Near-bed vertical flux formulas for mixed sediment transport in oscillatory sheet flow

Near-bed vertical sediment flux is essential for quantifying the intensity of vertical sediment exchange and serves as a key bottom boundary condition for sediment numerical models. This study develops an analytical framework to formulate the vertical flux for mixed sediment transport under oscillatory sheet flow and effectively addresses the existing limitations in this field. New formulas for downward settling, upward suspension, and total vertical sediment fluxes are derived from the mass conservation-based concentration profile of graded sediments using a size-fraction approach. These formulas account for sediment gradation effects, phase lag, acceleration effect, and asymmetric wave boundary layer dynamics. The hiding-exposure effects and the grading effects on fractional vertical sediment flux are illustrated based on vertical flux verification. Results show that the proposed formulas accurately capture the active layer, non-dynamical characteristics, and fractional behaviors under oscillatory sheet flow. The vertical sediment flux is related to both free stream velocity and acceleration, and the fractional behaviors show an anti-phase phenomenon and an anti-equal mobility state. The findings confirm that the proposed formulas can resolve the sheet flow layer with sufficient detail and are suitable to use as near-bed boundary conditions in numerical models of mixed sediment transport.