Symmetric rearrangement of groundwater-fed streams

Through history to growth dynamics: deciphering the evolution of spatial networks

Many ramified, network-like patterns in nature, such as river networks or blood vessels, form as a result of unstable growth of moving boundaries in an external diffusive field. Here, we pose the inverse problem for the network growth-can the growth dynamics be inferred from the analysis of the final pattern? We show that by evolving the network backward in time one can not only reconstruct the growth rules but also get an insight into the conditions under which branch splitting occurs. Determining the growth rules from a single snapshot in time is particularly important for growth processes so slow that they cannot be directly observed, such as growth of river networks and deltas or cave passages. We apply this approach to analyze the growth of a real river network in Vermont, USA. We determine its growth rule and argue that branch splitting events are triggered by an increase in the tip growth velocity.

Geodesic Loewner paths with varying boundary conditions

Equations of the Loewner class subject to non-constant boundary conditions along the real axis are formulated and solved giving the geodesic paths of slits growing in the upper half complex plane. The problem is motivated by Laplacian growth in which the slits represent thin fingers growing in a diffusion field. A single finger follows a curved path determined by the forcing function appearing in Loewner's equation. This function is found by solving an ordinary differential equation whose terms depend on curvature properties of the streamlines of the diffusive field in the conformally mapped 'mathematical' plane. The effect of boundary conditions specifying either piecewise constant values of the field variable along the real axis, or a dipole placed on the real axis, reveal a range of behaviours for the growing slit. These include regions along the real axis from which no slit growth is possible, regions where paths grow to infinity, or regions where paths curve back toward the real axis terminating in finite time. Symmetric pairs of paths subject to the piecewise constant boundary condition along the real axis are also computed, demonstrating that paths which grow to infinity evolve asymptotically toward an angle of bifurcation of π/5.

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.