Stochastic modeling in sediment dynamics: Exner equation for planar bed incipient bed load transport conditions

Even under flow equilibrium conditions, river bed topography continuously evolves with time, producing trains of irregular bed forms. The idea has recently emerged that the variability in the bed form geometry results from some randomness in sediment flux. In this paper, we address this issue by using the Exner equation and a population exchange model derived in an earlier paper. In this model, particle entrainment and deposition are idealized as population exchanges between the stream and the bed, which makes it possible to use birth‐death Markov process theory to track the number of moving grains. The paper focuses on nascent bed forms on initially planar beds, a situation in which the coupling between the stream and bed is weak. In a steady state, the number of moving particles follows a negative binomial distribution. Although this probability distribution does not enter the family of heavy‐tailed distributions, it may give rise to large and frequent fluctuations because the standard deviation can be much larger than the mean, a feature that is not accounted for with classic probability laws (e.g., Hamamori’s law) used so far for describing bed load fluctuations. In the large‐system limit, the master equation of the birth‐death Markov process can be transformed into a Fokker‐Planck equation. This transformation is used here to show that the number of moving particles can be described as an Ornstein‐Uhlenbeck process. An important consequence is that in the long term, the number of moving particles follows a Gaussian distribution. Laboratory experiments show that this approximation is correct when the mean number per unit length of stream, $\vec{N}$/L, is sufficiently large (typically two particles per centimeter in our experiments). The particle number fluctuations give rise to bed elevation fluctuations, whose spectrum falls off like $\omega^{-2}$ in the high‐frequency regime (with $\omega$ the angular frequency) and variance grows linearly with time. These features are in agreement with recent observations on bed form development (in particular, ripple growth).