Why an initially straight alluvial river begins to meander is a long-standing question of fluvial morphodynamics. Dey and Ali (2017) combined a linear stability analysis of a straight channel — yielding an onset wavenumber band for bed perturbations — with a phenomenological mechanism in which successive counter-rotating macro-turbulent eddies drive alternating erosion and deposition of the bed, and derived from turbulent-energy-cascade phenomenology a universal scaling law under which the longitudinal bed slope at the onset of meandering is a unique function of channel width, flow discharge and sediment grain size, corroborated by natural and model river data. Nine years later, however, the central ingredients remain unverified and uncompleted: the eddy mechanism has never been directly observed or resolved in a controlled experiment or simulation at meandering onset; the scaling law's proportionality constant is fitted rather than predicted from theory; and the depth-averaged stability analysis still does not represent secondary currents of Prandtl's second kind, erodible banks, suspended load, or the nonlinear evolution beyond onset, so no explicit emergent meander wavelength is predicted. The source-faithful open problem is to establish a verified and predictive theory of the onset of meandering of a straight alluvial river: confirm or refute the proposed mechanism, derive or independently test the scaling-law constant, and generalize the onset analysis to the physical ingredients it does not represent.

Background

An alluvial river flows over and moulds a bed of loose sediment that it transports. Many such rivers adopt a single-thread sinuous (meandering) planform, and why an initially straight channel loses stability to a sinuous course has been debated since the mid-twentieth century. Einstein and Shen (1964) proposed that turbulence-induced secondary circulation in nominally straight channels can initiate meandering. The classical and most developed explanation is bar instability: Callander (1969) showed by linear stability analysis — superposing small wavenumber-kk perturbations on a steady depth-averaged flow coupled to the bed through the Exner equation (the sediment mass balance governing bed elevation) — that an erodible straight bed is unstable to alternate bars, long wavelength bed undulations that oscillate from bank to bank, which then deflect the flow against the banks and can force a sinuous planform once banks erode.

Dey and Ali (2017) revisited the onset question from a different angle. They first performed a linear stability analysis of a straight alluvial channel with a bedload-dominated bed (sediment transported as rolling or saltating grains near the bed) and found that the natural perturbation modes confine themselves to an onset wavenumber band that depends on the flow regime, the aspect ratio (ratio of channel width to flow depth), the relative roughness (grain size relative to flow depth), and the Shields number (a dimensionless bed shear stress measuring how close the bed surface is to sediment entrainment). Second, they proposed a phenomenological mechanism for onset: turbulence in open channels organizes into macro-turbulent eddies (large-scale coherent structures with sizes comparable to the flow depth, arising from near-bed turbulence anisotropy associated with secondary currents of Prandtl's second kind), and neighbouring eddies rotating in opposite senses arrive in succession, generating alternating zones of erosion and deposition of the riverbed and hence a sinuous sediment-supply pattern along the channel. Third, using the phenomenology of the turbulent energy cascade (the Kolmogorov-type picture in which energy injected at large scales cascades down to dissipative scales), they derived a universal scaling law under which, at the onset of meandering, the longitudinal riverbed slope is a unique function of the river width, flow discharge and sediment grain size, with a proportionality constant fitted to measurements; this functional relationship was corroborated against data from natural and model rivers.

Several limitations remain unresolved. The counter-rotating macro-turbulent-eddy description is qualitative and has not been directly observed in the meandering-onset context; no experimental or spatially resolved numerical verification exists. The scaling-law constant is empirical, not predicted, and its possible dependence on factors such as sediment porosity or the choice of bedload formula is unknown. The stability analysis treats the banks as fixed (assuming bank erosion is slow compared with bed adjustment), includes only bedload, and — although the phenomenological mechanism invokes secondary currents of Prandtl's second kind (streamwise vortical circulations driven by turbulence anisotropy rather than bend curvature, occurring even in straight channels and plausibly important in narrow rivers) as the origin of the macro-turbulent eddies — the depth-averaged stability formulation does not represent them; the analysis defines the onset band by a small tolerance on the perturbation growth rate (reported as ±0.05\pm 0.05 about neutrality), and is purely linear, so it cannot describe the nonlinear growth of perturbations beyond onset or predict the emergent meander wavelength explicitly. Extension to fine or mixed-grain sediments, significant suspended load, unsteady flow and varying sediment supply is likewise open.

Problem Statement

Establish a verified and predictive theory of the onset of meandering of a straight alluvial river, building on Dey and Ali (2017). The scientific objective — a tested account of why and under what quantitative conditions a straight alluvial channel begins to meander — requires that all three of the following open questions, which remain unresolved, be resolved:

  1. Mechanism verification. Directly verify or refute, by controlled laboratory experiment or spatially resolved numerical simulation of turbulent flow over an erodible bed in a straight channel, the phenomenological onset mechanism: successive counter-rotating macro-turbulent eddies generating alternating erosion and deposition of the riverbed. This includes checking whether the unstable perturbations confine themselves to the predicted onset wavenumber band as a function of flow regime, aspect ratio, relative roughness and Shields number, and whether the eddy dynamics, rather than the classical depth-averaged bar instability alone, drive the alternating bed response.

  2. Predictive foundation of the onset scaling law. Derive from theory, or subject to rigorous independent test, the universal scaling law under which the longitudinal riverbed slope at onset is a unique function of channel width, flow discharge and sediment grain size: predict the proportionality constant (currently fitted to measurements) from the turbulent energy-cascade phenomenology or another principled closure, or determine its dependence on secondary factors such as sediment porosity or the choice of bedload transport formula, or refute the uniqueness claim with systematic data.

  3. Generalized onset analysis. Extend the depth-averaged linear stability analysis — which assumes fixed banks and bedload-only transport, and which does not represent secondary currents of Prandtl's second kind (invoked in the phenomenological mechanism but absent from the stability formulation) — to represent all three of the following effects, every one of which must be included: erodible banks, secondary currents of Prandtl's second kind, and suspended load. In addition, treat the nonlinear evolution of perturbations beyond onset, and determine both of the following: whether the onset wavenumber band and the onset criterion survive, and the emergent meander wavelength, predicted explicitly rather than only the onset condition.

References

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  2. Einstein HA, Shen HW (1964). A study on meandering in straight alluvial channels. Journal of Geophysical Research 69, 5239-5247. DOI 10.1029/JZ069i024p05239. https://doi.org/10.1029/JZ069i024p05239
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