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Fractional Atmospheric Pollutant Dispersion Equation in a Vertically Inhomogeneous Planetary Boundary Layer: an Analytical Solution Using Conformable Derivatives

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Abstract

This study aims to investigate the potential of fractional derivatives in the mathematical modelling of the dispersion of air pollutants. For this purpose, an analytical solution of the fractional two-dimensional advection–diffusion equation by combining generalized integral Laplace transform technique (GILTT) and conformable derivatives methods was proposed. Although the use of the conformable derivatives loses the non-local character contained in the fractional derivatives, fractional parameters remain in the solution. Thus, this procedure allows considering the anomalous behaviour in the diffusion process, resulting in a new methodology here called the \(\alpha\)-GILTT method. The concentrations calculated with the model were compared with ground-level crosswind-integrated concentrations data from the Copenhagen and Prairie Grass experiments. The statistical indices showed the best results for the moderately unstable Copenhagen experiment under conditions of low fractionality (values close to unity). However, for the strongly convective Prairie Grass experiment, the results showed greater dependence on the fractional parameters (integer-order: NMSE = 0.90, COR = 0.81, FAT2 = 0.63; non-integer order: NMSE = 0.56, COR = 0.89, FAT2 = 0.84). The results suggest that fractional parameters are dependent on atmospheric stability and open a new direction to improve the knowledge of the atmospheric pollutant dispersion processes.

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Data Availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

We are thankful for the support of SENAI CIMATEC and CNPq.

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  1. PPGMCTI, SENAI CIMATEC, Salvador, Brazil

    André Luiz Santos Soledade & Davidson Martins Moreira

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  1. André Luiz Santos Soledade
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Santos Soledade, A.L., Martins Moreira, D. Fractional Atmospheric Pollutant Dispersion Equation in a Vertically Inhomogeneous Planetary Boundary Layer: an Analytical Solution Using Conformable Derivatives. Water Air Soil Pollut 233, 395 (2022). https://doi.org/10.1007/s11270-022-05864-7

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