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Kobayashi & Madsen 1985
Kobayashi, N. and Madsen, O.S. (1985). Formation of ripples in erodible channels. Journal of Geophysical Research 90: doi: 10.1029/JC080i010p07332. issn: 0148-0227.

Formation of ripples and dunes in erodible channels is analyzed by using the hydrodynamic model developed by Kobayashi and Madsen (this issue). Applying the continuity equation of sediment, the propagation velocity of the wavy bed is expressed in terms of the net fluxes of bed load along the wavy bed and suspended sediment out of the bed load transport layer. The spatial variation of the bed load transport rate is derived by analyzing the balance of forces acting on a representative bed load particle on a sloping bed. The net flux of the suspended sediment out of the bed load transport layer is obtained by solving the diffusion equation for the suspended sediment near the bed. The stability analysis predicts that a perturbation that is either fairly short or long relative to the depth of flow will initially grow, depending on whether the flow is hydraulically smooth or fully rough. Thus the model explains why ripples are not observed in experiments involving fairly coarse sediments, since in these cases the flow is rough turbulent.

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Journal of Geophysical Research
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