Idealized river-dune evolution models replace the computationally expensive resolution of the recirculating flow behind dune crests with a parameterized flow-separation bubble, enabling dune-height and form-roughness predictions on flood-wave timescales. The framework's founding paper (Paarlberg et al., 2009) explicitly leaves its closure scheme open: a depth-independent eddy viscosity with partial-slip boundaries, sediment transport driven only by the turbulence-averaged bed shear stress, a possibly inconsistent coupling between the nonlinear bubble parameterization and the nonlinear hydrostatic flow solver, periodic domains in which dune length is fixed rather than emergent, and an empirical lee-slope threshold for separation.

Background

River dunes are sand waves, centimetres to metres high, that migrate along the beds of sandy rivers; through their form roughness (the additional flow resistance caused by pressure drag on the dune bodies) they control water levels during floods and navigable depths during low flows, motivating fast process-based models of dune evolution. Downstream of a dune crest the boundary layer detaches and forms a flow separation zone (FSZ): a recirculating eddy in which flow near the bed moves upstream, bounded above by a separation streamline. Resolving the turbulence inside the FSZ requires Reynolds-averaged Navier–Stokes (RANS) or large-eddy simulation (LES), which is too costly for flood-wave timescales. Paarlberg, Dohmen-Janssen, Hulscher and Termes therefore introduced a parameterization of flow separation over subaqueous dunes (2007): the separation streamline is prescribed as an artificial smooth bed shaped as a polynomial bubble behind any lee slope steeper than about −10∘-10^\circ, sediment avalanches down the lee face at the angle of repose (the steep slope at which granular material is stable, about 30∘30^\circ underwater), and the flow above the bubble is computed with a cheap depth-averaged-type model. Their 2009 morphodynamic model couples this to a hydrostatic flow model (pressure assumed vertically hydrostatic, so vertical accelerations are neglected), closed with a depth-independent eddy viscosity (the constant turbulent momentum diffusivity of mixing-length-type closures) and partial-slip boundary conditions (interpolating between no-slip and free-slip through calibration coefficients); bed load is computed from a Meyer-Peter–Müller-type formula (the standard excess-stress power-law sand-transport relation) with gravitational bed-slope effects, driven by the turbulence-averaged bed shear stress. The dune length is not simulated but fixed to the fastest-growing wavelength obtained from a linear stability analysis (the modal growth rate of small sinusoidal bed perturbations, maximized over wavelength), and periodic boundary conditions are used. The model reproduces observed dune heights, lengths, migration rates and times to equilibrium, and its Discussion (paragraphs 64–69) explicitly catalogs what remains unresolved: whether a spatially varying eddy viscosity — including the mixing signature that shear inside the FSZ imprints on the turbulence just above the bubble, where the same partial-slip condition as at the bed is applied — would change quantitative predictions; that driving sediment transport with the time-averaged bed shear stress neglects the turbulent stress fluctuations and stochastic bed-load dynamics that near-bed turbulence measurements and earlier free-surface simulations (e.g. Giri and Shimizu, 2006) show matter over dunes; that periodic boundaries plus the averaged-stress transport closure permit no creation of new crests, so dunes merge until one fills the domain, blocking wavelength selection, crest creation and long-term merging dynamics; and that the nonlinear feedback introduced by the bubble parameterization is combined with a flow solver that retains nonlinear convective terms, a coupling whose mathematical and physical consistency was never analyzed. The authors state these aspects 'remain for future research' within the model's stated purpose: fast dune and roughness prediction for river flood waves. Separately, the paper's Section 2.3 (paragraphs 20–21) flags the empirical −10∘-10^\circ separation threshold as a crude device given observations of intermittent separation over lower-angle lee slopes (Best and Kostaschuk, 2002): the dynamics of flow over low-angle dunes are not fully known, so 'it is therefore not possible to state at which exact leeside slope the flow will separate'.

Problem Statement

Within the parameterized-flow-separation river-dune evolution framework of Paarlberg et al. (2009) — whose stated objective is fast, process-based prediction of dune dimensions and form roughness on flood-wave timescales — resolve the closure questions that the source explicitly leaves open, while retaining the framework's low computational cost: (1) a turbulence closure replacing the depth-independent eddy viscosity, including a representation of how mixing and shear inside the flow separation zone alter the effective eddy viscosity above the separation streamline (where the 2009 model simply reuses the bed partial-slip condition), and a quantification of how this changes predicted fastest-growing wavelength, equilibrium dune dimensions and timescales; (2) a sediment-transport closure that incorporates the role of turbulent fluctuations of the bed shear stress and stochastic bed-load dynamics, rather than the turbulence-averaged stress alone; (3) an analysis and resolution of the potential inconsistency arising from feeding the nonlinear feedback of the parameterized separation bubble into a hydrostatic flow solver that retains nonlinear convective terms — either a consistent coupling, a demonstrated equivalence or bounded deviation against a resolved-physics reference, or a proof of inconsistency requiring reformulation; (4) a domain and boundary treatment (nonperiodic conditions, or an equivalent mechanism) under which dune wavelength selection, creation of new crests and long-term merging dynamics emerge from the model rather than being fixed in advance to the linear fastest-growing wavelength, including the change of the dominant wavelength once flow separation sets in; and (5) a physically grounded criterion for the onset and permanence of flow separation (replacing the fixed −10∘-10^\circ leeside threshold) valid across lee angles and flow strengths, in particular for intermittent separation over low-angle dunes. The five questions are independent and none is a prerequisite for another: resolving any one of them, or any consistent subset, is sufficient, and demonstrating that any one of them is immaterial for predicted dune dimensions and roughness likewise addresses the same scientific objective: making the cheap parameterized dune model physically consistent and predictive without resolving the FSZ turbulence directly.

References

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