Michael Robinson, a doctoral student in the graduate program in hydrology, was working on his dissertation but needed a better tool. The tool he was looking for was an accurate model of the length of tributaries relative to their location in the river. The problem: the tool didn’t exist.
Robinson and his dissertation supervisor, Associate Professor Joel Scheingross, recently published a paper in the journal The Proceedings of the National Academy of Sciences proposing a new model for tributary length.
“Our work does not contradict the old models in any way,” Robinson said. “We have simply created a new way to think about the geometry of river networks.”
The project came about when Robinson was trying to model watersheds or ridges, but the tools available were models developed in the mid-20th century. The models did not allow Robinson to model ridges, so he tried to develop a new model with help from Scheingross.
Robinson said previous studies that proposed methods for estimating the length and distance of tributaries were important for advancing the field.
“But the assumptions needed to predict ridges from this were not very satisfactory to me,” he said.
The researchers wanted to find out if there was a signal that could be used as an indicator of the length of tributaries depending on their location in the catchment. To find this signal, Robinson had to identify rivers in “happy” or “stable” mountain ranges.
Mountain ranges rise due to tectonic activity and are eroded by erosion. In a stationary system, the mountain range grows little or not at all on average over geological time (thousands to millions of years) because tectonic uplift is perfectly balanced by erosion. Stationary regions are much better suited to studying hydrogeological systems because the rivers are not actively changing much.
Robinson mapped rivers around the world and began using a high-performance computer to determine how the length and spacing of tributaries depended on their location along the main river. He created a function from the maps that shows that tributaries in stationary systems generally form a teardrop shape around the main river. Starting at the source of a river, tributaries tend to be short and closer together, but then become longer and farther apart before decreasing in length and becoming closer together near the river’s mouth.
Robinson was concerned about whether the model he proposed was a statistical inevitability, so he tested the model on rivers that had a history of disturbances.
“We got a different signal, which was nice because we wanted to show that it can break,” Robinson said. “It only works on ‘happy’ rivers.”
River systems are very different and require very different models to describe them accurately. However, for the typical “happy” catchment, Robinson has found a strong model with a mechanistic basis.
“We’re giving people the opportunity to say, ‘What is the function of the length and spacing of the tributaries in relation to their location?'” Robinson said.
“This work led by Mike represents a fundamental advance in our knowledge of river networks,” Scheingross said. “It shocks me that despite nearly a century of work on river networks, no one has been able to explain what determines the length of river tributaries and the spacing between tributaries. Mike’s contribution may seem somewhat abstract, but the discovery is actually quite significant. The ability to describe the structure of river tributary networks paves the way for us to predict how water, sediment and nutrients move through rivers, which can impact aquatic habitat, water quality, natural hazards and more.”
Robinson said the interdisciplinary potential of the tributary model is exciting. He is curious to see if a mathematician or other expert in networks can find similar patterns in other systems, such as the vascular systems of leaves or veins of bodies. If the model turns out to be relevant only to rivers, he hopes it can be used to determine whether branching patterns on other planets are due to moving fluids, for example.
