Probability distribution of bed particle instability

The probability distribution of the instability of bed particles is essential for application of statistical approaches to describe bedload transport. The instability generally depends on the near-bed flow conditions and characteristics of the particle packing on the bed surface. In this study, only irregularity of the bed particles is considered. The instability of each individual bed particle is characterised using an effective shear stress, beyond which the particle starts to move. The effective shear stresses vary randomly over the bed because of the randomness of the bed particles. For a flat bed comprised of uniform particles, the variation is demonstrated to be a narrow-banded random process with amplitudes equal to the magnitudes of the effective shear stress. A theoretical derivation shows that the probability density distribution of the effective shear stress follows the Rayleigh distribution. Subsequent analysis also indicates that the probability of the instability is proportional to the square of the bed shear stress for low rates of bedload transport.