Ratio Of Time Scales Research Articles

Numerical experiments on reaction front propagation in n-heptane/air mixture with temperature gradient

Usually different autoignition modes can be generated by a hot spot in which ignition occurs earlier than that in the surrounding mixture. However, for large hydrocarbon fuels with negative temperature coefficient (NTC) behavior, ignition happens earlier at lower temperature than that at higher temperature when the temperature is within the NTC regime. Consequently, a cool spot may also result in different autoignition modes. In this study, the modes of reaction front propagation caused by temperature gradient in a one dimensional planar configuration are investigated numerically for n-heptane/air mixture at initial temperature within and below the NTC regime. For the first time, different supersonic autoignition modes caused by a cool spot with positive temperature gradient are identified. It is found that the initial temperature gradient has strong impact on autoignition modes. With the increase of the positive temperature gradient of the cool spot, supersonic autoignitive deflagration, detonation, shock-detonation, and shock-deflagration are sequentially observed. It is found that shock compression of the mixture between the deflagration wave and leading shock wave produces an additional ignition kernel, which determines the autoignition modes. Furthermore, the cool spot is compared with the hot spot with temperature below the NTC regime. Similar autoignition modes are observed for the hot and cool spots. Different autoignition modes in the considered simplified configuration are summarized in terms of the normalized temperature gradient and acoustic-to-excitation time scale ratio. It is shown that the transition between different autoignition modes is not greatly affected by the NTC behavior. Therefore, our 1-D simulation indicates that like hot spot, the cool spot may also generate knock in engines when fuels with NTC behavior is used and the temperature is within the NTC regime.

MODELING ACTIVE GALACTIC NUCLEUS FEEDBACK IN COOL-CORE CLUSTERS: THE FORMATION OF COLD CLUMPS

We perform high-resolution (15-30 pc) adaptive mesh simulations to study the impact of momentum-driven AGN feedback in cool-core clusters, focusing in this paper on the formation of cold clumps. The feedback is jet-driven with an energy determined by the amount of cold gas within 500 pc of the SMBH. When the intra-cluster medium (ICM) in the core of the cluster becomes marginally stable to radiative cooling, with the thermal instability to the free-fall timescale ratio t_{TI}/t_{ff} < 3-10, cold clumps of gas start to form along the propagation direction of the AGN jets. By tracing the particles in the simulations, we find that these cold clumps originate from low entropy (but still hot) gas that is accelerated by the jet to outward radial velocities of a few hundred km/s. This gas is out of hydrostatic equilibrium and so can cool. The clumps then grow larger as they decelerate and fall towards the center of the cluster, eventually being accreted onto the super-massive black hole. The general morphology, spatial distribution and estimated H{\alpha} morphology of the clumps are in reasonable agreement with observations, although we do not fully replicate the filamentary morphology of the clumps seen in the observations, probably due to missing physics.

Modeling of Non-Equilibrium Homogeneous Turbulence in Rapidly Compressed Flows

The response of homogeneous turbulence to rapid mean-flow compression in idealized internal combustion engines and rapid compression machines is examined using a hierarchy of closure models for the Reynolds stress anisotropy. This hierarchy is based on a Reynolds stress anisotropy transport equation that is modeled from the exact transport equation for the anisotropy. The hierarchy of models includes a recently-developed non-equilibrium model, which is shown to be in good agreement with more computationally-complex fully differential models. Using this hierarchical approach, the flow physics addressed by each closure is identified and closure accuracy is shown to depend on the degree to which non-equilibrium turbulent flow effects are captured. We examine the evolution of the turbulence kinetic energy, Reynolds stresses, and anisotropy as a function of the degree of non-equilibrium in the flow, which is parameterized by the ratio of characteristic turbulence and mean-flow deformation time scales. By comparing model results to results obtained from rapid distortion theory and higher level closures, prescriptions are provided for the applicability of different closure models based on the magnitude of the non-equilibrium parameter. The theoretical analysis is complemented by comparisons of simulation results with previously established direct numerical simulations for a one-dimensional compression. Finally, we connect these prescriptions with experimental measurements of turbulent and mean-flow time scales for internal combustion engines operating at realistic conditions.

