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A new parameterization for the concentration flux using the fractional calculus to model the dispersion of contaminants in the Planetary Boundary Layer

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  • Goulart, A.G.
  • Lazo, M.J.
  • Suarez, J.M.S.

Abstract

In the present work, we propose a new parameterization for the concentration flux using fractional derivatives. The fractional order differential equation in the longitudinal and vertical directions is used to obtain the concentration distribution of contaminants in the Planetary Boundary Layer. We solve this model and we compare the solution against both real experiments and traditional integer order derivative models. We show that our fractional model gives very good results in fitting the experimental data, and perform far better than the traditional Gaussian model. In fact, the fractional model, with constant wind speed and a constant eddy diffusivity, performs even better than some models found in the literature where it is considered that the wind speed and eddy diffusivity are functions of the position. The results obtained show that the structure of the fractional order differential equation is more appropriate to calculate the distribution of dispersed contaminants in a turbulent flow than an integer-order differential equation. Furthermore, a very important result we found it is that there should be a relation between the order α of the fractional derivative with the physical structure of the turbulent flow.

Suggested Citation

  • Goulart, A.G. & Lazo, M.J. & Suarez, J.M.S., 2019. "A new parameterization for the concentration flux using the fractional calculus to model the dispersion of contaminants in the Planetary Boundary Layer," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 518(C), pages 38-49.
    • Handle: RePEc:eee:phsmap:v:518:y:2019:i:c:p:38-49
    DOI: 10.1016/j.physa.2018.11.064
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    References listed on IDEAS

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    1. Goulart, A.G.O. & Lazo, M.J. & Suarez, J.M.S. & Moreira, D.M., 2017. "Fractional derivative models for atmospheric dispersion of pollutants," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 477(C), pages 9-19.
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    Cited by:

    1. Jajarmi, Amin & Yusuf, Abdullahi & Baleanu, Dumitru & Inc, Mustafa, 2020. "A new fractional HRSV model and its optimal control: A non-singular operator approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 547(C).
    2. Rubayyi T. Alqahtani & Abdullahi Yusuf & Ravi P. Agarwal, 2021. "Mathematical Analysis of Oxygen Uptake Rate in Continuous Process under Caputo Derivative," Mathematics, MDPI, vol. 9(6), pages 1-19, March.
    3. Miglena N. Koleva & Lubin G. Vulkov, 2023. "Numerical Solution of Fractional Models of Dispersion Contaminants in the Planetary Boundary Layer," Mathematics, MDPI, vol. 11(9), pages 1-21, April.
    4. Goulart, A.G. & Lazo, M.J. & Suarez, J.M.S., 2020. "A deformed derivative model for turbulent diffusion of contaminants in the atmosphere," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 557(C).
    5. Jajarmi, Amin & Arshad, Sadia & Baleanu, Dumitru, 2019. "A new fractional modelling and control strategy for the outbreak of dengue fever," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 535(C).

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    2. Miglena N. Koleva & Lubin G. Vulkov, 2023. "Numerical Solution of Fractional Models of Dispersion Contaminants in the Planetary Boundary Layer," Mathematics, MDPI, vol. 11(9), pages 1-21, April.
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