APPLICATION OF FRACTIONAL DIFFERENTIAL EQUATION TO INTERPRET METAL UPTAKE DYNAMICS IN RIVERS FROM BEDFORM SCALE TO BASIN SCALE

Fractional differential equations (FDEs) provide promising phenomenological models to quantify various physical dynamics in hydrology exhibiting non-Fickian characteristics. Previous applications of FDEs in hydrologic processes, however, are mainly limited to non-reactive tracer transport through single-scale hydrological media, such as laboratory sand columns or field scale aquifers, while the feasibility of FDEs for non-Fickian dynamics within and across scales remains obscure. This study aims to fill this knowledge gap by developing a fractional-order advection-dispersion-reaction (fADR) equation to model metal uptake and oxidation in natural rivers varying from geomorphologic unit scale (10^-1~10⁰m) to watershed scale (>10³m). Theoretical analysis suggests that the order of the space and/or time fractional derivative, or the space/time truncation parameter, should evolve with scales, since larger-scale driving forces may be effective for the dynamics of the dissolved metal with an increasing travel distance. Application of the fADR model for literature data (hyporheic uptake of manganese observed in Pinal Creek, Arizona) shows that, however, the index of fADR model remains stable when crossing scales, while the effective transport parameters change significantly. Using our fADR model, we were better able to fit the earlier arrival of the data than the previously used model.