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Article

Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications

1
Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487, USA
2
State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, Nanjing 210098, China
3
School of Environment, Nanjing Normal University, Nanjing 210023, China
4
Bureau of Economic Geology, Jackson School of Geosciences, University of Texas Austin, Austin, TX 78713, USA
5
Atmospheric Sciences and Global Change, Pacific Northwest National Laboratory, Richland, WA 99352, USA
*
Author to whom correspondence should be addressed.
Academic Editor: Andrea Scapellato
Mathematics 2021, 9(7), 790; https://doi.org/10.3390/math9070790
Received: 24 February 2021 / Revised: 1 April 2021 / Accepted: 3 April 2021 / Published: 6 April 2021
(This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory)
Fractional calculus-based differential equations were found by previous studies to be promising tools in simulating local-scale anomalous diffusion for pollutants transport in natural geological media (geomedia), but efficient models are still needed for simulating anomalous transport over a broad spectrum of scales. This study proposed a hierarchical framework of fractional advection-dispersion equations (FADEs) for modeling pollutants moving in the river corridor at a full spectrum of scales. Applications showed that the fixed-index FADE could model bed sediment and manganese transport in streams at the geomorphologic unit scale, whereas the variable-index FADE well fitted bedload snapshots at the reach scale with spatially varying indices. Further analyses revealed that the selection of the FADEs depended on the scale, type of the geomedium (i.e., riverbed, aquifer, or soil), and the type of available observation dataset (i.e., the tracer snapshot or breakthrough curve (BTC)). When the pollutant BTC was used, a single-index FADE with scale-dependent parameters could fit the data by upscaling anomalous transport without mapping the sub-grid, intermediate multi-index anomalous diffusion. Pollutant transport in geomedia, therefore, may exhibit complex anomalous scaling in space (and/or time), and the identification of the FADE’s index for the reach-scale anomalous transport, which links the geomorphologic unit and watershed scales, is the core for reliable applications of fractional calculus in hydrology. View Full-Text
Keywords: fractional calculus; anomalous diffusion; multi-scale model; pollutant transport fractional calculus; anomalous diffusion; multi-scale model; pollutant transport
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MDPI and ACS Style

Zhang, Y.; Zhou, D.; Wei, W.; Frame, J.M.; Sun, H.; Sun, A.Y.; Chen, X. Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications. Mathematics 2021, 9, 790. https://doi.org/10.3390/math9070790

AMA Style

Zhang Y, Zhou D, Wei W, Frame JM, Sun H, Sun AY, Chen X. Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications. Mathematics. 2021; 9(7):790. https://doi.org/10.3390/math9070790

Chicago/Turabian Style

Zhang, Yong, Dongbao Zhou, Wei Wei, Jonathan M. Frame, Hongguang Sun, Alexander Y. Sun, and Xingyuan Chen. 2021. "Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications" Mathematics 9, no. 7: 790. https://doi.org/10.3390/math9070790

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