A fractional-order tempered-stable continuity model to capture surface water runoff

The dynamics of surface runoff exhibits scale-dependent anomalous behavior due to heterogeneity present within natural systems, including spatial variations in surface topography and soil hydraulic properties which may not be efficiently captured by traditional modeling approaches. This study proposes a fractional-order continuity equation to quantify the scale-dependent anomalous behavior of overland flow, where the influence of sub-scale heterogeneity on flow dynamics can be characterized using spatiotemporally nonlocal terms built upon fractional derivatives. Both Eulerian and Lagrangian solvers are developed and cross-verified to approximate the proposed physical model. Numerical experiments further show that, on one hand, the space-fractional diffusive term in the flow model does not lead to apparent early arrivals in the steep rising limb of a hydrograph. This is likely caused by the combined effects of uniformly distributed precipitation over the entire hillslope and the immediate arrival of surface runoff at the downslope portion of the hillslope, both of which can overshadow the leading front of superdiffusion. The time-fractional term in the model, on the other hand, can 1) distinguish mobile and immobile water packets, 2) account for the strong time-nonlocal influence of net recharge on the receding limb of a hydrograph, and 3) efficiently characterize a wide range of late-time behavior of flow according to the tempered stable law. The applicability of the physical model is tested using two local-scale surface runoff data sets. The fractional-order tempered-stable flow model therefore may capture the complex hydrological response to precipitation in the real-world land surface.