We now have the math to describe ‘matrix tides’ and other complex wave patterns seen in Qiantang River

BUFFALO, N.Y. — Last year, onlookers observed a startling site on China’s Qiantang River: waves forming a grid-like pattern.

Dubbed the “matrix tide,” this complex wave pattern was caused by the river’s famed tidal bores that surge upstream against the current. Specifically, two shockwave-like tidal bores, known as undular bores, that spread along two different directions like ripples on a pond and collided with each other.

This phenomenon is so complex that mathematicians don’t have the solutions to quantitatively describe it — that is, they didn’t until now.

New research by University at Buffalo and University of Colorado Boulder researchers has uncovered and characterized novel two-dimensional wave patterns — waves that propagate along two directions — whether they are in water or other settings like plasmas and condensed matter. 

Until now, longstanding governing equations could only be solved for one-dimensional cases where an undular bore propagates along a single direction, but this new study, published Aug. 5 in Physical Review Letters, uses numerical simulations to obtain solutions in two dimensions, describing what gives rise to patterns like the matrix tide.

“The equations become much more difficult to solve for two-dimensional waves and are very computationally intensive. That’s one of the reasons why this study hadn’t been done until now,” says Gino Biondini, PhD, professor of mathematics in the UB College of Arts and Sciences.

Biondini authored the study with former UB doctoral student Alexander Bivolcic, who is now an assistant professor at Embry-Riddle Aeronautical University, and Mark Hoefer, PhD, professor and chair of the Department of Applied Mathematics at the University of Colorado Boulder. 

The work was supported by the National Science Foundation.

Supercomputers crack the code

Also known as dispersive shock waves, undular bores consist of oscillations that propagate and spread. 

“The waves in an undular bore can be large and persist for a long time, making them a challenge to study mathematically,” Hoefer says. “But they don’t just attract mathematicians and physicists, surfers can ride a river bore for miles.”

In the 1960s, mathematician Gerald B. Whitham developed a mathematical framework for describing undular bores, but the equations could only handle cases where the wave traveled in a single direction — for example, down a narrow channel.

In the 1970s, Boris Kadomtsev and Vladamir Petviashvili derived an equation that provides a starting point for describing weakly two-dimensional undular bores, but the approach still had limitations, as the equation was too difficult to solve except in simple situations.

Biondini and his collaborators have been working to leverage today’s high-performance computing to solve these equations in two dimensions and accurately model an undular bore propagating along two different directions.

“This is the same process, in principle, that meteorologists use for weather prediction,” Biondini says. “They have equations that govern the weather and input measurements like temperature and pressure, but computers are needed to evolve the state numerically and get an approximate solution.”

The team relied on the supercomputing power at UB’s Center for Computational Research (CCR).

“If you ran one simulation of a wave on a laptop, it would take 22 hours. Using graphical processing units, or GPUs, we were able to reduce it down to about an hour per simulation,” Biondini says.

Some of the outcomes of the resulting simulations looked very similar to the matrix tide pattern observed in the Qiantang River in September of last year. The next step will be recreating the phenomena, whether in a water tank or in different physical systems, in order to experimentally validate the team’s predictions.

“For me, being a physicist turned mathematician, one of the biggest satisfactions is to use mathematics to describe something that happens in the real world,” Biondini says.

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