Cross-divide difference in channel-head steady-state elevation controls drainage-divide migration

It is generally agreed that the channel-head steady-state elevations ( $z_s$) across a drainage divide are different when the drainage divide is moving. However, whether it is the hillslope or river channel that absorbs the cross-divide difference in channel-head steady-state elevation ( $\Delta z_s$) remains unclear. These different views have consequences for both the methods used to measure drainage-divide stability and tectonic reconstructions from channel profiles. Two methods for determining drainage-divide stability include Gilbert metrics and χ-plots, which emphasise the role of hillslopes and river channels, respectively. Here, we address this issue by deducing equations for estimating $\Delta z_s$ and identifying the absorbers of $\Delta z_s$ using numerical simulations and two natural cases. Our results show that both hillslopes and river channels absorb parts of $\Delta z_s$ in each case; however, the proportion absorbed varies from case to case. When the hillslope absorbs a greater proportion of $\Delta z_s$, the river channel absorbs less, and vice versa. We suggest that both Gilbert metrics and χ-plots should be applied when evaluating drainage-divide stability; if either suggests the divide is unstable, then it is indeed unstable. Moreover, the river channel profiles on both sides of a drainage divide are in disequilibrium when the divide is moving, and the erosion rates are greater and less than the uplift rates at the expanding and shrinking sides, respectively. This underscores that drainage-divide migration can significantly hinder the extraction of uplift history from channel profiles.