A growth model driven by curvature reproduces geometric features of arboreal termite nests.

We present a simple three-dimensional model to describe the autonomous expansion of a substrate whose growth is driven by the local mean curvature of its surface. The model aims to reproduce the nest construction process in arboreal Nasutitermes termites, whose cooperation may similarly be mediated by the shape of the structure they are walking on, for example focusing the building activity of termites where local mean curvature is high. We adopt a phase-field model where the nest is described by one continuous scalar field and its growth is governed by a single nonlinear equation with one adjustable parameter d. When d is large enough the equation is linearly unstable and fairly reproduces a growth process in which the initial walls expand, branch and merge, while progressively invading all the available space, which is consistent with the intricate structures of real nests. Interestingly, the linear problem associated with our growth equation is analogous to the buckling of a thin elastic plate under symmetric in-plane compression, which is also known to produce rich patterns through nonlinear and secondary instabilities. We validated our model by collecting nests of two species of arboreal Nasutitermes from the field and imaging their structure with a micro-computed tomography scanner. We found a strong resemblance between real and simulated nests, characterized by the emergence of a characteristic length scale and by the abundance of saddle-shaped surfaces with zero-mean curvature, which validates the choice of the driving mechanism of our growth model.

Pattern formation in bubbles emerging periodically from a liquid free surface.

Patterns formed by centimeter scale bubbles on the free surface of a viscous liquid are investigated in a cylindrical container. These bubbles emerge periodically at the surface and interact with each other in the central zone. Their radial emission, due to interaction and radial surface flow, leads to the formation of a variety of patterns. Different star-like and spiral patterns appear spontaneously by increasing the bubble emergence frequency. It is found that these patterns are due to a constant angular shift in the bubble emission direction. Measurements of this angular shift show a supercritical bifurcation accompanied by a transition from a pattern of two opposed straight arms to spiral patterns. By applying the tools and concepts from the study of leaf arrangement in botany (phyllotaxis), the recognized patterns and the mechanism of the pattern formation are discussed. Close similarities to the leaf arrangement are found in the behavior of the angular shift and the patterns. These findings suggest that the observed patterns are formed by a packing mechanism of successively appearing elements (bubbles), which is similar to that of the leaves at the earliest stage of phyllotaxis.

Uphill solitary waves in granular flows.

We have experimentally observed uphill solitary waves in the surface flow on a granular material. A heap is constructed by injecting sand between two vertical glass plates separated by a distance much larger than the average grain size, with an open boundary. As the heap reaches the open boundary, solitary fluctuations appear on the flowing layer and move "up the hill" (i.e., against the direction of the flow). We explain the phenomenon in the context of stop-and-go traffic models.

Song of the dunes as a self-synchronized instrument.

Since Marco Polo it has been known that some sand dunes have the peculiar ability to emit a loud sound with a well-defined frequency, sometimes for several minutes. The origin of this sustained sound has remained mysterious, partly because of its rarity in nature. It has been recognized that the sound is not due to the air flow around the dunes but to the motion of an avalanche, and not to an acoustic excitation of the grains but to their relative motion. By comparing singing dunes around the world and two controlled experiments, in the laboratory and the field, we prove that the frequency of the sound is the frequency of the relative motion of the sand grains. Sound is produced because moving grains synchronize their motions. The laboratory experiment shows that the dune is not needed for sound emission. A velocity threshold for sound emission is found in both experiments, and an interpretation is proposed.

Four sided domains in hierarchical space dividing patterns.

The cracks observed in the glaze of ceramics form networks, which divide the 2D plane into domains. It is shown that, on the average, the number of sides of these domains is four. This contrasts with the usual 2D space divisions observed in Voronoi tessellation or 2D soap froths. In the latter networks, the number of sides of a domain coincides with the number of its neighbors, which, according to Euler's theorem, has to be six on average. The four sided property observed in cracks is the result of a formation process which can be understood as the successive divisions of domains with no later reorganization. It is generic for all networks having such hierarchical construction rules. We introduce a "geometrical charge," analogous to Euler's topological charge, as the difference from four of the number of sides of a domain. It is preserved during the pattern formation of the crack pattern.

Corridors of barchan dunes: Stability and size selection.

Barchans are crescentic dunes propagating on a solid ground. They form dune fields in the shape of elongated corridors in which the size and spacing between dunes are rather well selected. We show that even very realistic models for solitary dunes do not reproduce these corridors. Instead, two instabilities take place. First, barchans receive a sand flux at their back proportional to their width while the sand escapes only from their horns. Large dunes proportionally capture more sand than they lose, while the situation is reversed for small ones: therefore, solitary dunes cannot remain in a steady state. Second, the propagation speed of dunes decreases with the size of the dune: this leads, through the collision process, to a coarsening of barchan fields. We show that these phenomena are not specific to the model, but result from general and robust mechanisms. The length scales needed for these instabilities to develop are derived and discussed. They turn out to be much smaller than the dune field length. As a conclusion, there should exist further, yet unknown, mechanisms regulating and selecting the size of dunes.

Constitutive property of the local organization of leaf venation networks.

The leaf venation of dicotyledons forms complex patterns. In spite of their large variety of morphologies these patterns have common features. They are formed of a hierarchy of structures, which are connected to form a reticulum. Excellent images of these patterns can be obtained from leaves from which the soft tissues have been removed. A numerical image processing has been developed, specially designed for a quantitative analysis of this type of network. It provides a precise characterization of its geometry. The resulting data reveals a surprising property of reticula's nodes: the angles between vein segments are very well defined and it is shown that they are directly related by the radii of the segments. The relation between radii and angles can be expressed very simply using a phenomenological analogy to mechanics. This local organization principle is universal; all leaf venation patterns studied show the same behavior. The results are compared with physical networks such as fracture arrays or soap froth in terms of hierarchy and reorganization.

Selection of velocity profile and flow depth in granular flows.

The dynamics of a two-dimensional pile constituted by spherical grains organized in parallel layers is investigated theoretically. Only three effects are taken into account in the model: driving by gravity, nonlocal dissipation due to shocks, and trapping of grains by the bumps of the underneath layer. This is sufficient to recover the basic properties of granular avalanches: the transition between static and flowing state is hysteretic; the pile does not flow on the whole height but only in a layer at the surface; the velocity profile inside the flowing layer is approximately linear and is followed by a creep motion in the (quasi) static part. The flow height increases as a function of the pile angle and tends to infinity for a critical angle straight phi(infinity). The dependence of this critical angle with the static angle straight phi(s), the restitution coefficient rho, and the moment of inertia J, is investigated.

Dynamics of a grain on a sandpile model

The dynamics of a macroscopic grain rolling on an inclined plane composed of fixed identical grains is investigated both experimentally and theoretically. As real sand, the system exhibits an hysteretic transition between static and dynamical states: for angles smaller than straight phi(d), the roller always stops, for angles larger than straight phi(s), it spontaneously starts rolling down. But for angles between straight phi(d) and straight phi(s), it can be either at rest or in motion with a constant velocity. It is shown that the limit velocity is given by the equilibrium between gravity driving and dissipation by the shocks. Moreover, the rough plane acts as a periodic potential trap whose width and depth decrease when the angle is increased: the static angle straight phi(s) corresponds to the angle for which the trap disappears; the dynamical angle straight phi(d) to that for which the limit velocity is sufficient to escape from the trap. Finally, a continuous description of the force globally acting on the grain is proposed, which preserves this hysteretic behavior.