The equilibrium shape of turbulent alluvial rivers carrying bedload: extending the sediment-load shape model of Popović et al. beyond laminar flow

Popović, Devauchelle, Abramian and Lajeunesse (PNAS 2021) derived a physically based model showing that the balance between fluid stress and gravity on sediment grains, together with cross-stream diffusion of sediment, fixes the shape of straight laminar laboratory rivers and makes the sediment flux per unit width saturate at a value independent of water and sediment discharge (Parker's regime), so that a river responds to increasing sediment load only by widening. The model is confined to laminar Stokes flow, and the authors designate its extension to turbulent flow — the regime of all natural rivers — as the focus of future work. The first step, a validated turbulent closure for the cross-stream distribution of fluid stress, was completed by Popović, Devauchelle and Lajeunesse (JFM 2024), but the equilibrium shape of a turbulent river that actually carries sediment remains undetermined. The problem is to couple turbulent momentum transfer with sediment transport in this framework, solve for the equilibrium channel shape, establish whether the Parker and weak-transport regimes persist in turbulence, and thereby determine whether river geometry can be inverted into sediment discharge for natural rivers.

Background

An alluvial river builds its own bed and banks from the sediment it carries, so its width and cross-sectional shape are outcomes, not givens: a long-standing goal of fluvial geomorphology is to predict channel geometry from the water discharge $Q_w$ and sediment discharge $Q_s$ supplied by the landscape. Parker's classic model of the gravel river (J. Fluid Mech. 89, 127, 1978) proposed that a straight alluvial river splits into inert banks held close to the threshold of sediment motion — the critical stress at which grains begin to move — and a flat active bed, which forces the river to widen, rather than deepen, when the sediment load grows. This threshold behavior has since been supported at field scale (Phillips and Jerolmack, Science 352, 694, 2016; Phillips et al., Nat. Rev. Earth Environ. 3, 406–419, 2022), but a fully mechanistic derivation from the flow and sediment-transport equations was missing.

Popović, Devauchelle, Abramian and Lajeunesse (PNAS 118(49), 2021) supplied such a derivation for straight laboratory rivers in which the flow is laminar (viscous, Stokes regime) and sediment moves as bedload (rolling, sliding or saltating — hopping in short arcs — along the bed, modeled through a moving-grain density with a Boltzmann-like distribution). Their key ingredient is momentum diffusion across the stream: the shallow-water approximation, in which the local bed stress is simply proportional to the local depth, cannot drive sediment transport at all, because the stress would fall below the entrainment threshold over part of the bed. Retaining the leading cross-stream momentum-diffusion term — the first term of an expansion in $(D_{\max}/W)^2$, where $D_{\max}$ is the maximum depth and $W$ the width — yields a local, second-order boundary-value problem for the equilibrium shape. Solved and compared with the laminar flume experiments, it reproduces the observed shapes without tuning. Its central predictions are: (i) rivers exist only close to the threshold of sediment motion (bed stress only about 20% above critical, independent of both discharges); (ii) in the large-load Parker regime the sediment flux per unit width saturates at a discharge-independent maximum $q_s^{(P)} \approx 0.2\,q_\mu$, so added sediment discharge forces widening; (iii) in the opposite weak-transport regime the shape is independent of $Q_s$ while the flux grows with it; and (iv) in the Parker regime the sediment discharge can be inverted from pure geometry (width and depth) via algebraic relations involving the constants $q_s^{(P)}$ and $\tilde W_0/\tilde D_{\max,0}$ (Eq. 22 of that paper) — attractive in the field, where sediment discharge is notoriously hard to measure.

The catch is turbulence: natural rivers are turbulent, where no first-principles local stress closure exists, and the 2021 paper states that before the inversion can be applied to natural rivers, the theory must be extended to turbulent flow — explicitly designated as the focus of future work. As the first step, Popović, Devauchelle and Lajeunesse (J. Fluid Mech. 981, A24, 2024) used dimensional analysis and a depth-integrated momentum balance to derive a linear second-order ordinary differential equation for the cross-stream stress distribution $\tau(y)$ in a straight turbulent channel of arbitrary cross-section — a first-order correction to shallow-water theory at large aspect ratio (channel width much larger than depth), in the lineage of the Shiono–Knight depth-averaged model (a standard river-engineering description in which the downstream driving force is balanced against cross-stream turbulent momentum diffusion). It contains two parameters: a dimensionless momentum-diffusion parameter $\chi \sim \Lambda C_f^{-1/2}$, with $C_f$ the friction coefficient and $\Lambda \approx 0.3$, and a local-shape parameter $\alpha$ predicted to vanish in highly turbulent flow (the laminar limit is recovered at $\alpha=1$, $\chi=1/3$). The closure matches flume and field measurements of stress and velocity profiles to about 30%. The authors state explicitly that this is ‘a first step’ and that their model ‘suggests that it may be possible to understand equilibrium shape of alluvial, turbulent rivers as simply as in laminar ones’ — the coupled sediment-carrying problem itself remains unsolved, and the closure has not yet been extended to carry bedload.

Problem Statement

Determine the equilibrium shape of a straight alluvial river with turbulent flow that transports uniform, non-cohesive sediment as bedload, by coupling a cross-stream turbulent momentum-transfer closure (such as the stress equation of Popović, Devauchelle and Lajeunesse 2024, with parameters $\chi$ and $\alpha$) with a description of sediment entrainment, transport and cross-stream diffusion, in the spirit of the laminar shape model of Popović et al. 2021. The requested results, which jointly serve this single objective, are: (1) a well-posed free-boundary problem — one whose boundary, here the bank positions, is itself part of the solution — for the turbulent channel profile $D(y)$ (depth as a function of cross-stream coordinate) and its solution — closed-form where possible, numerical otherwise — reducing to the laminar 2021 model in the appropriate limit; (2) an asymptotic regime analysis establishing whether the Parker regime (channel confined close to the threshold of sediment motion, sediment flux per unit width saturating at a value independent of the water and sediment discharges, so that increasing sediment load is accommodated by widening) and the weak-transport regime (shape independent of sediment discharge) persist under turbulent momentum transfer, and if so with what values of the dimensionless constants ($q_s^{(P)}$, $\tilde W_0/\tilde D_{\max,0}$, and the degree of threshold proximity) that replace the laminar ones; and (3) a determination of whether, and with which sediment-dependent constants, the sediment discharge of a natural river can therefore be inverted from its measured width and depth. The objective is the single question whether the laminar regime structure — near-threshold confinement of the flow, a sediment flux per unit width saturating at a discharge-independent value in the Parker regime (with the weak-transport regime retaining a shape independent of sediment discharge), and a geometry from which sediment discharge is invertible — survives turbulent momentum transfer. Results (1) and (2) are the mandatory deliverables: the turbulent free-boundary problem with its solution, and the regime analysis establishing which structural components survive and with what dimensionless turbulent constants; result (3) is their applied consequence. Establishing this structure is a complete solution, and demonstrating with specific analysis or data that turbulence breaks it is equally complete: because the structural components are independently testable, a demonstrated failure of any one of them (for example, that the saturated flux or threshold proximity ceases to be discharge-independent) resolves the question, provided the free-boundary formulation and regime analysis needed to establish that failure are supplied.

References

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