Olivier Devauchelle

Ramification of stream networks

The geometric complexity of stream networks has been a source of fascination for centuries. However, a comprehensive understanding of ramification—the mechanism of branching by which such networks grow—remains elusive. Here we show that streams incised by groundwater seepage branch at a characteristic angle of 2π/5 = 72°. Our theory represents streams as a collection of paths growing and bifurcating in a diffusing field. Our observations of nearly 5,000 bifurcated streams growing in a 100 km2 groundwater field on the Florida Panhandle yield a mean bifurcation angle of 71.9° ± 0.8°. This good accord between theory and observation suggests that the network geometry is determined by the external flow field but not, as classical theories imply, by the flow within the streams themselves.

Cross-stream diffusion in bedload transport

We investigate experimentally the statistical properties of bedload transport induced by a steady, uniform, and laminar flow. We focus chiefly on lateral transport. The analysis is restricted to experiments where the flow-induced shear stress is just above the threshold for sediment transport. We find that, in this regime, the concentration of moving particles is low enough to neglect interactions between themselves. We can therefore represent bedload as a thin layer of independent walkers travelling over the bed surface. In addition to their downstream motion, the particles show significant fluctuations of their cross-stream velocity, likely due to the roughness of the underlying sediment bed. This causes particles to disperse laterally. Based on thousands of individual trajectories, we show that this lateral spreading is the manifestation of a random walk. The experiments are entirely consistent with Fickian diffusion.

Geometry of valley growth

Although amphitheatre-shaped valley heads can be cut by groundwater flows emerging from springs, recent geological evidence suggests that other processes may also produce similar features, thus confounding the interpretations of such valley heads on Earth and Mars. To better understand the origin of this topographic form, we combine field observations, laboratory experiments, analysis of a high-resolution topographic map and mathematical theory to quantitatively characterize a class of physical phenomena that produce amphitheatre-shaped heads. The resulting geometric growth equation accurately predicts the shape of decimetre-wide channels in laboratory experiments, 100 m-wide valleys in Florida and Idaho, and kilometre-wide valleys on Mars. We find that, whenever the processes shaping a landscape favour the growth of sharply protruding features, channels develop amphitheatre-shaped heads with an aspect ratio of n.

Bifurcation dynamics of natural drainage networks.

As water erodes a landscape, streams form and channellize the surficial flow. In time, streams become highly ramified networks that can extend over a continent. Here, we combine physical reasoning, mathematical analysis and field observations to understand a basic feature of network growth: the bifurcation of a growing stream. We suggest a deterministic bifurcation rule arising from a relationship between the position of the tip in the network and the local shape of the water table. Next, we show that, when a stream bifurcates, competition between the stream and branches selects a special bifurcation angle α=2π/5. We confirm this prediction by measuring several thousand bifurcation angles in a kilometre-scale network fed by groundwater. In addition to providing insight into the growth of river networks, this result presents river networks as a physical manifestation of a classical mathematical problem: interface growth in a harmonic field. In the final sections, we combine these results to develop and explore a one-parameter model of network growth. The model predicts the development of logarithmic spirals. We find similar features in the kilometre-scale network.

Longitudinal profile of channels cut by springs

We propose a simple theory for the longitudinal profile of channels incised by groundwater flow. The aquifer surrounding the stream is represented in two dimensions through Darcys law and the Dupuit approximation. The model is based on the assumption that, everywhere in the stream, the shear stress exerted on the sediment by the flow is close to the minimal intensity required to displace a sand grain. Because of the coupling of the stream discharge with the water table elevation in the neighbourhood of the channel head, the stream elevation decreases as the distance from the streams tip with an exponent of 2/3. Field measurements of steephead ravines in the Florida Panhandle conform well to this prediction.

