Experiments on straight laboratory rivers with laminar flow showed that the intensity of bedload transport across the channel follows a Boltzmann-like exponential distribution, and asked how this grain-scale statistical equilibrium selects the river's macroscopic cross section. For the laminar laboratory setting, that mechanistic derivation was subsequently supplied by a self-consistent theory coupling channel shape, flow stress, sediment flux, and lateral diffusion of grains. What remains open is the physically based closure of alluvial channel geometry in the turbulent regime relevant to natural rivers: a self-consistent equilibrium theory of the depth, width, slope, and aspect ratio of straight single-thread rivers carrying bedload in turbulent flow, together with the dependence of that geometry on water discharge, sediment discharge, grain size, fluid viscosity, and material parameters. The origin of the braiding instability that terminates single-thread transport when sediment supply becomes large is a related but separate open question, which the authors of the laminar theory treat as requiring an analysis beyond the shape theory itself.

Background

An alluvial river builds its own bed out of the sediment it transports, so its shape depends not only on water discharge but also on sediment supply. Grains travel mainly as bedload — dragged, rolled, and bounced along the bed — and begin to move only where the shear stress the flow exerts on the bed exceeds a critical value, a condition known as the threshold of sediment motion (the Shields criterion). If a river's bed were everywhere exactly at this threshold, the transverse cross section of the channel would take a cosine-like shape; this threshold-channel theory predicts that channel width scales with the square root of water discharge, in agreement with the classic empirical regime relations for rivers. Parker (1978) proposed that natural gravel rivers, though actively transporting sediment, are nevertheless confined to remain close to the threshold of motion. Abramian, Devauchelle and Lajeunesse (Phys. Rev. E 102, 053101, 2020) tested this physics in laboratory rivers: straight channels carved in a bed of uniform plastic grains (median diameter about 0.8 mm) and fed with a glycerol–water mixture (viscosity 10 cP) at Reynolds number near 10, so that the flow was laminar rather than turbulent. Tracking individually labelled grains, they found that at equilibrium the intensity of sediment transport across the channel follows a Boltzmann distribution: traveling grains diffuse laterally as they randomly collide with the bed, and this diffusion balances the downslope pull of gravity, so that the local sediment flux decays exponentially with elevation above the channel centerline, qs=qB eD/λq_s = q_B\,e^{D/\lambda}, where DD is the local flow depth and the diffusion length λ\lambda (analogous to a gas temperature in a gravitational field) scales with grain size, λ≈0.12 ds\lambda \approx 0.12\,d_s. That value comes from grain tracking in the same laminar glycerol–water flume (Abramian, Devauchelle, Seizilles and Lajeunesse, Phys. Rev. Lett. 123, 014501, 2019) — a single grain–fluid pair at a single water discharge — and that value has not been extended to turbulent bedload transport. The experiments also yielded an empirical square-root scaling of the active transport width with sediment discharge, WT∝Qs1/2W_T \propto Q_s^{1/2}, which the authors embedded in a single-parameter, discontinuous model with an actively transporting core and virtually idle banks. Their conclusion explicitly left open how the Boltzmann statistical equilibrium sets the cross section (depth, width, and their ratio, the aspect ratio), how to reconcile the model's discontinuous junction with the continuous Boltzmann profile — which requires computing the bed shear-stress distribution, including two-dimensional flow effects, more accurately than shallow-water (lubrication-type) approximations allow — how the scalings depend on water discharge, grain size, fluid viscosity, and material parameters (the experiments used one water discharge and one grain–fluid pair), and what causes the braiding instability (the splitting of a single channel into intertwined threads) that appears beyond a sediment supply of about 90 grains per second in the laminar flume and bounds the accessible supply. The subsequent work of Popović, Devauchelle, Abramian and Lajeunesse (PNAS 118, e2111215118, 2021) resolved the laminar core of this program: closing the loop shape → stress → sediment flux → Boltzmann distribution → shape yields a second-order boundary-value problem, solvable numerically and analyzable analytically, which reproduces the experiments without any tuning parameter, yields continuous cross sections and flux profiles, retains cross-stream momentum diffusion, and confirms that the flux intensity saturates within about 20% of the threshold of motion, forcing the river to widen as sediment discharge grows. A companion paper (Devauchelle, Popović and Lajeunesse, J. Fluid Mech. 932, A32, 2022) supplied the improved low-dimensional description of cross-stream viscous momentum transfer for shallow laminar flows that the 2020 paper had called for. For turbulent flow — the regime of all natural rivers — Popović, Devauchelle and Lajeunesse (J. Fluid Mech. 981, A24, 2024, DOI 10.1017/jfm.2023.1098) derived a linear second-order differential equation for the stress distribution across a straight open channel of arbitrary cross-sectional shape, with a dimensionless diffusion parameter χ∼Cf−1/2\chi \sim C_f^{-1/2} (CfC_f being the friction coefficient) estimated from eddy-scale arguments, validated against flume and field measurements; they state that this equation can serve as the basis for an equilibrium-shape theory of turbulent rivers that carry sediment, but the theory itself — coupling this closure with a sediment-transport law and the Boltzmann balance of traveling grains, and validating it — has not been published. What remains open, therefore, is the completion of the program in the turbulent regime: a self-consistent theory of the equilibrium cross-sectional shape of straight, single-thread, sediment-carrying rivers in turbulent flow, testable against turbulent flume or field data, together with the dependence of that shape on water discharge, sediment discharge, grain size, fluid viscosity, and material parameters. The mechanism of the braiding transition that bounds single-thread transport is a distinct open question, which the same authors describe as requiring an analysis beyond the scope of the shape theory.