Do not aim too high nor too low: Moderate expectation-based group formation promotes public cooperation on networks

We propose a public goods game model wherein individuals can adaptively adjust their social ties. Individuals have expectation towards the groups they are involved in. They drop out of these groups which do not satisfy their expectation and explore some new ones in the evolutionary process. The time scale ratio of natural selection to interaction determines how fast interaction happens relatively to selection. The results show that cooperation is greatly enhanced when interaction happens quickly. Furthermore, we find that there exists an optimal interval of intermediate expectation level most favoring cooperation. Robustness of the results has been checked by means of an extended model, where individuals’ expectation is subject to evolution as well.

Parameterization of Mean Residence Times in Idealized Rectangular Dead Zones Representative of Natural Streams

Three-dimensional Reynolds averaged Navier-Stokes modeling, validated against experimental data, is used to parameterize the flow features and time scales in idealized rectangular cavities for a wide range of width-to-length ratios, 0.4≤W/L≤1.1, and Reynolds number based on the depth, 5,000≤RD≤20,300, representative of isolated dead zones in small natural streams. The flow features for this parameter range are similar to open cavity flows and consist of a mixing layer spanning the entire length of the dead zone together with a single main recirculation region. The Langmuir time scale (ratio of dead-zone volume to discharge) based on the assumption of a well-mixed dead zone is found to be a function of the mean rotation time scale (inverse of average rotation rate) within the dead zone, the momentum thickness of the upstream boundary layer, and the dead-zone width. The entrainment coefficient, used to relate the exchange velocity to the average free- stream velocity, is shown to be directly related to the upstream boundary layer momentum thickness nondimensionalized by the width of the dead zone. Using passive tracer to quantify the mean residence time showed that the dead zone can be characterized by two perfectly mixed regions including a core or secondary region around the center of the eddy and a surrounding primary region that interacts directly with the free-stream through the mixing layer. A two-region model is developed to obtain time scales associated with the primary and secondary regions within the dead zone using an optimization procedure based on the computational data. The time scale associated with the primary region is representative of the Langmuir time scale and is found to be a strong function of the aspect ratio W/L and the Reynolds number. The secondary region time scale represents the long-time asymptotic behavior of the tracer concentration and is found to be a strong function of the dead- zone geometric parameters only.

Stability boundaries for resonant migrating planet pairs

Convergent migration allows pairs of planet to become trapped into mean motion resonances. Once in resonance, the planets' eccentricities grow to an equilibrium value that depends on the ratio of migration time scale to the eccentricity damping timescale, $K=\tau_a/\tau_e$, with higher values of equilibrium eccentricity for lower values of $K$. For low equilibrium eccentricities, $e_{eq}\propto K^{-1/2}$. The stability of a planet pair depends on eccentricity so the system can become unstable before it reaches its equilibrium eccentricity. Using a resonant overlap criterion that takes into account the role of first and second order resonances and depends on eccentricity, we find a function $K_{min}(\mu_p, j)$ that defines the lowest value for $K$, as a function of the ratio of total planet mass to stellar mass ($\mu_p$) and the period ratio of the resonance defined as $P_1/P_2=j/(j+k)$, that allows two convergently migrating planets to remain stable in resonance at their equilibrium eccentricities. We scaled the functions $K_{min}$ for each resonance of the same order into a single function $K_c$. The function $K_{c}$ for planet pairs in first order resonances is linear with increasing planet mass and quadratic for pairs in second order resonances with a coefficient depending on the relative migration rate and strongly on the planet to planet mass ratio. The linear relation continues until the mass approaches a critical mass defined by the 2/7 resonance overlap instability law and $K_c \to \infty$. We compared our analytic boundary with an observed sample of resonant two planet systems. All but one of the first order resonant planet pair systems found by radial velocity measurements are well inside the stability region estimated by this model. We calculated $K_c$ for Kepler systems without well-constrained eccentricities and found only weak constraints on $K$.

The Estuarine Circulation

Recent research in estuaries challenges the long-standing paradigm of the gravitationally driven estuarine circulation. In estuaries with relatively strong tidal forcing and modest buoyancy forcing, the tidal variation in stratification leads to a tidal straining circulation driven by tidal variation in vertical mixing, with a magnitude that may significantly exceed the gravitational circulation. For weakly stratified estuaries, vertical and lateral advection are also important contributors to the tidally driven residual circulation. The apparent contradiction with the conventional paradigm is resolved when the estuarine parameter space is mapped with respect to a mixing parameter M that is based on the ratio of the tidal timescale to the vertical mixing timescale. Estuaries with high M values exhibit strong tidal nonlinearity, and those with small M values show conventional estuarine dynamics. Estuaries with intermediate mixing rates show marked transitions between these regimes at timescales of the spring-neap cycle.

