Open problems in the threshold-dominated streamlet model of erosion: gully evolution in converging valleys, the logarithmic steady-state profile, lateral exchange between streamlets, and channelization stability

Pierre-Gilles de Gennes (J. Phys.: Condens. Matter 12, A499, 2000) modeled the erosion of an impermeable soil by overland flow as a set of downhill 'streamlets' obeying a Chezy-type turbulent flow law, with erosion acting only above a sharp critical shear stress. The threshold alone fixes the 'line of attack' separating eroded from uneroded ground — at distance $L = V_c^3/(k g \theta_0 p)$ from the crest on an inclined plane, and bounding a gully of half-width $\delta = Ra/L$ in a U-shaped valley — pins the flow velocity near threshold in a mature, slowly uplifting basin, and yields a logarithmic steady-state profile $z(x) = z_1 - \theta_0 L \ln(x/L)$, while valley formation is argued to require finite-amplitude rather than infinitesimal perturbations. The paper left four issues open: the widening and deepening dynamics of the gully in a converging valley; the robustness and experimental verification of the logarithmic profile (which field data on natural basins already disfavor) using weakly cohesive laboratory soils with low threshold velocity and fine-droplet rainfall; the treatment of lateral exchanges between streamlets; and whether the instability conclusion extends to complex topographies and to regimes with both erosion and sedimentation. No later work is known that resolves or refutes any of the four issues. The problem therefore remains open in its original, unreduced form.

Background

Erosion of upland landscapes by running water is often modeled by deterministic mass-balance equations in which the ground surface is lowered where flowing water exerts enough force to detach and carry soil, and raised where the transported load settles. A foundational analysis by Smith and Bretherton (1972) showed that in such models without an erosion threshold, a uniformly inclined planar hillslope is linearly unstable: any small perturbation that concentrates flow erodes faster and grows, so channels should form spontaneously from infinitesimal disturbances. Field geomorphology, however, shows that channel heads in real landscapes appear where the combination of upslope contributing area and local slope crosses a finite threshold (Montgomery and Dietrich, 1988; 1992), and detachment-limited simulation models of the 1990s — models in which the rate of surface lowering is set by the flow's ability to detach soil particles rather than by its capacity to carry them — likewise incorporated a critical stress for erosion onset (Howard, 1994).

Pierre-Gilles de Gennes's 2000 paper 'Effect of topographic convergence on erosion processes' asked what follows from taking such a threshold seriously in an analytically tractable setting. The model considers a basin of impermeable soil eroded purely by water flow (not by weathering) and adopts a 'streamlet' picture: overland flow — rainwater that runs off across the ground surface rather than infiltrating — is organized in thin threads that always run straight downhill. Each thread is treated as fully developed turbulent open-channel flow obeying a Chezy-type law — the classical empirical relation for uniform turbulent open-channel flow — $V^2 = k g h \theta$, where $V$ is the mean flow velocity, $h$ the water depth, $\theta = |\nabla z|$ the local ground slope, $g$ gravitational acceleration, and $k$ a numerical constant. Steady rainfall of rate $p$ feeding a thread implies, by water continuity, $V h\,\mathrm{d}l = p\,\mathrm{d}A$, where $\mathrm{d}A$ is the upstream area collected across an isolevel-line (constant-elevation contour) element $\mathrm{d}l$; combining the two gives the central relation $V^3 = k g \theta p\,(\mathrm{d}A/\mathrm{d}l)$, where $\mathrm{d}A/\mathrm{d}l$ — the upstream area gathered per unit length of contour — is called the collection factor and measures how strongly the topography converges the flow lines. Erosion acts only when the hydraulic shear stress $\tau = \rho g h \theta$ (with $\rho$ the water density) exceeds a critical value $\tau_c$, equivalently when $V$ exceeds the threshold velocity $V_c$ defined by $\rho V_c^2 = k \tau_c$; below threshold the erosion rate is exactly zero.

From this the paper derives four results. (1) On an inclined plane of slope $\theta_0$, the 'line of attack' — the border between eroded and uneroded ground — lies at distance $L = V_c^3/(k g \theta_0 p)$ from the crest. (2) In a U-shaped (parabolic) valley $z = z_0 - \theta_0 x + \theta_0 y^2/(2R)$, with $x$ the down-valley coordinate, $y$ the transverse coordinate, and $R$ the curvature radius of the isolevel lines at the valley axis, flow lines converge toward the axis, the collection factor grows like $Ra/y$, and the eroded region is a central gully of half-width $\delta = Ra/L$ bounded by lines parallel to the valley axis; the gully is narrower for stronger thresholds ($\delta \sim \tau_c^{-3/2}$) and for lighter rainfall. (3) In a mature one-dimensional basin subject to slow uniform uplift at rate $U$, the steady-state balance of erosion (taken proportional to the excess of $V^2$ above $V_c^2$), sedimentation, and uplift pins the flow velocity near threshold, $V \approx V_c$, so the slope must satisfy $\theta = \theta_0 L/x$ and the equilibrium elevation profile is logarithmic, $z(x) = z_1 - \theta_0 L \ln(x/L)$ (equation (19) of the source) — a prediction claimed to be independent of the detailed erosion–sedimentation law above threshold. (4) On a plane, no linear instability toward valley formation is expected within the uneroded strip $x < L$: to start erosion there, a perturbation would have to raise the collection factor well above the plane value $x$, which requires a valley of finite pre-existing amplitude — channelization in this model is a finite-amplitude, nonlinear process, unlike the infinitesimal instability of the threshold-free model of Smith and Bretherton.

