They are about as far apart as two things in science can be: a type of ocean wave that helps drive the El Niño climate cycle, and quantum materials that, thanks to a particularly strange bit of physics, have insulating interiors and conduct current along their surface. Yet, in a remarkable case of lateral thinking, the two disparate phenomena can be explained with the same topological mathematics of shapes with holes in them, a team of physicists reports.
“I’ve been trying to make the case that these two fields really are very closely connected,” says Brad Marston, a physicist at Brown University who led the study. In addition to explaining why ocean and atmospheric waves can become trapped at the equator, the study also suggests that condensed matter physics—the study of liquids and solids, such as the semiconductors that make up computer chips—and earth science could cross-pollinate in other ways, such as using topology to explain waves on other planets and moons, or in astrophysical disks of gas and dust.
For the new study, Marston teamed up with condensed-matter physicist Pierre Delplace and geophysicist Antoine Venaille at the Ecole Normale Supérieure de Lyon. The team applied condensed-matter theory to two types of waves, known as Kelvin and Yanai waves, that can propagate through the seas and air near Earth’s equator. Both are undulations with wavelengths hundreds or thousands of kilometers long that carry energy eastward along the equator, contributing to El Niño, tropical storm systems, and other weather patterns. They result from the interplay of two physical processes. First, gravity competes with buoyancy, causing colder air or water to sink and warmer air or water to rise and making the blobs of air or seawater independently bob up and down. Second, Earth’s rotation to the east creates the so-called Coriolis effect, which makes fluids moving over Earth’s surface veer in opposite directions depending on the hemisphere.
To see how these effects combine to drive the waves, Marston and his colleagues initially follow the same mathematical approach as Taroh Matsuno, an atmospheric scientist at the University of Tokyo who predicted the equatorially trapped waves in 1966. They simplify the vertical structure of the ocean or atmosphere and focus on a narrow latitude band, over which the Coriolis effect remains roughly constant. But then they take what most earth scientists would consider a step backward. They solve their equations not for equatorial waves, but for a more easily analyzed class of waves that occurs at higher latitudes.
This is a trick that condensed-matter physicists routinely pull. They switch to a simpler problem and then show it implicitly contains the answer to the original puzzle. Marston and colleagues study the waves not in ordinary space, but in an abstract space of all possible waves of different wavelengths and Coriolis effects. The equations for extremely long waves show two mathematical singularities, where the wave amplitude varies wildly with wavelength. These singularities are mathematical holes and also arise in topological insulators. Marston says there are two of them because Earth has two hemispheres, with opposite Coriolis effects.
As a result, the hemispheres behave a bit like two slabs of such a material, the researchers found in a paper published today in Science. Just as putting two electrically insulating materials together lets current flow along their surface, putting two hemispheres together results in waves at their interface—the equator—that die off with increasing latitude. And, as in the case of the material, the waves are robust—or, as physicists say, “topologically protected” by the singularities in the abstract space.
The topological approach reduces the problem to the barest facts, Marston says: Earth is rotating and has an equator where the Coriolis force switches direction. All the details of the dynamics fall away. “From a technical perspective, our calculation is much easier than what Matsuno did in 1966,” he says. Any rotating planet, however strange its shape, would have these waves; the team found that they emerged even for a hypothetical doughnut-shaped planet.
Marston and his colleagues say they hope to apply the same idea to other systems, such as disks of gas and dust around stars and black holes as well as the atmospheres of Venus and Titan, which also show equatorial waves. Inspired by their work, Cristina Marchetti, a theorist at Syracuse University in New York, and her colleagues recently studied how bird flocks and other collectives can experience Coriolis-like effects. “It is nice we started tuning our antennas towards identifying topological phenomena and studying their consequences,” says condensed-matter physicist Jiannis Pachos at the University of Leeds in the United Kingdom. But several geophysicists say they found the reasoning abstruse. “I must admit that I didn’t understand much of it,” says Matthew Wheeler of the Australian Bureau of Meteorology in Melbourne, who studies Kelvin waves. Perhaps the two communities are like two hemispheres that must find some place to meet in the middle.