Hansjörg Seybold

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.

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.

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.

Climate's watermark in the geometry of stream networks

Branching stream networks are a ubiquitous feature of the Earths surface, but the processes that shape them, and their dependence on the climate in which they grow, remain poorly understood. Research has mainly focused on climatic controls of channel incision rates, while the climatic influence on planform geometry has often been overlooked. Here we analyze nearly one million digitally mapped river junctions throughout the contiguous United States and show that branching angles vary systematically with climatic aridity. In arid landscapes, which are thought to be dominated by surface runoff erosion, junction angles average roughly 45° in the driest places. Branching angles are systematically wider in humid regions, averaging roughly 72°, which is the theoretically predicted angle for network growth in a diffusive field such as groundwater seepage. The correlation of mean junction angle with aridity is stronger than with topographic gradient, downstream concavity, or other geometric factors that have been proposed as controls of junction angles. Thus, it may be possible to identify channelization processes from stream network geometry in relict landscapes, such as those on Mars.

Mixture Ensembles for Data Assimilation in Dynamic Data-driven Environmental Systems

Abstract Many inference problems in environmental DDDAS must contend with high dimensional models and non-Gaussian uncertainties, including but not limited to Data Assimilation, Targeting and Planning. In this this paper, we present the Mixture Ensemble Filter (MEnF) which extends ensemble filtering to non-Gaussian inference using Gaussian mixtures. In contrast to the state of the art, MEnF embodies an exact update equation that neither requires explicit calculation of mixture element moments nor ad-hoc association rules between ensemble members and mixture elements. MEnF is applied to the chaotic Lorenz-63 model and to a chaotic soliton model that allows idealized and systematic studies of localized phenomena. In both cases, MEnF outperforms contemporary approaches, and replaces ad-hoc Gaussian Mixture approaches for non-Gaussian inference.

Symmetric rearrangement of groundwater-fed streams

Streams shape landscapes through headward growth and lateral migration. When these streams are primarily fed by groundwater, recent work suggests that their tips advance to maximize the symmetry of the local Laplacian field associated with groundwater flow. We explore the extent to which such forcing is responsible for the lateral migration of streams by studying two features of groundwater-fed streams in Bristol, Florida: their confluence angle near junctions and their curvature. First, we find that, while streams asymptotically form a 72° angle near their tips, they simultaneously exhibit a wide 120° confluence angle within approximately 10 m of their junctions. We show that this wide angle maximizes the symmetry of the groundwater field near the junction. Second, we argue that streams migrate laterally within valleys and present a new spectral analysis method to relate planform curvature to the surrounding groundwater field. Our results suggest that streams migrate laterally in response to fluxes from the surrounding groundwater table, providing evidence of a new mechanism that complements Laplacian growth at their tips.

Shapes of river networks

River network scaling laws describe how their shape varies with their size. However, the regional variation of this size-dependence remains poorly understood. Here we show that river network scaling laws vary systematically with the climatic aridity index. We find that arid basins do not change their proportions with size, while humid basins do. To explore why, we study an aspect ratio L⊥/L∥ between basin width L⊥ and basin length L∥. We find that the aspect ratio exhibits a dependence on climate and argue that this can be understood as a structural consequence of the confluence angle. We then find that, in humid basins, the aspect ratio decreases with basin size, which we attribute to a common hydrogeological hierarchy. Our results offer an explanation of the variability in network scaling exponents and suggest that the absence of self-similarity in humid basins can be understood as a morphological expression of subsurface processes.

Wetting controls of droplet formation in step emulsification

Significance Step emulsification is one of the few methods that enables upscaled production of droplets with a monodispersity desired for many industrial processes. However, the physical mechanisms of the droplet formation process are poorly understood. We study the droplet breakup by experiments and simulations and find a simple theory predicting the transition from dripping to jetting and the minimal contact angle of α = 2π/3 = 120° for which droplets form. These results have important consequences for the development and design of novel microfluidic systems and reactors that address the growing demand for tools to manipulate fluids at the submillimeter scale. The formation of droplets is ubiquitous in many natural and industrial processes and has reached an unprecedented level of control with the emergence of milli- and microfluidics. Although important insight into the mechanisms of droplet formation has been gained over the past decades, a sound understanding of the physics underlying this phenomenon and the effect of the fluid’s flow and wetting properties on the droplet size and production rate is still missing, especially for the widely applied method of step emulsification. In this work, we elucidate the physical controls of microdroplet formation in step emulsification by using the wetting of fluidic channels as a tunable parameter to explore a broad set of emulsification conditions. With the help of high-speed measurements, we unequivocally show that the final droplet pinch-off is triggered by a Rayleigh–Plateau-type instability. The droplet size, however, is not determined by the Rayleigh–Plateau breakup, but by the initial wetting regime, where the fluid’s contact angle plays a crucial role. We develop a physical theory for the wetting process, which closely describes our experimental measurements without invoking any free fit parameter. Our theory predicts the initiation of the Rayleigh–Plateau breakup and the transition from dripping to jetting as a function of the fluid’s contact angle. Additionally, the theory solves the conundrum why there is a minimal contact angle of α = 2π/3 = 120° for which droplets can form.

A free-boundary model of diffusive valley growth: theory and observation

Valleys that form around a stream head often develop characteristic finger-like elevation contours. We study the processes involved in the formation of these valleys and introduce a theoretical model that indicates how shape may inform the underlying processes. We consider valley growth as the advance of a moving boundary travelling forward purely through linearly diffusive erosion, and we obtain a solution for the valley shape in three dimensions. Our solution compares well to the shape of slowly growing groundwater-fed valleys found in Bristol, Florida. Our results identify a new feature in the formation of groundwater-fed valleys: a spatially variable diffusivity that can be modelled by a fixed-height moving boundary.

Ensemble Learning in Non-Gaussian Data Assimilation

The demand for tractable non-Gaussian Bayesian estimation has increased the popularity of kernel and mixture density representations. Here, using Gaussian Mixture Models (GMM), we posit that the reduction of total variance also remains an important objective in non-linear filtering, particularly in the presence of bias. We propose multi-objective estimation as an essential ingredient in data assimilation.