We illustrate the river-wall interactions with a conceptual model that forms the basis for our subsequent numerical analysis. Following a period in which an alluvial river is in equilibrium with its sediment supply (Figure 3a), vertical incision is initialized when an external forcing either reduces the river’s sediment load or increases its water discharge [Schumm, 1973; Leopold and Bull, 1979]. As the channel entrains material from its bed, it also easily erodes the shallow banks and freely migrates laterally. Vertical incision and lateral erosion continue and the random lateral migration of the stream forms fill-cut terraces [Bull, 1991; Limaye and Lamb, 2016] (Figure 3b). These fill-cut terraces result from the combination of external forcing (vertical incision) and autogenic dynamics (lateral migration). Progressively, the height of the valley walls increases, the lateral erosion rate of the river against the walls is reduced by the increasing amount of sediment they yield, and the vertical erosion is distributed across a smaller area, increasing net vertical incision [Nicholas and Quine, 2007]. As incision continues, the episodic collapse of the ever higher walls through undercut and run off erosion produces so much sediment that it exceeds the local sediment transport capacity at the channel margins and talus piles start accumulating at the toes of the cliffs (Figure 3c). The rivers of the north piedmont of the East Tian Shan, flowing on an unlithified fanglomerate [Avouac et al., 1993], illustrate this behavior well (Figure 1). Preceding the onset of incision 10–12 kyrs ago, the rivers migrated over an expansive and steep floodplain. Over 2km wide. This floodplain has since been reduced to a modern width of 200–300m, matching the bank full width of the channel after up to 300 m of channel entrenchment and a reduction of its slope by a factor ofalmost2[Poisson and Avouac, 2004]. The existing talus deposits reduce the width of the floodplain, forcing the channel to narrow with the shallowing slope [James, 1991; Surian and Rinaldi, 2003].

Instead of a continuous slowdown of vertical incision expected from a decrease in channel slope as the river profile nears its equilibrium in a 1-D setup, we expect here a sustained vertical incision rate and even in some cases an apparent acceleration of vertical incision rates. The faster rates are unrelated to any external forcing but, rather, caused by lateral inputs of sediment from valley walls that narrow the channel and increase the shear stress over the bed. Such an acceleration could be misinterpreted as resulting from some environmental change.

Finally, when the river is near its longitudinal equilibrium and can no longer incise vertically, it will still occasionally erode the valley walls and widen its floodplain (Figure3d).Asaresult, the youngest fill-cut terraces are destroyed first while the oldest terraces have a higher probability of preservation. A similar preservation bias toward older strath terraces has been demonstrated in the numerical modeling of Limaye and Lamb[2016]. Our model is aimed at constraining the timing and magnitude of these processes governing terrace creation and preservation.