The Shields parameter is , comparing bed shear stress with a grain-scale submerged weight per area. Its critical value depends on the assumed drag coefficient, static friction, lift, and contact geometry.
The sediment entrainment threshold is the bed shear stress at which resting grains begin moving. An elementary force balance using spherical grains, tangential quadratic drag, and static friction gives when the drag coefficient is defined relative to the bed shear velocity.
For locally bed-tangent drag force and an upslope angle , a spherical-grain force balance gives . The small-slope term is . A horizontal drag force instead changes the normal contact force and gives a different slope correction.
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The Shields parameter, often denoted by the Greek letter \( \tau^* \), is a dimensionless quantity used in sediment transport and fluid mechanics to characterize the initiation of sediment motion under flow conditions. It quantifies the ratio of the shear stress acting on the sediment bed to the gravitational forces acting on the sediment particles.