A stochastic formulation for erosion of cohesive sediments

The linear formulation for erosion E = M(?b??c), often applied in engineering applications, has two properties, which do not always comply with field and laboratory observations, they are as follows: (1) The erosion rate is zero below the critical bed shear stress ?c and increases linearly with bed shear stress ?b, when exceeding the critical bed shear stress; incipient motion (?b $\simeq$ ?c) is poorly represented. (2) The erosion rate is constant in time for constant values of M and ?c, whereas observations often suggest time dependency. In this paper we analyze the process of incipient motion and time dependency by using a stochastic forcing (bed shear stress) and a stochastic bed strength (critical bed shear stress). It is well known that the bed shear stress is not constant but varies due to turbulence. This stochastic nature of the turbulent motion is accounted for by a probability density distribution for the bed shear stress, which is based on the formulation of Hofland and Battjes (2006). This distribution is implemented in the linear erosion formulation. An analytical solution for the erosion rate is obtained, which only depends on the mean bed shear stress. A parametrization is made for efficient application in numerical models. The sediment in the bed is considered to be nonuniform. Therefore, it is subdivided into several classes, distinguished by the critical bed shear stress and not necessarily by the grain size. The variability of the critical bed shear stress is treated in a discritized way. Sediment balance equations are solved for each class. Considering different classes, the total erosion rate becomes time dependent, as the erosion depends on the availability of sediment. The model is applied to two annular flume data sets, Jacobs (2009) and Amos et al. (1992a). The results show that with a proper choice of the required parameters, the time dependence of the erosion rate and the concentration can be reproduced. We conclude that the occurrence of incipient motion can be explained from a stochastic forcing. Time?varying erosion rates can be explained from a stochastic bed strength distribution or from a vertical gradient in bed strength. The latter is, however, not likely and not measurable in the top layers of dense consolidated cohesive sediment beds