Sediment transport saturation

Sediment transport saturation is relaxation of a transported grain population toward the equilibrium sediment flux supported by local flow conditions. A change in bed shear stress does not immediately give a fully adjusted grain speed and population. Close to equilibrium, a steady model isFor constant $q_{\mathrm{sat}}$ and saturation length $L_{\mathrm{sat}}>0$, this gives $q=q_{\mathrm{sat}}+[q(0)-q_{\mathrm{sat}}]e^{-x/L_{\mathrm{sat}}}$. The spatial lag matters for bedform growth.^[ref]

A published wind tunnel test used a $1\mathrm{m}$ wide, $0.5\mathrm{m}$ high and $4.5\mathrm{m}$ long tunnel. Repeated measurements of bed elevation after stopping the flow supplied an erosion rate; integrating that rate inferred downstream flux by mass conservation. The fitted near-equilibrium relaxation length was distinguished from the earlier grain-ejection stage, so the total development distance was not treated as the same parameter.^[ref]

The equilibrium sediment flux is the transport supported locally by a prescribed bed shear stress. A common idealization is $q_{\mathrm{sat}}={\varphi }_b\chi (\tau -{\tau }_{\mathrm{th}}{)}_{+}^{\gamma }$. 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 $L_{\mathrm{sat}}{q}_x=q_{\mathrm{sat}}-q$; it introduces a phase lag between forcing and actual transport.