# Using Math To Better Understand River, Valley Networks

Lee Rannals for redOrbit.com — Your Universe Online

Researchers wrote in a paper published in the Proceedings of the National Academy of Sciences (PNAS) that they have used math to explain different characteristics of river and valley networks.

Rivers and valleys form intricate branching patterns, which have inspired some scientists to develop a theoretical understanding of river-network geometry.

MIT scientists have created a mathematical theory to discover a common angle at which valleys branch off. The theory predicts that rivers branch at an angle of 72 degrees.

To put the theory to test, the team measured 5,000 branching angles in the Florida Panhandle, and found that branching was indeed 72 degrees.

Dan Rothman, a professor of geophysics in MIT´s Department of Earth, Atmospheric and Planetary Sciences (EAPS), said the mathematical analysis may be applicable to other systems, like neuron dendrites and fungal filaments.

Another team of scientists published a report in the journal Nature that looked at another mathematical model of river networks. This model identifies a tipping point at which rivers branch.

The river may give rise to a dense network of tributaries, depending on a river's capacity to erode a landscape.

“We use mathematics to speed up time and help us understand how these systems evolve,” Taylor Perron, leader of the team with the paper published in Nature, said in a statement. “If you could speed up the clock, you would see that the landscape is a lot more dynamic.”

Rothman and his team looked to near the town of Bristol for their research. This area hosts a network of valleys cut into the landscape, and from an aerial view, you can see midsize branches running with water. Over time, the tips of the smallest valleys branch to create a denser valley network.

The team looked to the mechanics of groundwater flow to try and gain a better understanding of how these valleys branch. Groundwater flows under the surface, through material like porous sand. In the Florida Panhandle, groundwater may act to cut into a network of valleys.

Groundwater stored in the hills surrounding a valley slowly seeps out, and over time, the process slowly erodes the surrounding hills, which ultimately extends a valley and splits it in two.

Rothman and colleagues derived a mathematical expression to find the angle at which the split occurs. These paths generally curve either away from each other, when the angle between the stream is small, or toward each other, when the angle is large.

“What we show is that because of the properties of groundwater flow, one can understand something about the organization of this pattern,” Rothman said. “It opens a world into a really interesting geometry.”

Perron and his team examined the formation of river networks over land. They sought to find what governs the branching pattern that takes place.

The researchers of this study had to develop a simple mathematical model that represents the erosional mechanisms that act on a river network. With their model, the shape a river network takes is governed by a tug of war between two forces. The first force is the strength of river incision, or how quickly a river erodes. The second is the strength of soil creep, or how quickly soil from surrounding hills fills in a river valley.

The team found that as they turned up river incision, a river basin with a single river channel morphs into a network of branching channels at a very specific tipping point. This tipping point explains why larger rivers develop a network of tributaries.

Connecting the tipping point to specific erosional mechanisms allowed Perron and his team to understand why river basins in landscapes grow tributaries at different scales. They predicted that river basins should branch at a smaller size in environments where river incision is strong, or where soil creep is weak.

“Understanding how river networks originate and evolve is key to understanding how landscapes have evolved in the past, and how they will evolve in the future,” said Mikael Attal, a lecturer in landscape dynamics at the University of Edinburgh who was not a part of the research. “What is fascinating about these two papers is that they provide a physical explanation for the geometry of river networks using some very simple concepts. Studies such as these will help better parameterize models and help make more accurate predictions of what may happen in the future.”