Finite-amplitude saturation of river dune and antidune instabilities: closing the subcritical antidune branch and the role of suspended load

A turbulent open-channel flow over an erodible sand bed is linearly unstable to bedwaves: downstream-migrating dunes at subcritical Froude numbers and upstream-migrating antidunes at supercritical ones. Colombini and Stocchino (J. Fluid Mech. 611, 2008) closed the weakly nonlinear (Landau) amplitude equation at cubic order in a depth-resolved theory: for values of the ratio of the depth-averaged velocity to the shear velocity of practical interest the dune bifurcation is supercritical, so a stable finite-amplitude equilibrium exists and its predicted amplitude matches flume data, but the antidune bifurcation is subcritical, meaning the cubic truncation cannot saturate the branch and only signals coexistence of plane beds and antidunes below the linear threshold. The open problem is to complete this finite-amplitude theory: determine whether stable finite-amplitude antidune equilibria exist and where the plane–antidune coexistence region closes, through the routes the paper itself indicates — extending the expansion to fifth order and/or including suspended-sediment transport — together with the resulting shifts of the critical Froude numbers and of the tricritical points. The paper separately notes that how flow separation affects dune development as the steepness of the front increases remains open and can be handled satisfactorily only by a fully nonlinear numerical solution, and it declares the mutual interaction between antidunes and roll waves beyond its scope; these are related open directions rather than parts of the objective. The convective-versus-absolute character of the instabilities, posed in the same paper, has since been resolved by other authors and is not part of the target.

Background

Alluvial bedforms are sand waves that grow on the bed of a river or flume when the flowing water exerts a shear stress strong enough to move sediment. When the stress exceeds the threshold for grain motion, a perturbation of the bed interacts with the flow so that the bed shear stress is phase-shifted relative to the bed elevation; this feedback makes a flat bed linearly unstable to waves of a range of wavelengths. Two families matter here. Dunes migrate downstream and occur at Froude numbers $F=U/\sqrt{gD}$ (mean velocity $U$, flow depth $D$, gravity $g$) below unity; antidunes migrate upstream, or stand still, and are tied to the free surface, occurring near and above resonant conditions for surface gravity waves ($F$ around or above 1). Colombini and Stocchino (2008) studied the linear and weakly nonlinear stability of a uniform turbulent flow in an infinitely wide channel with an erodible bottom, using a rotational (depth-resolved, vorticity-carrying) mean-flow turbulence closure and a bedload transport formula coupled to the bed through the sediment mass balance (the Exner equation, which states that the divergence of the sediment flux balances the rate of bed rise). Sediment transport is characterized by the Shields parameter, a dimensionless measure of the shear stress relative to the submerged grain weight. Near the linear critical conditions they performed a multiple-scales (Stuart–Landau) expansion in the small amplitude $A$ of the critical bedwave, obtaining a Landau equation $dA/dt=\sigma A-\ell|A|^2A$: the linear growth rate $\sigma$ changes sign at criticality, and the Landau coefficient $\ell$ determines the character of the bifurcation. If $\ell$ has the stabilizing sign the bifurcation is supercritical: a stable equilibrium amplitude $|A|=\sqrt{\sigma/\ell}$ exists just beyond criticality; for dunes the coefficient has this sign throughout the range of the ratio of the depth-averaged velocity to the shear velocity of practical interest (equivalently, the dimensionless conductance coefficient $C=U/u_*$, set by grain roughness through the Keulegan relation), and the predicted amplitudes compared well with laboratory measurements. For antidunes the coefficient has the opposite sign: the bifurcation is subcritical, so finite-amplitude antidunes are predicted already below the linear threshold — consistent with the experimentally observed coexistence of plane beds and antidunes for the same flow and sediment parameters — but the cubic truncation then admits no stable small-amplitude equilibrium; saturation requires either the next (quintic, fifth-order) term of the expansion, which can turn the branch back and stabilize it, or additional physics. The leading candidate physics is suspended load: when grains travel in suspension rather than as bedload, an advection–diffusion equation for the sediment concentration replaces the local bedload flux relation, and the associated phase lags are known to alter linear thresholds and are expected to change the sign or magnitude of the nonlinear coefficients. Points where the cubic coefficient vanishes and higher-order terms control the bifurcation are called tricritical points; the 2008 theory locates them along both critical curves $F_{cd}$ (dunes) and $F_{ca}$ (antidunes), at such extreme values of the conductance coefficient ($C=11$ for antidunes, $C=24$ for dunes) that for values of $C$ of practical interest the dune bifurcation can consistently be taken as supercritical whereas the antidune bifurcation is subcritical. Two further limitations are acknowledged in the 2008 paper itself; they lie outside the objective stated below and are recorded here as related open directions. First, the perturbation expansion is valid only near the linear critical points, whereas natural dunes become steep enough that the flow separates from the crest and recirculates on the lee side, a fully nonlinear effect that modifies stress, transport, and equilibrium geometry; the paper states that how separation affects dune development as the steepness of the front increases remains open and can be handled satisfactorily only by means of a fully nonlinear numerical solution. Second, at higher Froude numbers the antidune mode coexists with roll waves — finite-amplitude waves on the free surface of supercritical shallow flows, which the paper notes provide a possible limiting mechanism for antidune formation — and the analysis of their mutual interaction was explicitly declared beyond the scope of that paper. The convective or absolute character of the dune and antidune instabilities (an instability is convective if growing wave packets are swept away while the bed eventually returns to its initial state locally, and absolute if perturbations amplify in place) was also left open in 2008, the paper noting that no formal proof had been provided yet to ascertain it; later work settled it, and it is not part of the objective.

