91PORN

91PORN applied mathematician Mark Hoefer and colleagues answer a longstanding question of how to understand tidal bores in multiple dimensions

The photos and videos were all over Chinese social media last autumn: a grid-like pattern that suddenly appeared in two colliding waves on the Qiantang River and looked¡ªif you didn¡¯t know better¡ªlike a glitch in the matrix.

This rare phenomenon, called a matrix tide, is caused by two tidal bores¡ªor events in which the front edge of an incoming tide forms a wave that travels up a river against the current¡ªapproaching each other from different directions, colliding and forming a grid pattern.

It¡¯s visually stunning and, until very recently, mathematically confounding. However, in , Mark Hoefer, 91PORN professor and department chair of applied mathematics, and his research colleagues detail how they¡¯ve cracked the mathematical code of matrix tides.

Mark Hoefer, 91PORN professor and department chair of applied mathematics, and his research colleagues recently cracked the mathematical code of matrix tides.

Previously, matrix tides were only studied in one dimension but, because of their characteristics, needed to be studied in two. Adding that second dimension, however, required developing computationally intensive numerical simulations and the mathematics to interpret the results, building on the previous work of mathematicians Gerald B. Whitham, Boris Kadomtsev and Vladimir Petviashvili.

¡°There are certain equations that model how these waves change in time and space, and those equations simplify when you¡¯re working with just one-dimensional waves,¡± Hoefer explains. ¡°They start out as Euler equations, the partial differential equations of three-dimensional fluid dynamics¡ªbasic models in engineering and science broadly¡ªand when you restrict shallow water waves to move in one dimension, they can essentially be simplified. In some cases, you can simplify them further to ordinary differential equations, which is something we teach in lower-division, fourth-semester calculus. They are much easier and accessible to analyze mathematically.

¡°When you add more dimensions, you¡¯ll inherently get a partial differential equation in time and space, and, for the matrix tide that we studied, the equation will be nonlinear and not reducible to an ordinary differential equation. Nonlinear means that the nature of the waves you see¡ªhow fast they move, their shape and the patterns they make¡ªall depend on how big they are. These are all factors that challenge the mathematical analysis of the patterns in these multidimensional, nonlinear waves.¡±

Studying the matrix tide

In some truly propitious timing, Hoefer and his colleagues Gino Biondini and Alexander Bivolcic at the University of Buffalo had been working on the question of multidimensional, nonlinear waves when Hoefer's wife, Jill, showed him a video that his mother-in-law had sent.

¡°I started this research because the general field of study I work in is waves,¡± Hoefer says, adding that he studies waves in a variety of applications, including the types whose expression can be seen in undular bores, which are tidal bores with smooth, wave-like profiles. ¡°Waves like undular bores arise in a variety of physical settings¡ªfrom waves in water, air, light and even matter in quantum mechanics¡ªand the fundamental mathematical reason why that¡¯s the case is that all of them are modeled by similar partial differential equations.¡±

For a long time, the study of these wave phenomena focused on analyzing them in one dimension, in which they move in one direction and there¡¯s no variation in the perpendicular or transverse direction. ¡°But my colleagues and I recognized that we really needed to extend their mathematical description to more than one dimension because the world is multidimensional,¡± Hoefer says.

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A numerical simulation of the matrix tide.

So, the researchers began studying undular bores in two dimensions. They had made good progress and had core results by fall 2024, which is when Hoefer¡¯s mother-in-law sent an Instagram video to his wife, saying, ¡°¡¯These waves are so cool, you¡¯ve got to show Mark!¡¯¡± he recalls. ¡°I thought, ¡®Whoa, this is awesome!¡¯ I immediately realized, ¡®Oh, these are the waves we¡¯re predicting in our mathematical analysis.¡¯¡±

Hoefer contacted former 91PORN applied mathematics PhD student Yifeng Mao, now a postdoctoral fellow at Scripps Institution of Oceanography and who is from China, and asked her to help him get to the bottom of the images and videos he was seeing on social media. She discovered that a tide association for the Qiantang River completed a tidal survey last fall, adding a new tide type to the eight previously identified ones. Piecing together that and other data, Hoefer and his colleagues identified the multidimensional waves they had been studying as what was seen on the river¡¯s surface in the matrix tide.

Expanding the model

Among the challenges in studying waves in undular bores is that while certain physical effects can be disregarded at the outset when studying other types of waves, must be considered with undular bores, Hoefer said. For example, when the wave oscillations are short enough, gravity causes them to move slower than longer waves.?This effect, called negative wave dispersion, can be set aside in the mathematical analysis of longer waves because there are principles that account for it.

¡°In this setting, though, those effects are things we can¡¯t neglect in our first pass-through,¡± he says. He and his colleagues used a supercomputer at the University of Buffalo¡¯s Center for Computational Research and graphical processing units to run many wave simulations in a few hours that would each take a day on a regular computer.

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A simulation of the Mach stem and colliding undular bores.

They used shallow-water wave models, in which fluid depth is much less than the horizontal wavelength. ¡°Counterintuitively, shallow water models can apply even in the open ocean,¡± Hoefer says. ¡°The reason is when you have something like a tsunami, where an earthquake suddenly shifts the ocean bottom and displaces huge amounts of water at the surface, it generates a wave that can be many, many miles wide. Fishermen may be on their boat and not know that a 200-mile wavelength wave is passing under them.?There, the tsunami wave is so long that dispersion can be neglected. It¡¯s only when it gets close to shore and the depth gets lower that the waves shorten, compressing the energy and creating destructively large waves. So, the same kind of dispersive wave model that describes near-shore tsunamis is what we used to describe this bore.¡±?

Hoefer and colleagues¡¯ mathematical analysis of two obliquely colliding undular bores predicts that, for a special collision angle, the biggest waves in the matrix tide are eight times the size of the original waves:? ¡°This critical angle prediction was borne out in our wave simulations and marks a fundamental change in the shape of the waves from a matrix tide to another pattern called a Mach stem,¡± he says.

Hoefer adds that the applications to describing these waves in more than one dimension extend beyond the surface of water¡ªto fiber and crystal optics, quantum mechanical Bose-Einstein condensates and magnetic materials, meteorology and other applications.

¡°We have a number of directions to go,¡± Hoefer says. ¡°We are looking for examples of the Mach stem from colliding undular bores. Maybe this will be the tenth tide type discovered during the next river survey.¡±??

On the mathematical modeling, Hoefer adds that the model he and his colleagues used "is what we would consider in the field to be the simplest model to describe this setting. Another thing we assumed was that the waves are not too big, so they¡¯re not breaking. But if you look at the Instagram videos of this phenomenon, you see them break. Another assumption we make in this model is that the variation in the direction that is transverse to wave propagation is not too large, so we want to quantify what that means and see if there are any other possible wave patterns.

¡°There are these assumptions in the model, so we want to gradually start adding more terms to the equations representing more physics and allow for more complications to see if new things happen.?This will make the mathematics harder, but the challenge and reward of predicting new physical phenomena from mathematical models is why I keep doing applied math research.¡±

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