Moment of Science: Mathematical law of rivers applies in both headwaters and river deltas

It’s been said that time is an ever-flowing river, yet how those rivers evolve over time depends on an assortment of things: How deep and long are the channels? What kind of sediments and rocks do they flow over and carry downstream? Where do they really start and end?

Doctors Tian Dong and Hongbo Ma’s works have focused on everything from sediment deposits to flash flood predictions, and they say the heads of most rivers conform to a rule developed in the late 1950s.

“We’re always curious about the shape and form of landscape, right?” Dr. Dong offers. “Over the years, scientists started to see this relationship between channel length -- how long the river channel in this drainage basin is -- to how big the basin is.”

Geomorphologist John T. Hack came up with a formula to describe that relationship between the length of the longest channel in a system (L), and the drainage area (A) -- and it’s more simple than you might expect:

L = 1.4 A^0.6

“You can easily calculate the area of the mountain,” Dr. Ma points out. “You can measure it on the map so you know the drainage area... [and] the channelized area in the mountain essentially provides the highway for the rainfall.”

Dr. Ma explains that 0.6 exponent is a delicate one. If it were 0.5, it basically means the whole thing just scales with the land as it fans out... but the closer to 1 it gets, the longer the basin gets as basically one single dominant channel.

Dr. Dong likes to think of those tributaries as the roots of a tree -- and the distributaries in a similar vein. “There’s this dominant topographic gradient, so the river kind of has to follow this broad direction; they don’t have room to migrate, or venture out laterally across the mountains. When the river meets a lake or ocean, it kind of naturally slows down, depositing its water, sediment, and nutrients... and starts building this network of [what] look like tree branches.”

The flowing braids of river deltas look impressive enough from the ground, let alone satellite data... and Hack developed his law just before the satellite era. It relies heavily on topography, starting in higher areas like mountain ranges and moving to lower... but where that flowing river ends, the dynamics are drastically different. The team replaced the “A” in Hack’s Law with a different type of area: “We call this a nourishment area, or land-building area,” says Dr. Dong. “Instead of collecting rainwater, now the ‘highway’ of delta is spreading out sediment.”

With a good amount of graph theory by their side, the team mapped nearly 6000 nourishment areas across 30 different deltas, noting where channels split off like tree branches, and what boundaries they created: “When you plot the land-building area to the length of the delta channel that builds this land, it shows the exact same exponent.”

L = 1.43 A^0.6

“When Tian showed the results to me,” Dr. Ma recalls, “I was completely shocked to see that.”

Even though the processes at the start and finish of the rivers are different, the team emerged with nearly the exact same formula as John Hack did nearly 70 years ago. It turns out, however, that not every delta obeys this law. Uniform ones with higher ground surrounding them are somewhat more contained, whereas composite deltas have more than one area-to-channel scaling law, and things get more complicated.

Dr. Ma says human engineering was designed to protect the people near coastal regions, but now it’s cutting off the sediment supply in certain deltas. That’s where their research could help. “How much land could a channel, potentially an engineered channel, build throughout time?” Dr. Ma asks. “We can figure out a pretty simple and straightforward way to predict [the outcome of] each of these restoration projects, to recover some of the wetland on the delta.”

For his part, Dr. Dong is excited to see if their discovery could “branch out” to other fields: “Blood vessels going from arteries, to capillaries in our kidney, or neurons in your brain... there’s tidal networks, and ice streams, there are also channel networks under the ocean. Is there something universal here governing these transport networks in nature? That’s kind of something that inspires us to keep working on this.”

This research was highlighted as a cover story for Science Magazine in April 2026 -- a first for a lead author from UT Rio Grande Valley (Dr. Tian Dong).

Latest Local News | First Alert Weather | Crime | National | 13abc Originals