Cyclic steps and roll waves in shallow water flow over an erodible bed

Abstract

AbstractThe St Venant equations in conjunction with a phenomenological law for erosion are used to explore the nonlinear dynamics of cyclic steps – linearly unstable bedform patterns which emerge when uniform flow over an erodible bed becomes supercritical. The instability saturates by blocking the overlying flow and creating hydraulic jumps just downstream of the steepest part of the steps. Near onset, steadily migrating, nonlinear step patterns are constructed and shown to suffer a short-wavelength secondary instability that ‘roughens’ the bed and renders the staircase patterns less regular and time-dependent. An eddy viscosity is needed to regularize both the onset of the primary steps and the secondary instabilities. Further beyond the critical Froude number, the steps block the flow sufficiently to arrest erosion significantly, creating complicated patterns mixing migrating steps and stationary bedforms. The reduction in flux also stabilizes roll waves – a second, hydrodynamic instability of uniform supercritical flow. It is further shown that roll waves are purely convective instabilities, whereas cyclic steps can be absolute. Thus, in the finite geometries of the laboratory or field, it may be difficult to excite roll waves. On the other hand, the complicated spatiotemporal patterns associated with the cyclic-step instability should develop naturally. The complicated patterns resulting from the secondary instability do not appear to have been observed experimentally, calling into question the validity of the model.