Random walkers usually diffuse according to Fick's law. On average, they move down the gradient of their concentration and, in the absence of external force, tend to distribute themselves uniformly. In some experiments, however, this familiar notion is at odds with observation. Sand grains, for instance, gather along the nodal lines of a vibrated elastic plate to form a Chladni figure, thus accumulating where fluctuations are weak – a fact that escapes the reach of Fick's law. On theoretical grounds, Büttiker [Zeitschrift für Physik B, 68, 1987] and Landauer [J Stat Phys, 53, 1988] proposed that particles submitted to a non-uniform temperature field would indeed gather where the temperature is low. They also predicted that, in the presence of a potential force, a non-uniform temperature could drive a steady current of particles, powered only by noise. Here, we present an experimental realization of these phenomena in a macroscopic system, which confirms the quantitative predictions of Büttiker.

When a horizontal plate vibrates strongly enough, it causes small particles such as sand grains to continually bounce on it and, over time, to diffuse across its surface. This phenomenon is the cause of the well-known Chladni figure, which is drawn by a higher density of grains gathering along the nodal lines of a resonating elastic plate. Using a heterogeneous, non-resonating plate, we investigate experimentally this type of diffusion. We find that, for the most part, is it comparable to classical molecular diffusion. We can define a temperature for the bouncing grains, and the system then obeys the fluctuation-dissipation theorem. We also recover Maxwell-Boltzmann statistics at equilibrium, when temperature is uniform. However, when temperature varies across the vibrating plate, the microscopic details of the grains' dynamics affect their macroscopic behavior: Fick's law, for instance, no longer applies. Instead, our experiments support a new transport relation that was recently proposed to represent diffusion in Chladni's experiment. Finally, we propose an expression for the heat flux associated to the non-equilibrium steady state predicted by this new relation, and test it against observations.

During a rainfall event, water infiltrates into the ground where it accumulates in porous rocks. This accumulation pushes the underlying groundwater towards neighbouring streams, where it runs to the sea. After the rain has stopped, the aquifer gradually releases its excess water, as the water table relaxes, until the next rain. In the absence of recharge, the water table would eventually reach its horizontal equilibrium position. The volume of groundwater stored above this level, which we call the active volume, sustains the river between two rainfall events. In this article, we use an experimental aquifer recharged by artificial rain to investigate how this active volume depends on the rainfall rate. Restricting our analysis to the steady-state regime, wherein the discharge into the stream balances rainfall, we explore a broad range of rainfall rates, for which the water table deforms significantly. We find that the active volume of water stored in the aquifer decreases with its depth. Using conformal mapping, we derive the flow equations and develop a numerical procedure that accounts for the active volume of groundwater in our experiments. In the case of an infinitely deep aquifer, the problem admits a closed-form solution, which provides a satisfying estimate of the active volume when the aquifer's depth is at least half its width. In the general case, a rougher estimate results from the energy balance of the dissipative groundwater flow.

Although Chladni patterns are widely used to visualize the modes of vibration of an elastic plate, the mechanism by which the grains form the pattern is still debated. In this Letter, we suggest that the pattern results from space-dependent diffusion: grains gather where their diffusivity is low. We test this hypothesis in experiments, wherein we generate Chladni patterns on a vibrating membrane. We measure the diffusivity of the grains as a function of the vibration amplitude and propose an expression of the diffusive flux based on our measurements. We find a steady-state distribution of the grains in agreement with observations. These findings could inspire new methods to manipulate and segregate particles.

A theory is presented for wave-driven propulsion of floating bodies driven into oscillation at the fluid interface. By coupling the equations of motion of the body to a quasipotential flow model of the fluid, we derive expressions for the drift speed and propulsive thrust of the body which in turn are shown to be consistent with global momentum conservation. We explore the efficacy of our model in describing the motion of SurferBot (Rhee et al., Bioinspir. Biomim., vol. 17, issue 5, 2022), demonstrating close agreement with the experimentally determined drift speed and oscillatory dynamics. The efficiency of wave-driven propulsion is then computed as a function of driving oscillation frequency and the forcing location, revealing optimal values for both of these parameters which await confirmation in experiments. A comparison with other modes of locomotion and applications of our model with competitive water sports is discussed in conclusion.

