[1] The initiation of channels and the factors that control their spacing is one of the most fundamental problems in geomorphology. Here we present an analysis of channelization that is downstream driven, so called because channelization is initiated at some point away from the divide where the Froude critical flow is achieved. By using two novel techniques, the ''frozen time approach'' and the ''momentary stability concept,'' we are able, for the first time, to investigate downstream-driven channelization that occurs on a plateau with an evolving base profile. The ''frozen time approach'' assumes that the time evolution of the base state can be ignored in the linearized equations, whereas the ''momentary stability concept'' assumes that an unsteady base state becomes momentarily unstable to a disturbance when the growth of the disturbance is faster than the evolution of the base state. The analysis shows that an arbitrary slope profile is momentarily unstable and incised by channels before it evolves into a steady base state if the topographic curvature at the location where the Froude critical flow is achieved is sufficiently small. As the curvature decreases further, dominant channel spacing increases slightly in a purely erosional case, while clear dominant channel spacing tends not to appear in an erosional-depositional case. It is also found from the analysis that in both cases, the channel spacing is on the order of 1000 times the Froude critical depth. The results are compared with field observations and experiments, and they show satisfactory agreement.

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[1] Because seepage erosion is generated by complex interactions with other processes, associated stream incision process is not well understood. In this study, some fundamental characteristics of incipient incision by seepage erosion were investigated by laboratory experiments and linear stability analysis. The experiments were conducted with various sediment layer depths and gradients. With similar discharges in the experiments, incision spacing decreases with increasing depth of the sediment layer and with increasing gradient, whereas incision width increases with increasing sediment layer depth. A linear stability analysis was performed using the Dupuit-Forchheimer equation and an expression of the planimetric retreat of the scarp. The retreat velocity of the scarp consists of two terms: (1) a power law function that describes the specific discharge in excess of a critical discharge and (2) a diffusion-like function that describes the incision edge shapes, in which the retreat rate is enhanced or reduced by the convexity and concavity of the edges, respectively. This analysis shows that the characteristic incision spacing becomes infinitely small when the effect of the edge shapes is excluded. Using the experimental data of incision spacings, the values of the diffusion-like coefficient in the second term were estimated. Since the weight of a failure block and hydraulic pressure are the driving forces in the slope stability analysis, a relationship was found between the diffusion-like coefficient and the combination of the two forces.

Channel head bifurcation is a key factor for generating complexity of channel networks. Here we investigate incipient channel head bifurcation using linear stability analysis. Channel heads are simplified as circular hollows, toward which surface sheet flow accelerates in the radial direction. Sinusoidal perturbations in the angular direction with different angular wave numbers k are imposed on the bed, and their growth rates truenormalΩ~ are computed. Because the channel head radius trueR~c is extending over time, the base state (circular hollow in the absence of perturbations) also evolves continuously. With the use of the momentary stability concept, the evolving base state is defined as momentarily unstable to the imposed perturbation if the disturbance is growing faster than the evolution of the base state. It was found that in the range of sufficiently small trueR~c, bifurcation cannot be initiated. As trueR~c increases, bifurcation starts to be possible with k≈ 3–5. A higher k implies bifurcation with a narrower channel junction angle (θ = 2π/k). The average junction angle of the Colorado High Plains for the smallest drainage area is about 85° with a standard deviation of 35° (Sólyom and Tucker, 2007). Our predicted angles (75°–120°) agree qualitatively with the observed angles. Finally, we propose a simple criterion to compute the threshold trueR~c for the onset of bifurcation.

E Input,0 (t p ) 207 (100%)DM,0 (t p ) 25 (12.1%) E Dissipated,0 (t p ) 70 (33.8%) E F,T,0 (t p ) 37 (17.9%) E F,V,0 (t p ) 8 (3.8%) DMs-district flowmeters, MWA-Metropolitan Waterworks Authority LAPPRASERT ET AL. | JULY 2018 • 110:7 | JOURNAL AWWA E25