Problem Statement

Within the depth-resolved morphodynamic framework of Colombini & Stocchino (2008) — rotational turbulent mean flow over an erodible bed with a bedload transport closure — complete the finite-amplitude equilibrium theory of river dunes and antidunes. The single objective is a predictive theory of finite-amplitude bedform equilibria and their bifurcation structure, whose open core is the subcritical antidune branch: at cubic order the theory yields no information on antidune saturation, and a conclusive answer on the observed coexistence of plane beds and antidunes (and on the possible hysteresis loop) cannot be sought within that formulation, since a higher-order bifurcation analysis would be required. Determine therefore whether stable finite-amplitude antidune equilibria exist, where the plane–antidune coexistence region closes, and how that coexistence region compares with flume and field observations, following the routes indicated by the 2008 paper: extending the weakly nonlinear multiple-scales expansion to fifth order to obtain the quintic Landau coefficients that can turn and stabilize the branch, and/or augmenting the model with suspended-sediment transport (an advection–diffusion equation for the concentration coupled to the bed evolution) and recomputing the linear thresholds and nonlinear coefficients. A solution that resolves the core questions above through either route alone — the fifth-order expansion or the suspension-augmented recomputation — satisfies this objective; the suspension-specific shifts stated next are required only when the suspension route is adopted. Within the suspension-augmented setting the same objective includes quantifying how suspended load shifts the linear critical Froude numbers $F_{cd}$ (dunes) and $F_{ca}$ (antidunes) — the paper expects suspension to raise $F_{ca}$ and to lower $F_{cd}$ as the Shields stress exceeds the incipient-suspension threshold $\theta_s$ — and how it repositions the tricritical points that in the bedload-only theory lie along both critical curves at extreme values of the conductance coefficient, stating the parameter dependence (Shields parameter, conductance coefficient $C=U/u_*$, roughness) of these shifts.

Two items noted by the source are related open directions rather than required components of the objective above. First, the 2008 theory describes finite-amplitude dune growth only while the steepness of the front at equilibrium remains small enough to prevent flow separation at the crest; how separation affects dune development as the steepness of the front increases remains open and, lying beyond the weakly nonlinear bedload framework, can be handled satisfactorily only by means of a fully nonlinear numerical solution. Second, the mutual interaction between the antidune and roll-wave modes at higher Froude numbers was explicitly declared beyond the scope of the 2008 paper. Determination of the convective or absolute character of the instabilities is not part of the target: it has been resolved for dunes (convective for all parameters) and antidunes (parameter-dependent, with both behaviors) by Vesipa et al. (2014).

References

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