Michael Robinson, a doctoral student in the Graduate Program of Hydrologic Sciences, was working on his dissertation, but he needed a better tool to get the job done. The tool he was looking for was an accurate model for river tributary length with respect to where the tributary lies on a river. The problem: the tool didn’t exist.
Robinson and his dissertation advisor, 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 doesn’t disagree with the old models,” Robinson said. “We’ve just given a new way to think about river network geometry.”
The project arose when Robinson was attempting to model drainage basin divides, or ridgelines, but the tools available were models developed in the mid-20th century. The models did not allow Robinson to model ridgelines, so he sought out, along with support from Scheingross, to generate a new model.
Robinson said the previous studies that proposed ways to estimate tributary length and spacing were important to advancing the field.
“But the assumptions needed to predict ridgelines from them didn’t feel very satisfactory to me,” he said.
The researchers wanted to identify whether there is some signal that can be used as a proxy for the length of tributaries as a function of their location in the watershed. To find that signal, Robinson needed to identify rivers in “happy” or “steady-state” mountain ranges.
Mountain ranges uplift due to tectonic activity and get torn down by erosion. In a steady-state system, the mountain range is averaging little to no growth on geologic time scales (thousands to millions of years) because the tectonic uplift is perfectly balanced by the erosion. Steady-state regions are much better to study hydrogeologic systems in because the rivers aren’t actively changing much.
Robinson mapped rivers all over the world and using a high-powered computer started to plot how tributary length and spacing are a function of its location along the main river. He came away from the maps with a function that shows in steady-state systems, tributaries generally form a teardrop shape around the main river. Starting at the source, or headwaters, of a river, the tributaries tend to be short and spaced closer together, but then lengthen and become spaced further apart, before starting to taper off in length and become closer spaced near the river outlet.
Robinson worried about whether the model he had proposed was a statistical inevitability, so he tested the model on rivers that had disruptive histories.
“We got a different signal, which was nice because we wanted to show that this can break,” Robinson said. “It only works on ‘happy’ rivers.”
River systems vary widely and need widely varied models to accurately describe them, but for the common “happy” watershed, Robinson found a strong model with mechanistic backing.
“We are giving people the ability to say, ‘How are tributary length and spacing functions of its location?’” Robinson said.
“This work that Mike led represents a basic, yet fundamental advancement to our knowledge on river networks,” Scheingross said. “It’s shocking to me that despite almost a century of work on river networks, no one has been able to explain what sets the length of river tributaries and the spacing between tributaries. Mike’s contribution may seem a bit abstract, but the finding is actually quite significant. Being able 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 have implications for aquatic habitat, water quality, natural hazards and more.”
Robinson said the interdisciplinary potential for 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, like vascular systems in leaves or veins in bodies. If the model is found to only be pertinent to rivers, he hopes the model can be used to determine whether branching patterns on other planets, for example, can be ascribed to moving liquid.
