Cellular Models for River Networks

A cellular model introduced for the evolution of the fluvial landscape is revisited using extensive numerical and scaling analyses. The basic network shapes and their recurrence especially in the aggregation structure are then addressed. The roles of boundary and initial conditions are carefully analyzed as well as the key effect of quenched disorder embedded in random pinning of the landscape surface. It is found that the above features strongly affect the scaling behavior of key morphological quantities. In particular, we conclude that randomly pinned regions (whose structural disorder bears much physical meaning mimicking uneven landscape-forming rainfall events, geological diversity or heterogeneity in surficial properties like vegetation, soil cover or type) play a key role for the robust emergence of aggregation patterns bearing much resemblance to real river networks.Comment: 7 pages, revtex style, 14 figure

Particle-hole symmetry in a sandpile model

In a sandpile model addition of a hole is defined as the removal of a grain from the sandpile. We show that hole avalanches can be defined very similar to particle avalanches. A combined particle-hole sandpile model is then defined where particle avalanches are created with probability $p$ and hole avalanches are created with the probability $1-p$. It is observed that the system is critical with respect to either particle or hole avalanches for all values of $p$ except at the symmetric point of $p_c=1/2$. However at $p_c$ the fluctuating mass density is having non-trivial correlations characterized by $1/f$ type of power spectrum.Comment: Four pages, our figure

A model of fasciculation and sorting in mixed populations of axons

We extend a recently proposed model (Chaudhuri et al., EPL 87, 20003 (2009)) aiming to describe the formation of fascicles of axons during neural development. The growing axons are represented as paths of interacting directed random walkers in two spatial dimensions. To mimic turnover of axons, whole paths are removed and new walkers are injected with specified rates. In the simplest version of the model, we use strongly adhesive short-range inter-axon interactions that are identical for all pairs of axons. We generalize the model to adhesive interactions of finite strengths and to multiple types of axons with type-specific interactions. The dynamic steady state is characterized by the position-dependent distribution of fascicle sizes. With distance in the direction of axon growth, the mean fascicle size and emergent time scales grow monotonically, while the degree of sorting of fascicles by axon type has a maximum at a finite distance. To understand the emergence of slow time scales, we develop an analytical framework to analyze the interaction between neighboring fascicles.Comment: 19 pages, 13 figures; version accepted for publication in Phys Rev

An Analytical and Numerical Study of Optimal Channel Networks

We analyze the Optimal Channel Network model for river networks using both analytical and numerical approaches. This is a lattice model in which a functional describing the dissipated energy is introduced and minimized in order to find the optimal configurations. The fractal character of river networks is reflected in the power law behaviour of various quantities characterising the morphology of the basin. In the context of a finite size scaling Ansatz, the exponents describing the power law behaviour are calculated exactly and show mean field behaviour, except for two limiting values of a parameter characterizing the dissipated energy, for which the system belongs to different universality classes. Two modified versions of the model, incorporating quenched disorder are considered: the first simulates heterogeneities in the local properties of the soil, the second considers the effects of a non-uniform rainfall. In the region of mean field behaviour, the model is shown to be robust to both kinds of perturbations. In the two limiting cases the random rainfall is still irrelevant, whereas the heterogeneity in the soil properties leads to new universality classes. Results of a numerical analysis of the model are reported that confirm and complement the theoretical analysis of the global minimum. The statistics of the local minima are found to more strongly resemble observational data on real rivers.Comment: 27 pages, ps-file, 11 Postscript figure

Typical properties of optimal growth in the Von Neumann expanding model for large random economies

We calculate the optimal solutions of the fully heterogeneous Von Neumann expansion problem with $N$ processes and $P$ goods in the limit $N\to\infty$. This model provides an elementary description of the growth of a production economy in the long run. The system turns from a contracting to an expanding phase as $N$ increases beyond $P$. The solution is characterized by a universal behavior, independent of the parameters of the disorder statistics. Associating technological innovation to an increase of $N$, we find that while such an increase has a large positive impact on long term growth when $N\ll P$, its effect on technologically advanced economies ($N\gg P$) is very weak.Comment: 8 pages, 1 figur

The emergence of topographic steady state in a perpetually dynamic self-organized critical landscape