Stability of bedforms in laminar flows with free surface: from bars to ripples

The present paper is devoted to the formation of sand patterns by laminar flows. It focuses on the rhomboid beach pattern, formed during the backswash. A recent bedload transport model, based on a moving-grains balance, is generalized in three dimensions for viscous flows. The water flow is modelled by the full incompressible Navier–Stokes equations with a free surface. A linear stability analysis then shows the simultaneous existence of two distinct instabilities, namely ripples and bars. The comparison of the bar instability characteristics with laboratory rhomboid patterns indicates that the latter could result from the nonlinear evolution of unstable bars. This result, together with the sensibility of the stability analysis with respect to the parameters of the transport law, suggests that the rhomboid pattern could help improving sediment transport models, so critical to geomorphologists.

Path selection in the growth of rivers

Significance The complex patterns of river networks evolve from interactions between growing streams. Here we show that the principle of local symmetry, a concept originating in fracture mechanics, explains the path followed by growing streams fed by groundwater. Although path selection does not by itself imply a rate of growth, we additionally show how local symmetry may be used to infer how rates of growth scale with water flux. Our methods are applicable to other problems of unstable pattern formation, such as the growth of hierarchical crack patterns and geologic fault networks, where dynamics remain poorly understood. River networks exhibit a complex ramified structure that has inspired decades of studies. However, an understanding of the propagation of a single stream remains elusive. Here we invoke a criterion for path selection from fracture mechanics and apply it to the growth of streams in a diffusion field. We show that, as it cuts through the landscape, a stream maintains a symmetric groundwater flow around its tip. The local flow conditions therefore determine the growth of the drainage network. We use this principle to reconstruct the history of a network and to find a growth law associated with it. Our results show that the deterministic growth of a single channel based on its local environment can be used to characterize the structure of river networks.

Shape and dynamics of seepage erosion in a horizontal granular bed

We investigate erosion patterns observed in a horizontal granular bed resulting from seepage of water motivated by observation of beach rills and channel growth in larger scale land forms. Our experimental apparatus consists of a wide rectangular box filled with glass beads with a narrow opening in one of the side walls from which eroded grains can exit. Quantitative data on the shape of the pattern and erosion dynamics are obtained with a laser-aided topography technique. We show that the spatial distribution of the source of groundwater can significantly impact the shape of observed patterns. An elongated channel is observed to grow upstream when groundwater is injected at a boundary adjacent to a reservoir held at constant height. An amphitheater (semicircular) shape is observed when uniform rainfall infiltrates the granular bed to maintain a water table. Bifurcations are observed as the channels grow in response to the groundwater. We further find that the channels grow by discrete avalanches as the height of the granular bed is increased above the capillary rise, causing the deeper channels to have rougher fronts. The spatiotemporal distribution of avalanches increase with bed height when partial saturation of the bed leads to cohesion between grains. However, the overall shape of the channels is observed to remain unaffected indicating that seepage erosion is robust to perturbation of the erosion front.

Forced dewetting on porous media

We study the dewetting of a porous plate withdrawn from a liquid bath. The contact angle is fixed to zero and the flow is assumed to be almost parallel to the plate (lubrication approximation). The ordinary differential equation involving the position of the water surface is analysed in phase space by means of numerical integration. We show the existence of a stationary moving contact line with zero contact angle below a critical value of the capillary number η U/γ. Above this value, no stationary contact line can exist. An analytical model, based on asymptotic matching is developed, which reproduces the dependence of the critical capillary number on the angle of the plate with respect to the horizontal (3/2 power law), provided the capillary length is much larger than the square root of the porous-medium permeability. In addition, it is shown that the classical lubrication equation leads not only to the well-known Landau–Levich–Derjaguin films, but also to a family of films for which thickness is not imposed by the problem parameters.

Recirculation cells in a wide channel

Secondary flow cells are commonly observed in straight laboratory channels, where they are often associated with duct corners. Here, we present velocity measurements acquired with an acoustic Doppler current profiler in a straight reach of the Seine river (France). We show that a remarkably regular series of stationary flow cells spans across the entire channel. They are arranged in pairs of counter-rotating vortices aligned with the primary flow. Their existence away from the river banks contradicts the usual interpretation of these secondary flow structures, which invokes the influence of boundaries. Based on these measurements, we use a depth-averaged model to evaluate the momentum transfer by these structures, and find that it is comparable with the classical turbulent transfer.