Origin of Hysteresis in Bed Form Response to Unsteady Flows

Field and laboratory studies indicate that changes in riverbed morphology often lag changes in water discharge. This lagged response produces hysteresis in the relationship between water discharge and bed form geometry. To understand these phenomena, we performed flume experiments to observe the response of a sand bed to step increases and decreases in water discharge. For an abrupt rise in discharge, we observed that bed forms grew rapidly by collision and merger of bed forms migrating with different celerities. Growth rate slowed as bed forms approached equilibrium with the higher discharge regime. After an abrupt discharge drop, bed form decay occurred through formation of smaller secondary bed forms, in equilibrium with the lower discharge, which cannibalized the original, relict features. We present a simple model framework to quantitatively predict time scales of bed form adjustment to flow changes, based on equilibrium bed form heights, lengths, and celerities at low and high flows. For rising discharge, the model assumes that all bed form collisions result in irreversible merger, due to a dispersion of initial celerities. For falling discharge, we derive a diffusion model for the decay of relict high-stage features. Our models predict the form and time scale of experimental bed form adjustments. Additional experiments applying slow and fast triangular flood waves show that bed form hysteresis occurs only when the time scale of flow change is faster than the modeled (and measured) bed form adjustment time. We show that our predicted adjustment time scales can also be used to predict the occurrence of bed form hysteresis in natural floods.