= 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 is
$$
L_{\rm sat}\frac{dq}{dx}=q_{\rm sat}-q.
$$
For constant $q_{\rm sat}$ and <saturation length> $L_{\rm sat}>0$, this gives $q=q_{\rm sat}+[q(0)-q_{\rm sat}]e^{-x/L_{\rm sat}}$. The spatial lag matters for <bedform> growth.<Measurements of the aeolian sand transport saturation length by Bruno Andreotti and others>[§1, equation (1): near-saturation relaxation and possible limiting mechanisms]{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.<Measurements of the aeolian sand transport saturation length by Bruno Andreotti and others>[§2, Wind tunnel measurements: geometry, elevation-based flux inference and separate ejection/relaxation stages]{ref}
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