The distribution of stress generated by a turbulent flow matters for many natural phenomena, of which rivers are a prime example. Here, we use dimensional analysis to derive a linear, second-order ordinary differential equation for the distribution of stress across a straight, open channel, with an arbitrary cross-sectional shape. We show that this equation is a generic first-order correction to the shallow-water theory in a channel of large aspect ratio. It has two adjustable parameters – the dimensionless diffusion parameter, $\chi$ , and a local-shape parameter, $\alpha$ . By assuming that the momentum is carried across the stream primarily by eddies and recirculation cells with a size comparable to the flow depth, we estimate $\chi$ to be of the order of the inverse square root of the friction coefficient, $\chi \sim C_f^{-1/2}$ , and predict that $\alpha$ vanishes when the flow is highly turbulent. We examine the properties of this equation in detail and confirm its applicability by comparing it with flume experiments and field measurements from the literature. This theory can be a basis for finding the equilibrium shape of turbulent rivers that carry sediment.

In modern democracies, the outcome of elections and referendums is often remarkably tight. The repetition of these divisive events are the hallmark of a split society; to the physicist, however, it is an astonishing feat for such large collections of diverse individuals. Many sociophysics models reproduce the emergence of collective human behavior with interacting agents, which respond to their environment according to simple rules, modulated by random fluctuations. A paragon of this class is the Ising model which, when interactions are strong, predicts that order can emerge from a chaotic initial state. In contrast with many elections, however, this model favors a strong majority. Here we introduce a new element to this classical theory, which accounts for the influence of opinion polls on the electorate. This brings about a new phase in which two groups divide the opinion equally. These political camps are spatially segregated, and the sharp boundary that separates them makes the system size dependent, even in the limit of a large electorate. Election data show that, since the early 1990s, countries with more than about a million voters often found themselves in this state, whereas elections in smaller countries yielded more consensual results. We suggest that this transition hinges on the electorate's awareness of the general opinion.

The soft part of the Earth's surface - the ground beneath our feet - constitutes the basis for life and natural resources, yet a general physical understanding of the ground is still lacking. In this critical time of climate change, cross-pollination of scientific approaches is urgently needed to better understand the behavior of our planet's surface. The major topics in current research in this area cross different disciplines, spanning geosciences, and various aspects of engineering, material sciences, physics, chemistry, and biology. Among these, soft matter physics has emerged as a fundamental nexus connecting and underpinning many research questions. This perspective article is a multi-voice effort to bring together different views and approaches, questions and insights, from researchers that work in this emerging area, the soft matter physics of the ground beneath our feet. In particular, we identify four major challenges concerned with the dynamics in and of the ground: (I) modeling from the grain scale, (II) near-criticality, (III) bridging scales, and (IV) life. For each challenge, we present a selection of topics by individual authors, providing specific context, recent advances, and open questions. Through this, we seek to provide an overview of the opportunities for the broad Soft Matter community to contribute to the fundamental understanding of the physics of the ground, strive towards a common language, and encourage new collaborations across the broad spectrum of scientists interested in the matter of the Earth's surface.

<p>We study the effect of an increase in flow discharge on the shape and growth of an experimental alluvial fan. The fan is built by a single-thread channel in which the flow occurs near the threshold of sediment motion. We first define a criterion that predicts the conditions under which a change in discharge leaves an inprint on the morphology of a fan. We then report on experimental runs which allow us to establish the relevance of this criterion. Experiments are carried out during which climatic changes are applied to the feeding channel of a fan. By playing on the initiation time of climate change, on the duration of the rise in flow, or on the total variation in discharge, we scan a range of configurations that allow us to qualitatively and quantitatively test our incision criterion. Qualitatively, we note that the dynamics of the fan seems altered only for values of the criterion which exceed the critical value of 1.5. In these situations, the channel stops moving and entranches. Quantitatively, we extract a characteristic time by autocorrelating spatio-temporal channel migration diagrams and show that this time correlates with the value of the incision criterion.</p>

Bedload transport, entrainment of coarse sediment by a river, is inherently a stochastic and intermittent process whose monitoring remains challenging. Here, we propose a new method to characterize bedload transport in the field. Using an uncrewed aerial vehicle (UAV) equipped with a high-resolution camera, we recorded yearly images of a bar of the Grande Rivière des Vieux-Habitants, a gravel bed river located on Basse-Terre Island (Guadeloupe, French West Indies). These images, combined with high-frequency measurements of the river discharge, allow us to monitor the evolution of the population of sediments of a diameter between 0.5 and 0.75 m on the riverbed. Based on this dataset, we estimate the smallest discharge that can move these boulders and calculate the duration of effective transport. We find that the transport of boulders occurs for approximately 10 h yr−1. When plotted as a function of the effective transport time, a given population of boulders decreases exponentially with an effective residence time of approximately 17 h. This exponential decay suggests that the probability of dislodging a grain from the bed is proportional to the number of grains at repose on the bed, an observation consistent with laboratory experiments. Finally, the residence time of bedload particles on a riverbed can be used to evaluate bedload discharge.

