From standing to breaking antidunes, a Hopf bifurcation.

Antidunes are bed morphologies often observed in steep slope mountain flows but also in small streams flowing on a sand beach. Linear stability analysis of the shallow water equations (SWE), when coupled to a sediment transport equation, predicts the growth of selected wavelength when the Froude number exceeds unity. If the bedform amplitude grows sufficiently (without being stabilized by nonlinear effects (Colombini and Stocchino (2008)), hydraulic jump can form on the lee side while a transcritical point, situated roughly on the crest of the dune, connects the sub and supercritical states. This has been described in the literature as cyclic steps. Their stationary travelling wave solution has already been theoretically and experimentally investigated (Balmforth and Vakil (2012); Taki and Parker (2005); Sun and Parker (2005)). In this talk, we provide numerical and theoretical evidence that a time dependent quasi-periodic solution of cyclic steps also exists in the SWE+Exner equations for a certain choice of the parameters. We compare this quasi-periodic phenomenon to breaking antidunes often observed in natural rivers.