Streams, stromatolites and the geometry of growth

Abstract

This collection of papers is about recognizing common geometric features in the dynamics shaping diverse phenomena in the natural world. In particular, we focus on two systems which grow in response to a diffusive flux. The first system is a microbial mat which overlays a layer of precipitated mineral. The microbial mat grows in response to the diffusion of nutrients while the mineral layer grows in response to the precipitation of dissolved ions which diffuse through the microbial mat. The second system is a network of streams that are fed by groundwater. In this case, groundwater flows through the aquifer and into the streams along the gradient of the pressure field, which, at equilibrium, diffuses through the aquifer. Here we show how a quantitative understanding of the shapes and scales of these two systems can be gained from physical and mathematical reasoning with few assumptions. We begin by considering the physical dimensions of systems shaped by diffusion. Guided by field observation and laboratory experiments of microbial mats, we identify two time scales important to the growth of these mats. We show how these processes shape the mat over different length scales and how these length scales are recognizable in the geometry of the mat. Next, we consider the shape of an interface growing in response to a diffusive flux. In microbial mats and streams, resources are focused toward regions of high curvature. We find that curvature-driven growth accurately predicts the shape of both fossilized microbial mats called stromatolites and the the landscape around a spring. Finally, we consider the geometric forms that arise when competition is mediated by diffusion. In particular, we show that when a growing stream bifurcates, competition between the nascent streams cause them to grow apart at an equilibrium angle of [alpha] = 2[pi]/5. The measured bifurcation angles of streams in a kilometer-scale network are in close agreement with this prediction.