Inspired by presentations by the authors during a workshop organized at the Princeton Center for Theoretical Science (PCTS) in January 2022, we present a perspective on some of the outstanding questions related to the "physics of the ground beneath our feet." These identified challenges are intrinsically shared with the field of Soft Matter but also have unique aspects when the natural environment is studied.

It has been shown experimentally that small droplets, bouncing on a vibrated liquid bath, can “walk” across the surface due to their interaction with their own wave-field. Gunwale bobbing is a life-size instance of this phenomena in which a person standing on the gunwales of a canoe propels it by pumping it into oscillation with the legs. The canoe moves forward by surfing the resulting wave-field. After an initial transient, the canoe achieves a cruising velocity which satisfies a balance between the thrust generated from pushing downwards into the surface gradients of the wave-field and the resistance due to a combination of profile drag and wave drag. By superposing the linear wave theories of Havelock (1919) for steady cruising and of Helmholtz for an oscillating source, we demonstrate that such a balance can be sustained. We calculate the optimal parameter values to achieve maximum canoe velocity. We compare our theoretical result to accelerometer data taken from an enthusiastic gunwale bobber. We discuss the similarities and differences between gunwale bobbing and hydrodynamic quantum analogues, and possible applications to competitive sports.

We study the effect of an increase in flow discharge on the shape and growth of an experimental alluvial fan using a small scale controlled experiment. The fan is built by a channel in which the flow is located near the threshold of sediment motion. Under these conditions, it is possible to define a simple criterion for the incision of a fan as a function of both the amplitude and duration of the change in fluid discharge. Our results show the relevance of this criterion and enable us to discuss the conditions under which a climate perturbation may or may not leave an imprint on the fan morphology and discuss its application to natural systems.

Abstract Titan's rich and dense atmosphere, composed mainly of methane and nitrogen, maintains a methane cycle that shapes its surface, like the water cycle on Earth. Methane precipitations erodes Titan's surface and forms complex river networks observed at all latitudes by the Cassini-Huygens mission. However, precipitation rates are poorly constrained and, in the absence of in situ measurements, understanding Titan’s global climate and meteorology is done primarily through climate models. Here, we apply a physics-based theory of river morphogenesis to provide new constraints on methane precipitation rates. In particular, we estimate the river discharges and precipitation rates required to shape two methane rivers located at the equator and south pole, which incise the surface of Titan. Our results show that the use of river morphology is relevant for inferring methane precipitation rates at various latitudes. Our estimates reduce the uncertainties from the climate models. In addition, our results reflect the climate variability between the equator and the south pole. Finally, this work sheds the light to unknown river on Earth: the giant gravel-bed rivers.

During rainfall, water infiltrates the soil, and percolates through the unsaturated zone until it reaches the water table. Groundwater then flows through the aquifer, and eventually emerges into streams to feed surface runoff. We reproduce this process in a two-dimensional laboratory aquifer recharged by artificial rainfall. As rainwater infiltrates, it forms a body of groundwater which can exit the aquifer only through one of its sides. The outlet is located high above the base of the aquifer, and drives the flow upwards. The resulting vertical flow component violates the Dupuit-Boussinesq approximation. In this configuration, the velocity potential that drives the flow obeys the Laplace equation, the solution of which crucially depends on the boundary conditions. Noting that the water table barely deviates from the horizontal, we linearize the boundary condition at the free surface, and solve the flow equations in steady state. We derive an expression for the velocity potential, which accounts for the shape of the experimental streamlines and for the propagation rate of tracers through the aquifer. This theory allows us to calculate the travel times of tracers through the experimental aquifer, which are in agreement with the observations. The travel time distribution has an exponential tail, with a characteristic time that depends on the aspect ratio of the aquifer. This distribution depends essentially on the geometry of the groundwater flow, and is weakly sensitive to the hydrodynamic dispersion that occurs at the pore scale.

We investigate the flow of water in a two-dimensional laboratory aquifer recharged by artificial rainfall. As rainwater infiltrates, it forms a body of groundwater which can exit the aquifer only through one of its sides. The outlet, located high above the aquifer bottom, drives the flow upwards. Noting that the water table barely departs from the horizontal, we linearize the boundary condition at the free surface, and solve the flow equations in steady state. We find an approximate expression for the velocity potential, which accounts for the shape of the streamlines, and for the propagation of dye through the aquifer. Based on this theory, we calculate the travel time of water through the experiment, and find that its distribution decays exponentially, with a characteristic time that depends on the shape of the aquifer. We find that the hydrodynamic dispersion that occurs at the pore scale does not matter much for this distribution, which depends essentially on the geometry of the groundwater flow.