Analytical models of heat conduction in fractured rocks

Discrete fracture network models routinely rely on analytical solutions to estimate heat transfer in fractured rocks. We develop analytical models for advective and conductive heat transfer in a fracture surrounded by an infinite matrix. These models account for longitudinal and transverse diffusion in the matrix, a two‐way coupling between heat transfer in the fracture and matrix, and an arbitrary configuration of heat sources. This is in contrast to the existing analytical solutions that restrict matrix conduction to the direction perpendicular to the fracture. We demonstrate that longitudinal thermal diffusivity in the matrix is a critical parameter that determines the impact of local heat sources on fluid temperature in the fracture. By neglecting longitudinal conduction in the matrix, the classical models significantly overestimate both fracture temperature and time‐to‐equilibrium. We also identify the fracture‐matrix Péclet number, defined as the ratio of advection timescale in the fracture to diffusion timescale in the matrix, as a key parameter that determines the efficiency of geothermal systems. Our analytical models provide an easy‐to‐use tool for parametric sensitivity analysis, benchmark studies, geothermal site evaluation, and parameter identification.

A Model for Spontaneous Imbibition as a Mechanism for Oil Recovery in Fractured Reservoirs

The flow of oil and water in naturally fractured reservoirs (NFR) can be highly complex and a simplified model is presented to illustrate some main features of this flow system. NFRs typically consist of low-permeable matrix rock containing a high-permeable fracture network. The effect of this network is that the advective flow bypasses the main portions of the reservoir where the oil is contained. Instead capillary forces and gravity forces are important for recovering the oil from these sections. We consider a linear fracture which is symmetrically surrounded by porous matrix. Advective flow occurs only along the fracture, while capillary driven flow occurs only along the axis of the matrix normal to the fracture. For a given set of relative permeability and capillary pressure curves, the behavior of the system is completely determined by the choice of two dimensionless parameters: (i) the ratio of time scales for advective flow in fracture to capillary flow in matrix $$\alpha =\tau ^f/\tau ^m$$ ; (ii) the ratio of pore volumes in matrix and fracture $$\beta =V^m/V^f$$ . A characteristic property of the flow in the coupled fracture–matrix medium is the linear recovery curve (before water breakthrough) which has been referred to as the “filling fracture” regime Rangel-German and Kovscek (J Pet Sci Eng 36:45–60, 2002), followed by a nonlinear period, referred to as the “instantly filled” regime, where the rate is approximately linear with the square root of time. We derive an analytical solution for the limiting case where the time scale $$\tau ^{m}$$ of the matrix imbibition becomes small relative to the time scale $$\tau ^{f}$$ of the fracture flow (i.e., $$\alpha \rightarrow \infty $$ ), and verify by numerical experiments that the model will converge to this limit as $$\alpha $$ becomes large. The model provides insight into the role played by parameters like saturation functions, injection rate, volume of fractures versus volume of matrix, different viscosity relations, and strength of capillary forces versus injection rate. Especially, a scaling number $$\omega $$ is suggested that seems to incorporate variations in these parameters. An interesting observation is that at $$\omega =1$$ there is little to gain in efficiency by reducing the injection rate. The model can be used as a tool for interpretation of laboratory experiments involving fracture–matrix flow as well as a tool for testing different transfer functions that have been suggested to use in reservoir simulators.

Lagrangian Intermittent Modelling of a Turbulent Lifted Methane-Air Jet Flame Stabilized in a Vitiated Air Coflow

