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Soil water sustains life on Earth, and how to quantify water equilibrium and kinetics in soil remains a challenge for over a century despite significant efforts. For example, various models were proposed to interpret non‐Darcian flow in saturated soils, but none of them can capture the full range of non‐Darcian flow. To unify the different models into one overall framework and improve them if needed, this technical note proposes a theory based on the tempered stable density (TSD) assumption for the soil‐hydraulic property distribution, recognizing that the underlying hydrologic processes all occur in the same, albeit very complex and not measurable at all the relevant scales, soil‐water system. The TSD assumption forms a unified fractional‐derivative equation (FDE) using subordination. Preliminary applications show that simplified FDEs, with proposed hydrological interpretations and TSD distributed properties, effectively capture core equilibrium and kinetic water processes, spanning non‐Darcian flow, water retention, moisture movement, infiltration, and wetting/drying, in the soil‐water system with various degrees and scales of system heterogeneity. Model comparisons and evaluations suggest that the TSD may serve as a unified density for the properties of a broad range of soil‐water systems, driving multi‐rate mass, momentum, and energy equilibrium/kinetic processes often oversimplified by classical models as single‐rate processes.

Hydrologically mediated hot moments (HM‐HMs) of transient anomalous diffusion (TAD) denote abrupt shifts in hydraulic conditions that can profoundly influence the dynamics of anomalous diffusion for pollutants within heterogeneous aquifers. How to efficiently model these complex dynamics remains a significant challenge. To bridge this knowledge gap, we propose an innovative model termed “the impulsive, tempered fractional advection‐dispersion equation” (IT‐fADE) to simulate HM‐HMs of TAD. The model is approximated using an L1‐based finite difference solver with unconditional stability and an efficient convergence rate. Application results demonstrate that the IT‐fADE model and its solver successfully capture TAD induced by hydrologically trigged hot phenomena (including hot moments and hot spots) across three distinct aquifers: (a) transient sub‐diffusion arising from sudden shifts in hydraulic gradient within a regional‐scale alluvial aquifer, (b) transient sub‐ or super‐diffusion due to convergent or push‐pull tracer experiments within a local‐scale fractured aquifer, and (c) transient sub‐diffusion likely attributed to multiple‐conduit flow within an intermediate‐scale karst aquifer. The impulsive terms and fractional differential operator integrated into the IT‐fADE aptly capture the ephemeral nature and evolving memory of HM‐HMs of TAD by incorporating multiple stress periods into the model. The sequential HM‐HM model also characterizes breakthrough curves of pollutants as they encounter hydrologically mediated, parallel hot spots. Furthermore, we delve into discussions concerning model parameters, extensions, and comparisons, as well as impulse signals and the propagation of memory within the context of employing IT‐fADE to capture hot phenomena of TAD in aquatic systems.

Spatiotemporal fractional‐derivative models (FDMs) have been increasingly used to simulate non‐Fickian diffusion, but methods have not been available to define boundary conditions for FDMs in bounded domains. This study defines boundary conditions and then develops a Lagrangian solver to approximate bounded, one‐dimensional fractional diffusion. Both the zero‐value and nonzero‐value Dirichlet, Neumann, and mixed Robin boundary conditions are defined, where the sign of Riemann‐Liouville fractional derivative (capturing nonzero‐value spatial‐nonlocal boundary conditions with directional superdiffusion) remains consistent with the sign of the fractional‐diffusive flux term in the FDMs. New Lagrangian schemes are then proposed to track solute particles moving in bounded domains, where the solutions are checked against analytical or Eulerian solutions available for simplified FDMs. Numerical experiments show that the particle‐tracking algorithm for non‐Fickian diffusion differs from Fickian diffusion in relocating the particle position around the reflective boundary, likely due to the nonlocal and nonsymmetric fractional diffusion. For a nonzero‐value Neumann or Robin boundary, a source cell with a reflective face can be applied to define the release rate of random‐walking particles at the specified flux boundary. Mathematical definitions of physically meaningful nonlocal boundaries combined with bounded Lagrangian solvers in this study may provide the only viable techniques at present to quantify the impact of boundaries on anomalous diffusion, expanding the applicability of FDMs from infinite domains to those with any size and boundary conditions.

The operational time distribution (OTD) defines the time for bed‐load sediment spent in motion, which is needed to characterize the random nature of sediment transport. This study explores the influence of bed clusters and size gradation on OTD for non‐uniform bed‐loads. First, both static and mobile bed armouring experiments were conducted in laboratorial flumes to monitor the transport of mixed sand/gravel sediments. Only in the mobile armouring experiment did apparent bed clusters develop, because of stable feeding and a longer transport period. Second, a generalized subordinated advection (GSA) model was applied to quantify the observed dynamics of tracer particles. Results show that forthe static armour layer(without sediment feed), the best‐fit OTD assigns more weight to the large displacement of small particles, likely because of the size‐selective entrainment process. The capacity coefficient in the GSA model, which affects the width of the OTD, is space dependent only for small particles whose dynamics can be significantly affected by larger particles and whose distribution is more likely to be space dependent in a mixed sand and gravel system. However, the OTD forthe mobile armour layer(with sediment recirculation) exhibited longer tails for larger particles. This is because the trailing edge of larger particles is more resistant to erosion, and their leading front may not be easily trapped by self‐organized bed clusters. The strong interaction between particle–bed may cause the capacity coefficient to be space‐dependent for bed‐load transport along mobile armour layers. Therefore, the combined laboratory experiments and stochastic model analysis show that the OTD may be affected more by particle–bed interactions (such as clusters) than by particle–particle interactions (e.g. hiding and exposing), and that the GSA model can quantify mixed‐size sand/gravel transport along river beds within either static or mobile armour layers. Copyright © 2016 John Wiley & Sons, Ltd.