Bedform Stability: There Are Not Two, But Three Flow-Regimes

Bedforms are classically categorized into lower and upper flow-regimes, separated by the critical Froude number (Fr=1). Three main bedform types occur in unidirectional flows: dunes (Fr<1), antidunes (1<Fr<1.5), and cyclic steps (Fr>1.5). With the recognition of cyclic steps as a separate bedform, these were implicitly attributed to the upper flow-regime. However, cyclic steps are dynamically very different from antidunes, which raises the question whether both types belong genetically to the same flow-regime. Hydrodynamics suggests three characteristic velocities: U: flow velocity, W: celerity of energy-waves, and V: celerity of mass-waves. These can be combined into two dimensionless parameters: Froude number: Fr=U/W, indicating whether energy-waves migrate upstream, and Vedernikov number: Ve=V/W, indicating whether mass-waves can outrun the flow, generating roll waves and flow instability. An important phenomenon occurs at the supercritical to subcritical transition: the hydraulic jump, which classically marks a separation from lower to upper flow-regime at Fr=1. However, a less well-known but important transition occurs at Fr=1.5, namely the transition from undular (1<Fr<1.5) to breaking hydraulic jumps (Fr>1.5). The Vedernikov number can be expressed in terms of the Froude number: Ve=1.5xFr. Therefore, Ve>1 coincides with Fr>1.5, indicating that only supercritical flows can show instability phenomena. Notably, the occurrence of these two fundamental flow transitions (at Fr=1 and Fr=1.5) coincides with the flow conditions discriminating the stability fields of antidunes and cyclic steps. Thus, we propose here that there are not two, but three flow regimes: stable subcritical flow (dunes), stable supercritical flow (antidunes), and unstable supercritical flow (cyclic steps).