The present manuscript reports results of numerical simulations of turbulent lifted jet flames of methane with special emphasis placed on the autoignition effects. The impact of dilution with burned gases on the flame stabilization is analyzed under the conditions of a laboratory jet flame surrounded by a vitiated air coflow. In this geometry, mass flow rates, temperature levels and exact chemical composition of hot products mixed with air are fully determined. The effects of both finite rate chemistry and partially premixed combustion are taken into account within a Lagrangian intermittent framework. Detailed chemistry effects are incorporated through the use of chemical time scales, which are tabulated as functions of the mixture fraction. The concept of residence time (or particle age) and the transport equation for the mean scalar dissipation rate of the mixture fraction fluctuations, i.e., \(\widetilde{\varepsilon}_Z=\overline{\rho \mathcal{D} (\partial Z'' / \partial x_i) (\partial Z'' / \partial x_i) }/\overline{\rho}\), are also considered. This allows to improve the description of turbulent mixing including both the large scales engulfment processes through the consideration of the residence time scale, as well as the small scales molecular mixing processes the intensity of which is set by the integral scalar (mixing) time scale \(\tau_Z=\widetilde{Z''^2}/\widetilde{\varepsilon}_Z\). Numerical simulations of the turbulent diluted jet flame of methane studied by Cabra and his co-workers are performed. The flame liftoff height is reasonably well-captured and the computational results display a fairly satisfactory level of agreement with experimental data. They also confirm that the consideration of an additional transport equation for the mean scalar dissipation rate may significantly improve the computational results. The investigation is supplemented by a sensitivity analysis of the scalar to turbulence time scale ratio Cmix = τZ/τt often referred to as the mixing constant. Finally, the manuscript ends with the application of the proposed closure to experimental conditions which are characterized by variations of (i) the jet exit velocity, (ii) coflow velocity and (ii) coflow temperature.

Origins of Activity Enhancement in Enzyme Cascades on Scaffolds

The concept of "metabolic channeling" as a result of rapid transfer of freely diffusing intermediate substrates between two enzymes on nanoscale scaffolds is examined using simulations and mathematical models. The increase in direct substrate transfer due to the proximity of the two enzymes provides an initial but temporary boost to the throughput of the cascade and loses importance as product molecules of enzyme 1 (substrate molecules of enzyme 2) accumulate in the surrounding container. The characteristic time scale at which this boost is significant is given by the ratio of container volume to the product of substrate diffusion constant and interenzyme distance and is on the order of milliseconds to seconds in some experimental systems. However, the attachment of a large number of enzyme pairs to a scaffold provides an increased number of local "targets", extending the characteristic time. If substrate molecules for enzyme 2 are sequestered by an alternative reaction in the container, a scaffold can result in a permanent boost to cascade throughput with a magnitude given by the ratio of the above-defined time scale to the lifetime of the substrate molecule in the container. Finally, a weak attractive interaction between substrate molecules and the scaffold creates a "virtual compartment" and substantially accelerates initial throughput. If intermediate substrates can diffuse freely, placing individual enzyme pairs on scaffolds is only beneficial in large cells, unconfined extracellular spaces or in systems with sequestering reactions.

A new approach to model turbulent lifted CH4/air flame issuing in a vitiated coflow using conditional moment closure coupled with an extinction model

In this article, conditional moment closure model (CMC) with detailed chemistry is used to model lifted turbulent methane flame in a high temperature and vitiated coflow and to predict flame lift-off height. The flow and mixing field are predicted by a 2D in-house code employing a k–ε turbulence model (RANS) with modified constant Cε2. The first-order CMC model on its own could not capture the behavior of the lifted flame. Large eddy simulations (LES) coupled with second-order CMC model would be a promising alternative but the objective here was to improve low-cost simulations based on RANS and first-order CMC to address realistic problems. Hence, an extinction model has been incorporated in the first-order CMC to improve its predictions and is referred in this paper as CMCE. In the CMCE model, flame is assumed to be extinguished when the ratio of flow time scale to the chemical time scale falls below a critical value. Predicted lift-off height by the CMCE model agrees very well with the experimental results. There is a significant improvement in temperature and species distributions in both axial and radial directions with the implementation of the CMCE model. Further, the model is extended to predict the flame lift-off height for various coflow temperatures and jet velocities by using scaling ratios. With these modifications, the lift-off heights predicted by the CMCE model match well with the experimental results for a wide range of jet velocities and coflow temperatures. Results from both CMC and CMCE models are compared against the experimental data to show the importance of the extinction model. Flame stabilization process indicates that flame stabilizes on the contour of mean stoichiometric mixture fraction where axial mean velocity equals the turbulent burning velocity.

Nonuniversal Power-Law Spectra in Turbulent Systems

Turbulence is generally associated with universal power-law spectra in scale ranges without significant drive or damping. Although many examples of turbulent systems do not exhibit such an inertial range, power-law spectra may still be observed. As a simple model for such situations, a modified version of the Kuramoto-Sivashinsky equation is studied. By means of semianalytical and numerical studies, one finds power laws with nonuniversal exponents in the spectral range for which the ratio of nonlinear and linear time scales is (roughly) scale independent.

