Nonlinear saturation of the laminar free-surface bar instability and a quantitative theory of the rhomboid beach pattern

When a thin laminar film drains across a sandy beach after a wave (the backswash), diamond-shaped bedforms called rhomboid patterns often appear. Linear stability analysis of the full free-surface Navier–Stokes equations coupled to a three-dimensional bedload transport law (Devauchelle et al., J. Fluid Mech. 642, 2010) shows that oblique bar-type modes are unstable under these conditions and reproduces the measured pattern angles reasonably well, supporting the interpretation of rhomboid patterns as the laminar counterpart of river alternate bars. However, the analysis is strictly linear: the nonlinear saturation of the bar instability, the saturated amplitude and migration velocity of the pattern, a turbulent three-dimensional counterpart, and the experimental constraint of the empirical transport coefficients all remain open. The problem asks for these missing pieces so that the theory predicts, rather than only initiates, fully developed rhomboid patterns.

Background

Rhomboid patterns are centimeter-scale diamond-shaped structures that form on sandy beaches when the water flowing back to the sea after a wave (the backswash) drains as a thin laminar film over the sand. They can also be produced in laboratory flumes. Bars and ripples are the two generic instabilities of an erodible bed: ripples are bedwaves transverse to the flow (their wavevector is parallel to the mean flow direction), while bars are modes oblique to it; alternate bars in rivers are the turbulent, large-scale member of this family. Bedload is sediment transport by grains rolling, sliding, and hopping along the bed without being suspended. Its conservation is expressed by the Exner equation, which states that the rate of change of bed elevation equals minus the divergence of the volumetric sediment flux. The Froude number $\mathit{Fr}$ is the ratio of flow inertia to gravity, the Bond number $\mathit{Bo}$ the ratio of inertia to surface tension, and the Shields number $\mathit{Sh}$ the ratio of fluid shear stress to the submerged weight of a grain (the driving vs. resisting force on bed grains); $S$ denotes the bed slope. Devauchelle, Malverti, Lajeunesse, Lagrée, Josserand and Nguyen Thu-Lam (J. Fluid Mech. 642, 2010) modeled the water as a laminar incompressible Navier–Stokes flow with a free surface (a Nusselt film, the parabolic-velocity laminar film down an inclined plane, as base state) over a flat tilted granular bed. Sediment transport was described by the erosion–deposition law of Charru (2006) and Charru & Hinch (2006): grains are eroded from the fixed bed only above a threshold Shields number $\theta_t$, they redeposit under gravity at a rate set by a coefficient $c_d$, they drift with a velocity proportional (through a coefficient $c_u$) to the local shear rate of the flow, and they travel in a moving layer of thickness proportional ($c_g$) to the grain diameter; the ratio $c_u/c_d$ sets the deposition length over which the sediment flux relaxes toward equilibrium with the bed shear stress. Generalizing this law to three dimensions required no new empirical coefficient. A linear stability analysis of this coupled system showed two coexisting instabilities — transverse ripples and oblique bars — whose interaction as $\mathit{Fr}$ decreases explains the observed angle of rhomboid patterns, and the predicted opening angles compared acceptably with the laboratory measurements of Devauchelle et al. (2010), far better than a shallow-water (Saint-Venant) approximation. The authors concluded that beach rhomboid patterns are the laminar counterparts of alternate bars in rivers, and that the pattern, being easy to reproduce experimentally, can serve as a test case for three-dimensional sediment transport laws, whose transverse bed-slope behavior remains a modeling challenge. The paper is nevertheless strictly linear: the authors state they believe, but do not prove, that nonlinear saturation does not significantly change the pattern's angle and wavelength. Companion experiments (Devauchelle et al., J. Geophys. Res. Earth Surf. 115, 2010) over slopes 0.008–0.052 also observed the bed remaining flat at high slope and discharge despite active sediment transport, and the angle discrepancy at extreme parameter values reflects high sensitivity to the empirical parameters $\theta_t$, $c_u/c_d$ and $c_g$ of the transport law.

Problem Statement

Within the model framework of Devauchelle et al. (2010) — a laminar incompressible Navier–Stokes film with a free surface over an erodible granular bed, coupled to the three-dimensional generalization of the Charru (2006) bedload transport law and the Exner sediment balance — determine the finite-amplitude nonlinear evolution of the bar instability found by their linear analysis, and thereby predict the fully developed rhomboid pattern: its saturated wavelength and opening angle, its amplitude, and its migration velocity, as functions of the control parameters ($S$, $\mathit{Fr}$, $\mathit{Bo}$, $\mathit{Sh}$) and of the transport coefficients ($\theta_t$, $c_u/c_d$, $c_g$). The single overarching objective is to turn the linear initiation theory into a quantitative, testable theory of the saturated pattern, with two linked components requested by the source: (i) a three-dimensional stability analysis with a turbulent flow and a free surface, coupled to a grain-scale turbulent bedload law, able to interpret rhomboid-pattern experiments in turbulent flumes and to clarify where the laminar theory ceases to apply; and (ii) the use of the predicted saturated amplitude and migration velocity, compared with targeted low-Reynolds-number flume measurements, as two new quantitative constraints on the empirical coefficients of the bedload transport law — including whether the observed absence of patterns at the highest experimental slopes reflects parameter sensitivity or a genuine limitation of the model.

The overarching objective and the two linked components (i) and (ii) are jointly required, and a partial result on any single one of them does not by itself resolve the task. The task is complete only when each part returns an explicit reported result: the saturated wavelength, opening angle, amplitude and migration velocity as functions of the control parameters and of the transport coefficients (overarching objective); the solution of the three-dimensional turbulent free-surface stability problem together with an explicit statement of the Reynolds-number range in which the laminar result is or is not recovered (component (i)); and the confrontation of the predicted amplitude and migration velocity with the low-Reynolds-number flume measurements, reported as the ranges of the coefficients $\theta_t$, $c_u/c_d$ and $c_g$ that they admit and as a definite answer to whether the observed high-slope absence of patterns follows within the model (component (ii)).

References

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