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*AN ANALOGY AS EASY AS WRAPPING A HOCKEY STICK AROUND A DONUT :*

## Abstract

Topology sheds new light on the emergence of unidirectional edge waves in a variety of physical systems, from condensed matter to artificial lattices. Waves observed in geophysical flows are also robust to perturbations, which suggests a role for topology. We show a topological origin for two celebrated equatorially trapped waves known as Kelvin and Yanai modes, due to the Earth’s rotation that breaks time-reversal symmetry. The non-trivial structure of the bulk Poincaré wave modes encoded through the first Chern number of value 2 guarantees existence for these waves. This invariant demonstrates that ocean and atmospheric waves share fundamental properties with topological insulators, and that topology plays an unexpected role in the Earth climate system.

DAN PISUT

# Waves that drive global weather patterns finally explained, thanks to inspiration from bagel-shaped quantum matter

“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 ...

Marston and his colleagues 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 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, ... 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.

**UPDATE Stoat obects:**

[Seems believable, though naturally I haven’t checked the maths. It doesn’t obviously shed any new light on Kelvin or Rossby waves, though -W]

[I see RS covers this too. (Re)reading it from that, perhaps the flow is the other way: the insulators folk can perhaps learn from all the stuff that’s already been done on K/R -W]

[But (without having ploughed through the details) Waves that drive global weather patterns finally explained, thanks to inspiration from bagel-shaped quantum matter is clearly bullshit -W]