Probabilistic exner sediment continuity equation for mixtures with no active layer

The Exner equation of sediment continuity is the foundation of river morphodynamics. Generalization of this equation to mixtures of grain sizes has required the introduction of an active layer (i.e., a buffer layer between the sediment moving in the water column and the immobile substrate below). The active layer is defined to be a well-mixed layer, with no vertical structure, that encompasses those grains available to exchange directly with the moving sediment. The sediment in the substrate below exchanges with the active layer only as the bed aggrades or degrades. The active layer concept is a useful one that has served the research community well for 3 decades. However, the division of the erodible bed into a discrete active layer and substrate must represent only an approximation of a more general formulation that contains no active layer and in which parameters pertaining to the entrainment from and deposition to the bed vary continuously with depth below the sediment-water interface. Here the probability density function of bed elevation is used to derive a general Exner equation of sediment continuity with no discrete layers. The formulation is applicable to both sediment mixtures and tracers in uniform sediment. Although the treatment requires more information than that of the active layer approach, it offers the prospect of a better understanding of how streams create a stratigraphic record of their activities through deposition.