We conducted a series of four physical modeling experiments of mountain growth at differing rates of uplift and three distinct climates ranging from relatively wet to relatively dry. The spatial and temporal pattern of landscape behavior is characterized by ∼f−1 scaling in sediment discharge and power law scaling in the magnitude and frequency of ridge movement in all four experiments. We find that internally generated self-organized critical (SOC) processes generate dynamically stable catchment geometries after ∼1 relief depths of erosion: these regularly spaced catchments have an average outlet-spacing ratio of 2.16, well within the range of values reported in field studies. Once formed, large catchment bounding ridges oscillate about a critically balanced mean location, with occasional large-scale changes in catchment size. Ridge movement appears to be driven by the competition for discharge as landslides push ridges back and forth. These dynamics lead to the emergence of a complex twofold scaling in catchment dynamics that is fully established by 1.8 relief depths of erosion; at this stage, a clear threshold has emerged separating two distinct scaling regimes, where large ridge mobility is insensitive to relief and small ridge mobility is relief dependent. Overall, we demonstrate that the development of dynamically stable large-scale landforms is related to the emergence of a complex-system hierarchy in topographic dynamics. Once formed, these landscapes do not evolve; statistical properties such as average topography and discharge become stationary while topography remains highly dynamic at smaller length scales

Network Structures from Selection Principles

We present an analysis of the topologies of a class of networks which are optimal in terms of the requirements of having as short a route as possible between any two nodes while yet keeping the congestion in the network as low as possible. Strikingly, we find a variety of distinct topologies and novel phase transitions between them on varying the number of links per node. Our results suggest that the emergence of the topologies observed in nature may arise both from growth mechanisms and the interplay of dynamical mechanisms with a selection process.Comment: 4 pages, 5 figure

Contextualizing Wetlands Within a River Network to Assess Nitrate Removal and Inform Watershed Management

Aquatic nitrate removal depends on interactions throughout an interconnected network of lakes, wetlands, and river channels. Herein, we present a network‐based model that quantifies nitrate‐nitrogen and organic carbon concentrations through a wetland‐river network and estimates nitrate export from the watershed. This model dynamically accounts for multiple competing limitations on nitrate removal, explicitly incorporates wetlands in the network, and captures hierarchical network effects and spatial interactions. We apply the model to the Le Sueur Basin, a data‐rich 2,880 km2 agricultural landscape in southern Minnesota and validate the model using synoptic field measurements during June for years 2013–2015. Using the model, we show that the overall limits to nitrate removal rate via denitrification shift between nitrate concentration, organic carbon availability, and residence time depending on discharge, characteristics of the waterbody, and location in the network. Our model results show that the spatial context of wetland restorations is an important but often overlooked factor because nonlinearities in the system, e.g., deriving from switching of resource limitation on denitrification rate, can lead to unexpected changes in downstream biogeochemistry. Our results demonstrate that reduction of watershed‐scale nitrate concentrations and downstream loads in the Le Sueur Basin can be most effectively achieved by increasing water residence time (by slowing the flow) rather than by increasing organic carbon concentrations (which may limit denitrification). This framework can be used toward assessing where and how to restore wetlands for reducing nitrate concentrations and loads from agricultural watersheds.This research was funded by NSF grant EAR-1209402 under the Water Sustainability and Climate Program (WSC): REACH (REsilience under Accelerated CHange)NSF grant EAR-1242458 under Science Across Virtual Institutes (SAVI): LIFE (Linked Institutions for Future EarthA.T.H. acknowledges support provided by NSF grant EAR- 1415206 under the Science, Engineering and Education for Sustainability (SEES

Two dimensional modulational instability in photorefractive media

We study theoretically and experimentally the modulational instability of broad optical beams in photorefractive nonlinear media. We demonstrate the impact of the anisotropy of the nonlinearity on the growth rate of periodic perturbations. Our findings are confirmed by experimental measurements in a strontium barium niobate photorefractive crystal.Comment: 8 figure

Observation of dipole-mode vector solitons

We report on the first experimental observation of a novel type of optical vector soliton, a {\em dipole-mode soliton}, recently predicted theoretically. We show that these vector solitons can be generated in a photorefractive medium employing two different processes: a phase imprinting, and a symmetry-breaking instability of a vortex-mode vector soliton. The experimental results display remarkable agreement with the theory, and confirm the robust nature of these radially asymmetric two-component solitary waves.Comment: 4 pages, 8 figures; pictures in the PRL version are better qualit