Adaptive tag switching reinforces the coevolution of contingent cooperation and tag diversity

Most of the previous studies concerning the similarity-based interaction have assumed that the change of tags just happens in the imitation stage. Individuals actually can adjust their tags whenever the environments related to these tags grow nasty. We institute a spatial model to investigate the effect of the coevolution of tag and strategy on the evolution of cooperation in the context of the Prisoner's Dilemma game. Interactions just happen between tag-identical neighbors. Individuals exploited by defectors change their current tags at a certain cost. The time-scale ratio controls how fast interaction happens relatively to selection. Results show that whenever individuals have enough chance to adapt to the environment, cooperation is greatly improved even for quite large temptation to defect. Intensive exploration reveals that both little and large costs of tag switching can further favor the establishment of cooperation. Our work may add more into the literature concerning games on adaptive networks.

Small‐scale permeability heterogeneity has negligible effects on nutrient cycling in streambeds

Aquatic sediment hosts coupled pore water flow and biogeochemical reactions that mediate water quality across the fluvial corridor including adjacent alluvial aquifers. However, the effect of small‐scale variations in sediment hydraulic properties on nutrient transformation occurring within the sediment is poorly understood. We show through numerical flow and transport simulations, for two realistic heterogeneous permeability cases typical of lowland rivers and two idealized homogeneous ones, that there is little difference in the reactive transport fields, the bulk reaction rates, and nutrient sink/source function despite stark differences in flow fields. This is because the reactions are ultimately controlled by characteristic or bulk residence times that are similar for the heterogeneous and homogeneous cases. This is a promising result for predictive models based solely on relative ratios of residence times and reactive time scales.

A Novel Methodology for Chemical Time Scale Evaluation with Detailed Chemical Reaction Kinetics

Interaction between turbulent mixing and chemical kinetics is the key parameter which determines the combustion regime: only understanding such interaction may provide insight into the physics of the flame and support the choice and/or development of modeling tools. Turbulence-chemistry interaction may be evaluated through the analysis of the Damköhler number distribution, which represents the flow to chemical time-scale ratio. Large Damköhler values indicate mixing controlled flames. On the other hand, low Damköhler values corresponds to slow chemical reactions: reactants and products are quickly mixed by turbulence, so the system behaves like a perfect stirred reactor. The calculation of the Damköhler number requires the definition of proper flow and chemical time-scales. For turbulent conditions, various flow time-scales can be used, such as the integral and Kolmogorov time-scales. Chemical time-scale calculation poses some issues. In the literature, several examples of Damköhler calculation are reported, but in most cases a global chemical reaction rate is used to estimate the chemical time scale. A method for considering more complex kinetic schemes is proposed by Fox [Computational Models for Turbulent Reacting Flows; Cambridge University Press: Cambridge, U. K., 2003], who defines the chemical time-scale in terms of the inverse of the eigenvalues from the decomposition of the chemical source term Jacobian matrix. The present work aims at developing a procedure for the calculation of the chemical time-scale (and thus of the Damköhler number) with complex kinetics starting from the analysis of the Jacobian matrix of the chemical species source terms. Emphasis is given on the dimension of the Jacobian matrix, as it is not fully understood how the species for the time-scale calculation should be chosen. In other words, one can refer to the full set of species (thus all species will have the same “weight”) but also to a subset of them. The main concept is to perform a preliminary analysis, based on Principal Variables (PV), to determine the relative importance of the chemical species, in order to select an optimal subset for the chemical time-scale calculation. The procedure is illustrated and applied for the Moderate and Intense Low-oxygen Dilution (MILD) combustion, as this kind of regime shows a strong coupling between turbulence and chemistry, mainly because of slower reaction rates (due to the dilution of reactants) in comparison with conventional combustion. The methodology is further validated on a DNS data set modeling a CO/H2 jet flame.

