Stability of an erodible bed in various shear flows

The 2D laminar quasi-steady asymptotically simplified and linearized flow with a simplified mass transport of sediments is solved over a slowly erodible bed in various laminar basic shear flow ( steady, oscillating or decelerating). The simplified mass transport equation includes the two following phenomena: flux of erosion when the skin friction goes over a threshold value, and a non local effect coming either from an inertial effect or from a slope effect. It is shown that the bed is always unstable for small wave numbers. Examples of long time evolution in various shear regimes are presented, wave trains of ripples are created and merge into a unique bump. This coarsening process is such that the maximum wave length obeys a power law with time.