Quantitative dynamics of the self-constraining feedback in low-energy meandering rivers: temporal trajectories, the conjectured threshold hysteresis, and boundary conditions

Many low-energy meandering rivers have tortuous planforms with sharp bends and anomalously low channel mobility. Candel, Makaske, Kijm, Kleinhans, Storms and Wallinga (2020) proposed a conceptual explanation — self-constraining: the river's own fine-grained, erosion-resistant deposits (residual channel-fills, counterpoint deposits, cohesive middle point-bar and overbank units) accumulate in the banks, progressively raising bank strength, increasing planform tortuosity, and suppressing migration whenever river energy stays below a critical threshold for eroding cohesive banks. The paper, corroborated by a 47-reach global dataset and a palaeogeographic reconstruction of the Dommel River (Netherlands), conjectures an initially exponential increase in tortuosity and a hysteretic two-state cycle between self-constrained and actively migrating behaviour, but could quantify none of this from field snapshots. The open problem is to convert this conceptual model into a quantitative, tested one: determine the temporal trajectories of bank cohesiveness, planform tortuosity, and channel mobility (expected via coupled numerical modelling experiments); confirm or refute the existence and shape of the conjectured hysteresis loop (no river has yet been documented crossing the critical threshold); and delimit the boundary conditions (notably strong aggradation or incision) under which the model remains valid.

Background

Meandering rivers classically show sinuous planforms with rounded bends that grow, migrate, and are cut off. A large subset — low-energy meandering rivers, i.e. rivers whose driving energy is too small to erode their (partly cohesive) banks at appreciable rates — instead display tortuous planforms (here quantified by the fraction of bends that are 'sharp', with curvature radius divided by channel width $R_\mathrm{curv}/w < 2.0$) and very limited lateral migration. River energy is commonly measured by potential specific stream power $\omega_\mathrm{pot}$ (W/m²), the power per unit bed area available from discharge and valley slope; earlier discriminators showed that below a critical stream power rivers cannot erode cohesive banks. In the source's parameterization, bank erosion-resistance is proxied by the average silt-plus-clay (SC) fraction of the river banks, and the critical threshold is quantitative: their Eq. (2) gives the critical bed shear stress for eroding cohesive banks as $\tau_c = 0.1 + 0.18\,SC + 0.0028\,SC^2 - 2.34\times10^{-5}\,SC^3$ (in Pa, with SC in %), and their Eq. (3) converts $\tau_c$ into the minimum required potential specific stream power $\omega_\mathrm{crit}$ using the Chézy coefficient $C = 36\ \mathrm{m^{0.5}/s}$ (the median of the rivers analysed). Palaeogeographic study of the Holocene Dommel River (Netherlands) by Candel et al. (2020) revealed that the river's own deposits progressively armor its banks: residual channel-fills (fine-grained, organic plugs left in abandoned channels after neck cut-off), counterpoint deposits (fine silt-loam and clay-loam with organic layers — thick deposits of structureless or layered fine sediments and organics deposited from suspension along the margins of the valley or on the upstream side of concave bends, in flow-separation zones associated with sharp bends), and cohesive middle point-bar and overbank units are preferentially preserved while easily erodible sandy point-bar deposits are reworked. Because these resistant units reach channel-depth thickness and consolidate over time, bank strength grows autogenically — a positive feedback the authors named self-constraining: increasing bank cohesiveness forces sharper bends (higher tortuosity), sharper bends generate more counterpoint and channel-fill deposits, and average channel mobility falls. The source defines a self-constraining river as 'a river of which the channel mobility is increasingly hampered over time by self-formed erosion-resistant banks'; average channel mobility is the reach-scale lateral channel activity that their Figure 12D plots conceptually against time, and for which the snapshot dataset provides no quantitative metric. A global dataset of 47 meandering reaches confirmed that reaches with energy below a critical threshold have cohesive banks and tortuous planforms (the dataset is restricted to alluvial rivers, laterally connected with their floodplain, not strongly incising or aggrading, and with no sign of strong modification by humans). From these field snapshots the authors derived a conceptual model (their Figures 12 and 13) predicting (i) an initially exponentially increasing self-constraining response that later saturates as mobility declines, and (ii) two system states separated by an energy threshold: a self-constrained low-mobility state and an actively migrating state, with break-out requiring river energy to rise well above the threshold for eroding cohesive banks. The existence of two states implies a hysteresis loop — the river's planform response depends on whether energy is rising or falling — proposed by analogy with Late-Glacial to Holocene river metamorphosis in temperate regions. The source delimits applicability by prerequisites: river energy insufficient to erode cohesive depositional units but sufficient to erode non-cohesive material; a heterogeneous cohesive/non-cohesive sediment load; lateral connection with the floodplain without strong aggradation or incision (rates should be 'relatively limited', since aggradation or incision may lead to overtopping or undercutting of the erosion-resistant deposits and reset the self-constraining); and sufficient undisturbed time (channel belts not frequently abandoned by avulsion). Crucially, however, the dataset is a set of modern snapshots: the temporal trajectories of bank cohesiveness, tortuosity, and mobility could not be quantified, no known river has been documented in the act of crossing the threshold, the generality of the Dommel's particular resistant-deposit assemblage has not been systematically assessed — the source notes that overbank deposits, fine-grained middle point-bar deposits or oblique accretion deposits (fine sediments accreted on point-bar surfaces during overbank flow) may be more dominant in other low-energy systems, but asserts that they 'have the same effect on the river planform' — and strong aggradation or incision can overtop or undercut the resistant layers, resetting or invalidating the feedback. The question is therefore how, and under what conditions, this influential conceptual model actually behaves dynamically.

Problem Statement

Turn the conceptual self-constraining model of Candel et al. (2020) into a quantitative, tested account of low-energy meandering-river dynamics, by answering three jointly linked questions: (1) Trajectories — determine quantitatively how bank cohesiveness (proxied in the source by the average SC-fraction of the banks and the ratio of non-cohesive to cohesive banks), planform tortuosity (fraction of bends that are 'sharp', $R_\mathrm{curv}/w < 2.0$), and average channel mobility co-evolve in time for a self-constraining river satisfying the prerequisites above (the source proposes, but cannot test, an initially exponentially increasing response that saturates as mobility declines); the anticipated route is coupled numerical modelling experiments in which the floodplain stratigraphy co-evolves with the channel, with autogenic formation and preservation of erosion-resistant deposits and consolidation-driven strengthening of banks. (2) Threshold and hysteresis — establish whether two distinct system states (self-constrained versus actively migrating) really exist and are separated by the critical potential specific stream power $\omega_\mathrm{crit}$ for eroding cohesive banks, and determine the existence and shape of the conjectured hysteresis loop, including its dependence on the direction and rate of energy change; a documented threshold crossing means a river's energy, reconstructed, monitored or simulated, passing $\omega_\mathrm{crit}$ in either direction — upward, producing a transition from the self-constrained to the actively migrating state, or downward, producing the reverse transition — evidence the source explicitly notes is lacking ('convincing evidence of rivers that have crossed this threshold is presently lacking'). (3) Boundary conditions — define, through modelling or field analysis, the boundary conditions beyond which the self-constraining sequence is reset or the conceptual model ceases to be valid, notably the strong aggradation or incision that overtops or undercuts the erosion-resistant deposits (for which the source requires 'relatively limited' rates as a prerequisite of applicability), thereby delimiting the parameter space of river energy and bank erosion-resistance in which self-constraining operates (the source's second stated aim).

References

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