The paper closes by listing four unresolved issues, which are the subject of this record. They should be read against adjacent work that already existed in the 1990s: Loewenherz-Lawrence (1994) performed a linear stability analysis of coupled shallow overland flow and advective sediment transport, finding that the cross-slope pressure gradient within the flowing water film forces a minimum transverse spacing of incipient rills — a lateral-coupling effect absent from the streamlet picture — and Izumi and Parker (1995) combined an erosion threshold with the full shallow-water equations for channel inception, finding classical linear stability analysis inadequate for selecting the spacing of incipient drainage basins and developing an upstream-driven theory instead. Later landscape-evolution research has treated erosion thresholds mainly within stream-power incision models, in which the rate of channel-bed lowering is taken proportional to a power of the product of water discharge and channel slope (DiBiase and Whipple, 2011; Lague, 2014), while gully widening and deepening are studied mainly through field monitoring and empirical regression (Poesen et al., 2003; Wells et al., 2013) and through small-scale laboratory landform experiments combining artificial rainfall with imposed uplift (Ouchi, 2015). None of these streams of work adopts the de Gennes streamlet model itself.

Problem Statement

The problem is to complete and empirically test the threshold-dominated streamlet erosion theory of de Gennes (2000), within the model exactly as defined there: Chezy-type streamlet flow $V^2 = k g h \theta$ ($V$ mean flow velocity, $h$ water depth, $\theta$ local ground slope, $k$ a numerical constant of the flow law, $g$ gravitational acceleration), rainfall continuity $V h\,\mathrm{d}l = p\,\mathrm{d}A$ ($p$ steady rainfall rate; $\mathrm{d}A$ the upstream area collected across an isolevel-line element $\mathrm{d}l$), the combined relation $V^3 = k g \theta p\,(\mathrm{d}A/\mathrm{d}l)$ with collection factor $\mathrm{d}A/\mathrm{d}l$, a sharp erosion threshold $\rho V_c^2 = k \tau_c$ ($\rho$ water density, $\tau_c$ critical shear stress, $V_c$ the threshold flow velocity) below which the erosion rate vanishes, and an erosion–sedimentation–uplift balance for the ground height $z$ in which the erosion term above threshold is proportional to $V^2 - V_c^2$ and sedimentation returns sediment to the surface. The source identifies four unresolved issues, which together determine whether the model's distinctive predictions — a sharply bounded eroded region, a gully of calculable width, and a logarithmic steady-state profile — are internally consistent and physically realized:

(1) Gully evolution in a converging valley. Determine the dynamics of the gully that the model predicts will form in a U-shaped valley of initial width $2\delta$ (half-width $\delta = Ra/L$, where $R$ is the curvature radius of the valley's isolevel lines at the axis, $a$ the lateral offset of a starting flow line, and $L$ the line-of-attack distance on a plane of slope $\theta_0$ under the same rainfall): how it widens and deepens under continued rainfall and removal of the eroded material, and what form it approaches in the long run.

(2) Status of the logarithmic mature-basin profile. Establish whether the steady-state logarithmic profile $z(x) = z_1 - \theta_0 L \ln(x/L)$, with $x$ the downslope distance from the crest, $\theta_0$ the reference slope, $L$ the line-of-attack distance, and $z_1$ a constant fixed by the downstream boundary (equation (19) of the source) is robust — in particular whether the velocity-pinning argument $V \approx V_c$ survives erosion laws more general than the linear excess-stress form and a sediment budget derived self-consistently from the erosion–sedimentation equation — and test the prediction experimentally in the regime the source specifies: weakly cohesive laboratory soils with low threshold velocity $V_c$, rainfall applied as fine droplets so that raindrop-impact erosion does not contaminate the flow-threshold signal, in a setting where the line of attack, gully width, and approach to a steady profile under imposed uplift can be measured. The source itself notes that the predicted profile differs significantly from field observations of natural basins, which is precisely why a dedicated weak-soil experiment is called for.

(3) Lateral exchanges between streamlets. Analyze the effect of lateral exchange of water and sediment between neighboring streamlets — neglected in the deterministic streamlet picture, where only diffusion widths are estimated in the source's appendix — on the predicted line of attack and on the steady-state profile.

(4) Stability of the threshold model. Determine whether the conclusion that no linear instability drives valley formation on an inclined plane (valid within the uneroded region $x < L$, with $x$ the downslope distance from the crest) extends to more complex topographies and to regimes in which both erosion and sedimentation are active — that is, provide a stability or finite-amplitude analysis of the threshold model in those settings, replacing the source's heuristic two-sentence argument.

The common scientific objective is the completion and empirical test of this one threshold-dominated erosion theory; the four components are the gaps the source itself identifies in that theory. The empirical clause of component (2) is a weak-soil experiment — its protocol, the calibration of the threshold velocity $V_c$ and the flow-law constant $k$, and the quantitative, falsifiable signatures (the line of attack, the gully width, and the steady-state profile) by which the logarithmic prediction would be confirmed or refuted.

References

1. Pierre-Gilles de Gennes (2000). Effect of topographic convergence on erosion processes. Journal of Physics: Condensed Matter 12(8A), A499–A505. DOI: 10.1088/0953-8984/12/8a/369. https://doi.org/10.1088/0953-8984/12/8a/369
2. Pierre-Gilles de Gennes (2003). Effect of topographic convergence on erosion processes. In: Simple Views on Condensed Matter, World Scientific, pp. 553–559. DOI: 10.1142/9789812564849_0060. https://doi.org/10.1142/9789812564849_0060 (reprint of the primary source)
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