Backwater number scaling of alluvial bed forms

Abstract

The backwater number, Bw, compares the backwater length scale to the length scale of alluvial bed forms. We derive theory to show that Bw plays an important role in determining the behavior and scaling of morphodynamic systems. When Bw ≫ 1, spatial patterns in deposition and erosion derive from flow accelerations associated with changes in flow depth, and bed evolution is akin to a kinematic wave. When Bw ≪ 1, the spatial pattern of shear stress is determined by variations in energy slope, and alluvial beds experience topographic dispersion. This theory is confirmed using a numerical model and data compiled from the literature. We present a data set of Bw for bed forms ranging from dunes to river deltas, including field and experimental measurements. For field-scale measurements, we find that dunes have Bw > 49, braid bars exist in the range Bw = [7.1,17], meanders have a range Bw = [7.1,18], and river mouth deposition ranges over Bw = [7.4,29]. Further, alluvial morphologies that are easily recreated in the laboratory (dunes and avulsions) have overlapping field and laboratory Bw ranges. In contrast, alluvial forms that have traditionally been difficult to recreate (meanders and river mouth processes) have field Bw that are difficult to match in laboratory settings. Large experimental Froude numbers are shown to reduce experimental Bw and incite diffusional behavior. Finally, we demonstrate the utility of Bw scaling for estimating fundamental scales in sedimentary systems.