The equilibrium sediment flux is the transport supported locally by a prescribed bed shear stress. A common idealization is . Linearization above the sediment entrainment threshold requires perturbations small compared with the positive excess stress.
The saturation length measures downstream relaxation of a transported grain population toward its equilibrium sediment transport flux. The elementary steady relaxation law is ; it introduces a phase lag between forcing and actual transport.
The Exner equation is local sediment mass conservation expressed in terms of solid volume: accumulation raises the bed and flux divergence lowers it. Here is the solid volume fraction of the packed bed and is solid volume flux per unit width.
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.
Combining the Exner equation, sediment transport saturation, bed shear response, and inclined-bed sediment threshold gives a competition between upstream forcing and downstream relaxation. With , , , and positive wavenumber , the growth rate is . Thus permits long-wave growth when , while a finite saturation length stabilizes shorter waves.
For a small sinusoidal bedform, a fluid-dynamical closure can be written for . The real coefficient is in phase with the bed height and describes an upstream phase lead. Neither coefficient follows from the Exner equation or sediment relaxation alone; they require a flow calculation or measurement.
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Sediment transport refers to the movement of solid particles (sediment) due to forces exerted by fluid flow, which can be water (in rivers, lakes, and oceans) or air (in deserts and other arid environments). This process plays a crucial role in shaping landscapes, forming sedimentary rocks, and influencing ecosystems.