Mathematical Model Using Fractional Derivatives Applied to the Dispersion of Pollutants in the Planetary Boundary Layer
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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.
KeywordsAdvection–diffusion equation Decomposition method Planetary boundary layer Pollutant dispersion
We thank CNPq and FAPESB for financial support.
- 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
- 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
- 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
- He JH (2006) Recent development of the homotopy perturbation method. Topol Methods Nonlinear Anal 31(2):205–209Google Scholar
- 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
- Podlubny I (1999) Fractional differential equations. Academic Press, Cambridge, p 340Google Scholar
- Wang YX, Si HY, Mo LF (2008) Homotopy perturbation method for solving reaction-diffusion equations. Math Probl Eng. Article ID 795838Google Scholar
- 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