Low-pass filtered continuum streambed and bedload sediment mass balance laws for an alluvial, gravel-bed stream

In an alluvial, gravel-bed stream governed by a plane-bed bedload transport regime, the physicochemical properties, size distribution, and granular architecture of the sediment grains that constitute the streambed surface influence many hydrodynamic, geomorphic, chemical, and ecological processes. Consequently, the abilities to accurately characterize the morphology and model the morphodynamics of the streambed surface and its interaction with the bedload above and subsurface below are necessary for a more complete understanding of how sediment, flow, organisms, and biogeochemistry interact. We report on our progress in the bottom-up development of low-pass filtered continuum streambed and bedload sediment mass balance laws for an alluvial, gravel-bed stream. These balance laws are assembled in a four stage process. First, the stream sediment-water system is conceptually abstracted as a nested, multi-phase, multi-species, structured continuum. Second, the granular surface of an aggregate of sediment grains is mathematically defined. Third, an integral approach to mass balance, founded in the continuum theory of multiphase flow, is used to formulate primordial, differential, instantaneous, local, continuum, mass balance laws applicable at any material point within a gravel-bed stream. Fourth, area averaging and time-after-area averaging, employing planform, low-pass filtering expressed as correlation or convolution integrals and based on the spatial and temporal filtering techniques found in the fields of multiphase flow, porous media flow, and large eddy simulation of turbulent fluid flow, are applied to smooth the primordial equations while maximizing stratigraphic resolution and preserving the definitions of relevant morphodynamic surfaces. Our approach unifies, corrects, contextualizes, and generalizes prior efforts at developing stream sediment continuity equations, including the top-down derivations of the surface layer (or "active layer") approach of Hirano [1971a,b] and probabilistic approach of Parker et al. [2000], as well as the bottom-up, low-pass filtered continuum approach of Coleman & Nikora [2009] which employed volume and volume-after-time averaging. It accommodates partial transport (e.g., Wilcock & McArdell [1997], Wilcock [1997a,b]). Additionally, it provides: (1) precise definitions of the geometry and kinematics of sediment in a gravel-bed stream required to collect and analyze the high resolution spatial and temporal datasets that are becoming ever more present in both laboratory and field investigations, (2) a mathematical framework for the use of tracer grains in gravel-bed streams, including the fate of streambed-emplaced tracers as well as the dispersion of tracers in the bedload, (3) spatial and temporal averaging uncompromised by the Reynolds rules necessary to assess the nature of scale separation, and (4) a kinematic foundation for hybrid Langrangian-Eulerian models of sediment morphodynamics.