Exner equation: A continuum approximation of a discrete granular system

In contrast to using a standard control volume approach, general statements of sediment mass balance are derived herein from spatial averaging of the sub-particle-scale differential equation of solid mass conservation. The general form of the Exner equation for sediment continuity that is obtained enables analyses in terms of size fractions and also in terms of individual or successive layers, where layer interfaces (e. g., for the bed surface and for bed and suspended loads) are defined on the basis of isosurfaces of sediment concentration (volume fraction) or other sediment properties (e. g., densities or transport rates) within regions of constant concentration. The presented expressions highlight the averaged nature of variables and also the effects of the scales of consideration on definition and interpretation of the macroscopic (mixture-scale) sediment and layer properties (e. g., averaged densities, volume concentrations or fractions, velocities, transport modes and rates, interfaces, and bed layers). For appropriate simplifications, the general form of the Exner equation is shown to reduce to give more specific conventional expressions, revealing the assumptions implicit in these equations.