Nonlinear sand bedform dynamics in a viscous flow

We investigate theoretically the nonlinear evolution of a sand bedform sheared by a laminar viscous flow. On the basis of the hydrodynamic equations coupled with a sediment transport law, we derive a closed nonlinear and nonlocal equation for the spatiotemporal evolution of the bedform profile in the case of an unbounded flow. The numerical resolution of this equation shows that the bedform coarsens indefinitely in the course of time. During the coarsening process, the wavelength scales as the cube of the vertical extension w as a result of the nonlinear interactions. Interestingly, in the case of a bounded flow, we argue that coarsening is interrupted when the flow perturbation induced by the bedform extends over the whole flow depth h, and we predict that the final wavelength λ{f} and vertical extension w{f} should scale respectively as (γ/ν)h³ and h (where ν is the fluid viscosity and γ the flow shear rate).