A generalized Damköhler number for classifying material processing in hydrological systems

Abstract. Assessing the potential for transfer of pollutants and nutrients across catchments is of primary importance under changing land use and climate. Over the past decade the connectivity/disconnectivity dynamic of a catchment has been related to its potential to export material; however, we continue to use multiple definitions of connectivity, and most have focused strongly on physical (hydrological or hydraulic) connectivity. In contrast, this paper constantly focuses on the dynamic balance between transport and material transformation, and defines material connectivity as the effective transfer of material between elements of the hydrological cycle. The concept of exposure timescales is developed and used to define three distinct regimes: (i) which is hydrologically connected and transport is dominated by advection; (ii) which is hydrologically connected and transport is dominated by diffusion; and (iii) which is materially isolated. The ratio of exposure timescales to material processing timescales is presented as the non-dimensional number, NE, where NE is reaction-specific and allows estimation of relevant spatial scales over which the reactions of interest take place. Case studies within each regime provide examples of how NE can be used to characterise systems according to their sensitivity to shifts in hydrology and gain insight into the biogeochemical processes that are signficant under the specified conditions. Finally, we explore the implications of the NE framework for improved water management, and for our understanding of biodiversity, resilience and chemical competitiveness under specified conditions.

Nanoparticle agglomeration in an evaporating levitated droplet for different acoustic amplitudes

Radiatively heated levitated functional droplets with nanosilica suspensions exhibit three distinct stages namely pure evaporation, agglomeration, and finally structure formation. The temporal history of the droplet surface temperature shows two inflection points. One inflection point corresponds to a local maximum and demarcates the end of transient heating of the droplet and domination of vaporization. The second inflection point is a local minimum and indicates slowing down of the evaporation rate due to surface accumulation of nanoparticles. Morphology and final precipitation structures of levitated droplets are due to competing mechanisms of particle agglomeration, evaporation, and shape deformation. In this work, we provide a detailed analysis for each process and propose two important timescales for evaporation and agglomeration that determine the final diameter of the structure formed. It is seen that both agglomeration and evaporation timescales are similar functions of acoustic amplitude (sound pressure level), droplet size, viscosity, and density. However, we show that while the agglomeration timescale decreases with initial particle concentration, the evaporation timescale shows the opposite trend. The final normalized diameter can be shown to be dependent solely on the ratio of agglomeration to evaporation timescales for all concentrations and acoustic amplitudes. The structures also exhibit various aspect ratios (bowls, rings, spheroids) which depend on the ratio of the deformation timescale (tdef) and the agglomeration timescale (tg). For tdef&amp;lt;tg, a sharp peak in aspect ratio is seen at low concentrations of nanosilica which separates high aspect ratio structures like rings from the low aspect ratio structures like bowls and spheroids.

Energy dissipation in unsteady turbulent pipe flows caused by water hammer

Energy dissipation and turbulent kinetic energy production and its dissipation in unsteady turbulent pipe flows due to water hammer phenomena are numerically studied. For this purpose, the two-dimensional governing equations of water hammer are solved using the method of characteristics. A k–ω turbulence model which is accurate for two-dimensional boundary layers under adverse and favorable pressure gradients is applied. The numerical results are in good agreement with the experimental data. Through an order of magnitude analysis, two dimensionless parameters have been identified which can be used for the evaluation of viscous and turbulent shear stress terms. The influence of these non-dimensional parameters on pressure oscillations, wall-shear-stress, dissipation rate as well as profiles of velocity, turbulent production and dissipation are investigated. The non-dimensional parameter P, which represents time scale ratio of turbulence diffusion in the radial direction to the pressure wave speed, is used to study the structure and strength of turbulence. It is found that for the case of P≈1, for which the values of the non-dimensional groups are larger, the peaks of turbulence energy production and dissipation move rapidly away from the wall and turbulence structure is significantly changed. For the case of P≫1, for which the values of non-dimensional parameters are smaller, these variations are found to be small.

The root of branching river networks

Branching river networks are one of the most widespread and recognizable features of Earth's landscapes and have also been discovered elsewhere in the Solar System. But the mechanisms that create these patterns and control their spatial scales are poorly understood. Theories based on probability or optimality have proven useful, but do not explain how river networks develop over time through erosion and sediment transport. Here we show that branching at the uppermost reaches of river networks is rooted in two coupled instabilities: first, valleys widen at the expense of their smaller neighbours, and second, side slopes of the widening valleys become susceptible to channel incision. Each instability occurs at a critical ratio of the characteristic timescales for soil transport and channel incision. Measurements from two field sites demonstrate that our theory correctly predicts the size of the smallest valleys with tributaries. We also show that the dominant control on the scale of landscape dissection in these sites is the strength of channel incision, which correlates with aridity and rock weakness, rather than the strength of soil transport. These results imply that the fine-scale structure of branching river networks is an organized signature of erosional mechanics, not a consequence of random topology.