Bedforms in a turbulent stream.Part 2: Formation of ripples by primary linear instability and of dunes by non-linear pattern coarsening

TL;DR

This paper investigates the formation mechanisms of ripples and dunes in turbulent streams, combining flow computation, sediment transport modeling, and field experiments to distinguish linear instability origins of ripples from non-linear dune formation.

Contribution

It provides a comprehensive analysis linking flow dynamics, sediment transport, and pattern formation, clarifying the linear and non-linear processes behind bedform development.

Findings

Ripples form via primary linear instability controlled by flow and sediment parameters.

Dunes result from non-linear pattern coarsening, not primary linear instability.

Free surface effects stabilize certain bedform wavelengths, influencing dune formation.

Abstract

It is widely accepted that both ripples and dunes form in rivers by primary linear instability, the wavelength of the former scaling on the grain size, that of the latter being controled by the water depth. We revisit here this problem, using the computation of the turbulent flow over a wavy bottom performed in Part 1. The details of the different mechanisms controlling sediment transport are encoded into three quantities: the saturated flux, the saturation length and the threshold shear stress. Theses quantities are modelled in the case of erosion and momentum limited bed loads. This framework allows to give a clear picture of the instability in terms of dynamical mechanisms. The relation between the wavelength at which ripples form and the flux saturation length is quantitatively derived. Inverting the problem, experimental data is used to determine the saturation length as a function of grain size and shear velocity. Finally, using the systematic expansion of the flow field with respect to the corrugation amplitude, we discuss the non-linear selection of ripple aspect ratio. Investingating the effect of a free surface on the linear instability, we show that the excitation of standing waves at the surface has a stabilising effect, independently of the details of the flow and sediment transport models. Consequently, the shape of the dispersion relation is such that dunes can not result from a primary linear instability. We present the results of field experiments performed in the natural sandy Leyre river, which evidence the formation of ripples by a linear instability and the formation of dunes by a non-linear pattern coarsening limited by the free surface. We show that mega-dunes form when the sand bed presents heterogeneities such as a wide distribution of grain sizes.