Hydraulics of floods upstream of horseshoe canyons and waterfalls

Horseshoe waterfalls are ubiquitous in natural streams, bedrock canyons, and engineering structures. Nevertheless, water flow patterns upstream of horseshoe waterfalls are poorly known and likely differ from the better studied case of a one-dimensional linear step because of flow focusing into the horseshoe. This is a significant knowledge gap because the hydraulics at waterfalls controls sediment transport and bedrock incision, which can compromise the integrity of engineered structures and influence the evolution of river canyons on Earth and Mars. Here we develop new semiempirical theory for the spatial acceleration of water upstream of, and the cumulative discharge into, horseshoe canyons and waterfalls. To this end, we performed 110 numerical experiments by solving the 2-D depth-averaged shallow-water equations for a wide range of flood depths, widths and discharges, and canyon lengths, widths and bed gradients. We show that the upstream, normal flow Froude number is the dominant control on lateral flow focusing and acceleration into the canyon head and that focusing is limited when the flood width is small compared to a cross-stream backwater length scale. In addition, for sheet floodsmuch wider than the canyon, flow focusing into the canyon head leads to reduced discharge (and drying in cases) across the canyon sidewalls, which is especially pronounced for canyons that are much longer than they are wide. Our results provide new expectations for morphodynamic feedbacks between floods and topography, and thus canyon formation.