Similarity solutions for fluvial sediment fining by selective deposition

Current models for downstream sediment sorting by selective deposition generally perform well at describing observed sorting data. However, since most were developed initially for application to modern rivers, they are typically formulated in terms of hydraulic and bed-surface variables that are not readily measurable in the sedimentary record. Moreover, their algebraic complexity obscures some of the underlying simplicity of the segregation process. Here we show how a pair of hydraulically based sorting models developed by Parker et al. can be reformulated, with minimal loss of accuracy, in terms of the size distribution of the supplied sediment and the downstream depositional mass balance. By invoking constant dimensionless shear stress within either the gravel or sand regimes, reach-scale, short-term details of hydraulics and sediment transport are summarized via a pair of dimensionless relative mobility fimetions, one for gravel and one for sand. Our approach yields simplified similarity solutions in which the long-term longitudinal grain-size distribution of the substrate and the relative mobility functions can be collapsed into self-similar forms in which only local mean and standard deviation of sizes in transport are used as scaling parameters. The formulation we propose offers a simple means to explore the impact of controlling variables on fining profiles and can be easily incorporated in long-term, basin-scale numerical stratigraphic models, avoiding the necessity of modeling the details of hydraulics and sediment transport. The model involves a minimum number of physically based parameters, the numerical values of which can be determined from the spatial distribution of rate of deposition, dimensionless shear stress, and the coefficient of variation of the supply gravel or sand size distributions.