Problem Statement

Construct a self-consistent, physically based equilibrium theory for the cross-sectional shape of straight, single-thread alluvial rivers that carry sediment as bedload in turbulent flow — the regime of natural rivers — closing the loop between channel shape, the distribution of flow-induced shear stress on the bed (including two-dimensional, cross-stream momentum-transfer effects as in the 2024 turbulent stress equation), a sediment-transport law, and the Boltzmann-like lateral equilibrium between cross-stream diffusion of traveling grains and gravity. The theory should predict the equilibrium depth, width, slope, and aspect ratio from the water discharge, sediment discharge, grain size, fluid viscosity, and densities, and so should yield the dependence of the channel geometry on these inputs rather than reproducing a single fixed-discharge, single-material experimental condition; it should be testable against turbulent flume or field data. The laminar results — the experiments of Abramian et al. (2020), the theory of Popović et al. (PNAS 2021), and the grain-diffusion length λ ≈ 0.12 d_s measured in those laminar glycerol–water experiments — are background and a consistency reference for the turbulent closure rather than prescribed ingredients of it: in particular, the lateral diffusion length of traveling grains (its grain-size scaling and any dependence on fluid and grain properties) is left to be determined or justified within the theory. Likewise, the laminar theory's prediction that a river's aspect ratio, rather than its total width, tracks its sediment discharge is a candidate consequence to be tested in the turbulent regime, not a result the theory is required to reproduce — the same laminar model contains a weak-transport regime in which the bed shape, and hence the aspect ratio, is essentially independent of sediment discharge.

References

  1. A. Abramian, O. Devauchelle, E. Lajeunesse, 'Laboratory rivers adjust their shape to sediment transport', Physical Review E 102, 053101 (2020). DOI: 10.1103/PhysRevE.102.053101, https://journals.aps.org/pre/abstract/10.1103/PhysRevE.102.053101 (open-access full text: https://hal.science/hal-02987014)
  2. Predrag Popović, Olivier Devauchelle, Anaïs Abramian, Eric Lajeunesse, 'Sediment load determines the shape of rivers', Proceedings of the National Academy of Sciences 118, e2111215118 (2021). DOI: 10.1073/pnas.2111215118, arXiv:2203.03758, https://arxiv.org/abs/2203.03758
  3. O. Devauchelle, P. Popović, E. Lajeunesse, 'Viscous transfer of momentum across a shallow laminar flow', Journal of Fluid Mechanics 932, A32 (2022). DOI: 10.1017/jfm.2021.1021, https://doi.org/10.1017/jfm.2021.1021
  4. Predrag Popović, Olivier Devauchelle, Eric Lajeunesse, 'Momentum transfer across an open-channel, turbulent flow', Journal of Fluid Mechanics 981, A24 (2024). DOI: 10.1017/jfm.2023.1098, https://doi.org/10.1017/jfm.2023.1098
  5. Gary Parker, 'Self-formed straight rivers with equilibrium banks and mobile bed. Part 2. The gravel river', Journal of Fluid Mechanics 89, 127–146 (1978). DOI: 10.1017/S0022112078002505, https://doi.org/10.1017/S0022112078002505
  6. A. Abramian, O. Devauchelle, G. Seizilles, E. Lajeunesse, 'Boltzmann distribution of sediment transport', Physical Review Letters 123, 014501 (2019). DOI: 10.1103/PhysRevLett.123.014501, arXiv:1907.01880, https://arxiv.org/abs/1907.01880