Regime closure for disturbed gravel-bed rivers: the reference Shields stress–slope relation and transient channel adjustment under unsteady water and sediment supply

Regime theory predicts the equilibrium bankfull geometry (width, depth, slope) of alluvial channels from water and sediment supply, but it rests on two elements that Kaless, Mao and Lenzi (2014) identified as unresolved when they applied three regime models to human-disturbed gravel-bed rivers in northeast Italy: (i) the exact functional relationship between the reference shear stress for gravel transport and channel slope found empirically by Mueller, Pitlick and Nelson (2005) — including how bank strength and sediment supply modulate it — and (ii) the assumption of equilibrium with a single formative discharge, which breaks down for reaches in transient adjustment after gravel mining, channelization, or flow regulation. No study or combination of studies supplies the closure. The open problem is stated as: establish the functional relation (or demonstrate and quantify its conditionality) between reference/bankfull Shields stress, slope, and bank strength, and embed it in a regime-type closure valid for transient reaches under unsteady discharge and sediment supply, with the differential adjustment timescales of width, depth, and slope tested against field trajectories.

Background

Regime theory (also called hydraulic-geometry theory) seeks to predict the size and shape a river channel would attain in equilibrium with the water discharge and sediment load supplied to it: given a bankfull discharge (the flow that just fills the channel to the top of its banks, typically reached about every 1–2 years) and a sediment supply, it predicts bankfull width, mean depth, and bed slope. Classic formulations range from empirical power laws to physically based 'rational regime' equations that couple flow resistance and bed-material transport relations with mass and momentum conservation, such as Ikeda, Parker and Kimura (1988) and the quasi-universal bankfull relations of Parker, Wilcock, Paola, Dietrich and Pitlick (2007). A separate branch replaces part of the physics with an extremal hypothesis — the idea that the channel adjusts its geometry to extremize some quantity such as sediment-transport capacity — as in Millar (2005), whose theoretical regime equations for gravel-bed rivers with stable banks include bank strength through a friction angle of the bank material, so that stronger (e.g., vegetated or cohesive) banks yield narrower channels. In gravel-bed rivers the bed is typically coarse and armoured, and sediment moves mostly as bed load only when the flow is strong enough. The key transport variable is the dimensionless Shields stress, defined as $\tau^* = \tau / [(\rho_s - \rho) g D]$, where $\tau$ is bed shear stress, $\rho_s$ and $\rho$ are sediment and water densities, $g$ is gravity, and $D$ is a characteristic grain size (usually the median surface grain size $D_{50}$). Because observable motion begins gradually, practical work uses the reference shear stress $\tau^*_r$: the Shields stress at a small reference dimensionless transport rate ($W^* = 0.002$, where $W^*$ normalizes volumetric bed-load transport rate by grain size and flow intensity). Mueller, Pitlick and Nelson (2005) analysed 45 gravel streams and rivers and found that $\tau^*_r$ is not a universal constant near the textbook value 0.045 but increases systematically, though with much scatter, with reach-averaged channel slope $S$. This matters because regime equations consume $\tau^*_r$ (or the related critical stress) as an input: an uncertain, slope-dependent threshold propagates directly into predicted equilibrium width, depth, and slope. Kaless, Mao and Lenzi (2014) tested three regime models against 142 published reaches plus new data from Patagonia and the Italian Alps and used Millar's model to interpret recent changes of the Piave and Brenta rivers, which lost sediment to decades of in-channel gravel mining and have been widening and reworking their beds since mining ceased. Their concluding discussion flags four linked unresolved issues: regime models assume equilibrium while these rivers are in transient adjustment, and unsteady sediment fluxes and variable discharge are absent from the closures; the controls on the Piave's post-mining widening (temporary adjustment versus a new equilibrium) are unknown, partly for lack of measured reach-scale sediment budgets; the hypothesis that width and depth adjust quickly while slope requires much longer times has been supported only indirectly; and the exact functional relationship between $\tau^*_r$ and slope, and its applicability across different bank-strength conditions, remains an open question requiring laboratory and field experiments. Later literature (2008–2025) treats these issues as follows: mechanistic models now explain part of the slope dependence of entrainment thresholds, large compilations show the competing slope-based regressions are ambiguous and possibly spurious, sediment supply and flow history have emerged as additional controls, and stochastic models show single-discharge regime predictions diverge from variable-flow reality.

Problem Statement

The problem, as posed by Kaless, Mao and Lenzi (2014) for disturbed gravel-bed rivers (instantiated on the Piave and Brenta), is to close regime theory for non-equilibrium reaches by solving two jointly necessary components. Component 1 (threshold closure): determine the exact functional relationship between the reference shear stress $\tau^*_r$ (and the related bankfull Shields stress $\tau^*_{bf}$) and reach-averaged channel slope $S$ — or, if no unique relation exists, establish and quantify how it is conditioned by bank strength and sediment supply — with the applicability of the relation across bank-strength conditions (weak gravel banks through vegetated or cohesive banks, as parameterized for example by the bank friction angle in Millar's regime equations) determined. Component 2 (transient closure): incorporate this threshold closure into regime-type equations so that they apply to reaches out of equilibrium after disturbance, i.e., admitting unsteady sediment fluxes and variable (non-single-formative) discharge, and thereby encode and allow direct field testing of the hypothesis that channel width and depth adjust quickly while slope requires substantially longer time spans.

References

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