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Pure and Applied Geophysics

, Volume 176, Issue 4, pp 1797–1806 | Cite as

A Model Using Fractional Derivatives with Vertical Eddy Diffusivity Depending on the Source Distance Applied to the Dispersion of Atmospheric Pollutants

  • Paulo Henrique Farias Xavier
  • Erick Giovani Sperandio Nascimento
  • Davidson Martins MoreiraEmail author
Article
  • 90 Downloads

Abstract

This work presents an analytical solution of the two-dimensional advection–diffusion equation of fractional order, in the sense of Caputo and applied it to the dispersion of atmospheric pollutants. The solution is obtained using Laplace decomposition and homotopy perturbation methods, and it considers the vertical eddy diffusivity dependency on the longitudinal distance of the source with fractional exponents of the same order of the fractional derivative (K\(\propto\)xα). For validation of the model, simulations were compared with data from Copenhagen experiments considering moderately unstable conditions. The best results were obtained with α = 0.98, considering wind measured at 10 m, and α = 0.94 with wind measured at a height of 115 m.

Keywords

Decomposition method advection–diffusion equation planetary boundary layer pollutant dispersion 

Notes

Acknowledgements

We thank the financial support of CNPq and FAPESB.

References

  1. Crank, J. (1979). The mathematics of diffusion (p. 414). Oxford: Oxford University Press.Google Scholar
  2. Debnath, L. (2003). Recent applications of fractional calculus to science and engineering. International Journal of Mathematics and Mathematical Sciences, 2003(54), 3413–3442.CrossRefGoogle Scholar
  3. Degrazia, G. A., Moreira, D. M., & Vilhena, M. T. (2001). Derivation of an eddy diffusivity depending on source distance for a vertically inhomogeneous turbulence in a convective boundary layer. Journal of Applied Meteorology, 40, 1233–1240.CrossRefGoogle Scholar
  4. Essa, K. S. M., Etman, S. M., & Embaby, M. (2007). New analytical solution of the dispersion equation. Atmospheric Research, 84, 337–344.CrossRefGoogle Scholar
  5. Ganji, D. D., & Rafei, M. (2006). Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation by homotopy perturbation method. Physics Letters A, 356(2), 131–137.CrossRefGoogle Scholar
  6. Ghorbani, A. (2009). Beyond Adomian polynomials: He polynomials. Chaos, Solitons and Fractals, 39(3), 1486–1492.CrossRefGoogle Scholar
  7. Goulart, A. G. O., Lazo, M. J., Suarez, J. M. J., & Moreira, D. M. (2017). Fractional derivative models for atmospheric dispersion of pollutants. Physica A, 477, 9–19.CrossRefGoogle Scholar
  8. Gryning, S. E., Holtslag, A. M. M., Irwin, J., & Sivertsen, B. (1987). Applied dispersion modelling based on meteorological scaling parameters. Atmospheric Environment, 21, 79–89.CrossRefGoogle Scholar
  9. Gryning, S. E., & Lyck, E. (1984). Atmospheric dispersion from elevated sources in an urban area: Comparison between tracer experiments and model calculations. Journal of Climate and Applied Meteorology, 23(4), 651–660.CrossRefGoogle Scholar
  10. Hanna, S. R. (1989). Confidence limit for air quality models as estimated by bootstrap and jackknife resampling methods. Atmospheric Environment, 23, 1385–1395.CrossRefGoogle Scholar
  11. He, J. H. (1999). Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178, 257–262.CrossRefGoogle Scholar
  12. He, J. H. (2009). An elementary introduction to the homotopy perturbation method. Computers and Mathematics with Applications, 57(3), 410–412.CrossRefGoogle Scholar
  13. Jumarie, G. (2008). Fourier’s transformation of fractional order via Mittag-Leffler function and modified Riemann-Liouville derivatives. Journal of Applied Mathematics & Informatics, 26, 1101–1121.Google Scholar
  14. Moreira, D. M., Moraes, A. C., Goulart, A. G., & Albuquerque, T. T. (2014). A contribution to solve the atmospheric diffusion equation with eddy diffusivity depending on source distance. Atmospheric Environment, 83, 254–259.CrossRefGoogle Scholar
  15. Moreira, D. M., & Moret, M. (2018). A new direction in the atmospheric pollutant dispersion inside of the planetary boundary layer. Journal of Applied Meteorology and Climatology, 57(1), 185–192.CrossRefGoogle Scholar
  16. Moreira, D. M., & Vilhena, M. T. (2009). Air pollution and turbulence: Modeling and applications (p. 354). Boca Raton: CRC Press.CrossRefGoogle Scholar
  17. Moreira, D. M., Vilhena, M. T., Buske, D., & Tirabassi, T. (2009). The state-of-art of the GILTT method to simulate pollutant dispersion in the atmosphere. Atmospheric Research, 92, 1–17.CrossRefGoogle Scholar
  18. Moreira, D. M., Vilhena, M. T., Tirabassi, T., Buske, D., & Cotta, R. M. (2005). Near source atmospheric pollutant dispersion using the new GILTT method. Atmospheric Environment, 39, 6289–6294.CrossRefGoogle Scholar
  19. Pimentel, L. C. G., Perez-Grerrero, J. S., Ulke, A. G., Duda, F. P., & Heilbron Filho, P. F. L. (2014). Assessment of the unified analytical solution of the steady-state atmospheric diffusion equation for stable conditions. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 470, 20140021.CrossRefGoogle Scholar
  20. Podlubny, I. (1999). Fractional differential equations (p. 340). Cambridge: Academic.Google Scholar
  21. Rounds, W. (1955). Solutions of the two-dimensional diffusion equation. American Geophysical Union, 36, 395–405.CrossRefGoogle Scholar
  22. Sharan, M., & Modani, M. (2006). A two-dimensional analytical model for the dispersion of air-pollutants in the atmosphere with a capping inversion. Atmospheric Environment, 40, 3479–3489.CrossRefGoogle Scholar
  23. Sharan, M., Singh, M. P., & Yadav, A. K. (1996). A mathematical model for the dispersion in low winds with eddy diffusivities as linear functions of downwind distance. Atmospheric Environment, 30, 1137–1145.CrossRefGoogle Scholar
  24. Tirabassi, T., Buske, D., Moreira, D. M., & Vilhena, M. T. (2008). A two-dimensional solution of the advection-diffusion equation with dry deposition to the ground. Journal of Applied Meteorology and Climatology, 47, 2096–2104.CrossRefGoogle Scholar
  25. Wang, Y.X., Si, H.Y. & Mo, L.F. (2008). Homotopy perturbation method for solving reaction-diffusion equations. Mathematical Problems in Engineering Article ID 795838.Google Scholar
  26. Yeh, G., & Huang, C. (1975). Three-dimensional air pollutant modelling in the lower atmosphere. Boundary-Layer Meteorology, 9, 381–390.CrossRefGoogle Scholar
  27. Yildirim, A., & Kocak, H. (2009). Homotopy perturbation method for solving the space-time fractional advection-dispersion equation. Advances in Water Resources, 32(12), 1711–1716.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Paulo Henrique Farias Xavier
    • 1
  • Erick Giovani Sperandio Nascimento
    • 1
  • Davidson Martins Moreira
    • 1
    Email author
  1. 1.SENAI CIMATECSalvadorBrazil

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