In a shallow channel, the flow transfers most of its momentum vertically. Based on this observation, one often neglects the momentum that is transferred across the stream – the core assumption of the shallow-water theory. In the context of viscous flows, this approximation is referred to as the ‘lubrication theory’, in which one assumes that the shear stress exerted by the fluid on the substrate over which it flows is proportional to its velocity. Here, we revise this theory to account for the momentum that viscosity transfers across a shallow laminar flow, while keeping the problem low-dimensional. We then test the revised lubrication theory against analytical and numerical solutions of the exact problem. We find that, at a low computational cost, the present theory represents the actual flow more accurately than the classical lubrication approximation. This theoretical improvement, devised with laboratory rivers in mind, should also apply to other geophysical contexts, such as ice flows or forming lava domes.

Understanding how rivers adjust to the sediment load they carry is critical to predicting the evolution of landscapes. Presently, however, no physically based model reliably captures the dependence of basic river properties, such as its shape or slope, on the discharge of sediment, even in the simple case of laboratory rivers. Here, we show how the balance between fluid stress and gravity acting on the sediment grains, along with cross-stream diffusion of sediment, determines the shape and sediment flux profile of laminar laboratory rivers that carry sediment as bedload. Using this model, which reliably reproduces the experiments without any tuning, we confirm the hypothesis, originally proposed by Parker [G. Parker, J. Fluid Mech. 89, 127-146 (1978)], that rivers are restricted to exist close to the threshold of sediment motion (within about 20%). This limit is set by the fluid-sediment interaction and is independent of the water and sediment load carried by the river. Thus, as the total sediment discharge increases, the intensity of sediment flux (sediment discharge per unit width) in a river saturates, and the river can transport more sediment only by widening. In this large discharge regime, the cross-stream diffusion of momentum in the flow permits sediment transport. Conversely, in the weak transport regime, the transported sediment concentrates around the river center without significantly altering the river shape. If this theory holds for natural rivers, the aspect ratio of a river could become a proxy for sediment discharge-a quantity notoriously difficult to measure in the field.

The coupling of sediment transport with the flow that drives it shapes the bed of alluvial rivers. The channel steers the flow, which in turns deforms the bed through erosion and sedimentation. To investigate this process, we produce a small river in a laboratory experiment by pouring a viscous fluid on a layer of plastic sediment. This laminar river gradually reaches its equilibrium shape. In the absence of sediment transport, the combination of gravity and flow-induced stress maintains the bed surface at the threshold of motion (Seizilles et al., 2013). If we impose a sediment discharge, the river widens and shallows to accommodate this input. Particle tracking reveals that the grains entrained by the flow behave as random walkers. Accordingly, they diffuse towards the less active areas of the bed (Seizilles et al., 2014). The river then adjusts its shape to maintain the balance between this diffusive flux, which pushes the grains towards the banks, and gravity, which pulls them towards the center of the channel. This dynamical equilibrium results in a peculiar Boltzmann distribution, in which the local sediment flux decreases exponentially with the elevation of the bed (Abramian et al., 2019). As the sediment discharge increases, the channel gets wider and shallower. Eventually, it destabilizes into multiple channels. A linear stability analysis suggests that it is diffusion that causes this instability, which could explain the formation of braided rivers (Abramian, Devauchelle, and Lajeunesse, 2019). References: * Abramian, A., Devauchelle, O., and Lajeunesse, E., “Streamwise streaks induced by bedload diffusion,” Journal of Fluid Mechanics 863, 601–619 (2019). * Abramian, A., Devauchelle, O., Seizilles, G., and Lajeunesse, E., “Boltzmann distribution of sediment transport,” Physical review letters 123, 014501 (2019). * Seizilles, G., Devauchelle, O., Lajeunesse, E., and M ́etivier, F., “Width of laminar laboratory rivers,” Phys. Rev. E. 87, 052204 (2013). * Seizilles, G., Lajeunesse, E., Devauchelle, O., and Bak, M., “Cross-stream diffusion in bedload transport,” Phys. of Fluids 26, 013302 (2014).

An alluvial river builds its own bed with the sediment it transports; its shape thus depends not only on its water discharge but also on the sediment supply. Here we investigate the influence of the latter in laboratory experiments. We find that, as their natural counterpart, laboratory rivers widen to accommodate an increase of sediment supply. By tracking individual particles as they travel downstream, we show that, at equilibrium, the river shapes its channel so that the intensity of sediment transport follows a Boltzmann distribution. This mechanism selects a well-defined width over which the river transports sediment, while the sediment remains virtually idle on its banks. For lack of a comprehensive theory, we represent this behavior with a single-parameter empirical model which accords with our observations.