Flooding as a sub-critical instability in open channels

In flood events caused by a gradual increase in the flow rate of a watercourse, the rise in water level is often abrupt, while the fall in level is delayed. We show that such behavior can be demonstrated by considering stationary flows at a high Reynolds number in a prismatic open channel: Several geometries of the channel cross section lead to a subcritical instability that results in a discontinuous rise in the level when the flow rate exceeds a critical value $F_i$, and in a fall, also discontinuous, when the flow rate returns below a value $F_{\mathcal{o}}$ lower than $F_i$. This hysteretic behavior originates from the interplay between gravity, which drives the flow downstream, and turbulent friction with the channel wall. The potential existence of several solutions arising from this bifurcation requires careful consideration in flood simulations.