1–17CrossRefGoogle Scholar
  26. Moreira DM, Moraes AC, Goulart AG, Albuquerque TT (2014) A contribution to solve the atmospheric diffusion equation with eddy diffusivity depending on source distance. Atmos Environ 83:254–259CrossRefGoogle Scholar
  27. Nascimento ES, Moreira DM, Albuquerque TTA (2017) The development of a new model to simulate the dispersion of rocket exhaust clouds. Aerosp Sci Tech 69:298–312CrossRefGoogle Scholar
  28. Nascimento ES, Souza NBP, Kitagawa YKL, Moreira DM (2018) Simulated dispersion of the gas released by the SpaceX Falcon 9 rocket explosion. J Spacecr Rockets 55:1–9CrossRefGoogle Scholar
  29. Nieuwstadt FTM (1980) Application of mixed-layer similarity to the observed dispersion from a ground-level source. J Appl Meteorol 19(2):157–162CrossRefGoogle Scholar
  30. Nigmatullin RR (1986) The realization of the generalized transfer equation in a medium with fractal geometry. Phys Star Sol B 133:425–430CrossRefGoogle Scholar
  31. Pimentel LCG, Perez-Grerrero JS, Ulke AG, Duda FP, Heilbron Filho PFL (2014) Assessment of the unified analytical solution of the steady-state atmospheric diffusion equation for stable conditions. Proc R Soc A 470:20140021CrossRefGoogle Scholar
  32. Podlubny I (1999) Fractional differential equations. Academic Press, Cambridge, p 340Google Scholar
  33. Rounds W (1955) Solutions of the two-dimensional diffusion equation. Am Geophys Union 36:395–405CrossRefGoogle Scholar
  34. Sharan M, Modani M (2006) A two-dimensional analytical model for the dispersion of air-pollutants in the atmosphere with a capping inversion. Atmos Environ 40(19):3479–3489CrossRefGoogle Scholar
  35. Sharan M, Singh MP, Yadav AK (1996) A mathematical model for the dispersion in low winds with eddy diffusivities as linear functions of downwind distance. Atmos Environ 30(7):1137–1145CrossRefGoogle Scholar
  36. Tirabassi T, Buske D, Moreira DM, Vilhena MT (2008) A two-dimensional solution of the advection–diffusion equation with dry deposition to the ground. J Appl Meteorol Climatol 47:2096–2104CrossRefGoogle Scholar
  37. Wang YX, Si HY, Mo LF (2008) Homotopy perturbation method for solving reaction-diffusion equations. Math Probl Eng. Article ID 795838Google Scholar
  38. Weil J (1988) Dispersion in the convective boundary layer. In: Venkatram A, Wyngaard J (eds) Lectures on air pollution modeling. American Meteorological Society, Massachusetts, p 390Google Scholar
  39. Wortmann S, Vilhena MT, Moreira DM, Buske D (2005) A new analytical approach to simulate the pollutant dispersion in the PBL. Atmos Environ 39(12):2171–2178CrossRefGoogle Scholar
  40. Yeh G, Huang C (1975) Three-dimensional air pollutant modelling in the lower atmosphere. Boundary-Layer Meteorol 9:381–390CrossRefGoogle Scholar
  41. Yildirim A, Kocak H (2009) Homotopy perturbation method for solving the space-time fractional advection–dispersion equation. Adv Water Resour 32(12):1711–1716CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Palmira Santana Acioli
    • 1
  • Frederico Andrade Xavier
    • 1
  • Davidson Martins Moreira
    • 1
    Email author
  1. 1.Integrated Campus of Manufacturing and Technology - SENAI CIMATECSalvadorBrazil

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