Rossby Waves

I was reading a paper recently that mentioned Rossby waves. I had vague memories of learning about these things when I took Physical Oceanography as an undergrad, but that was a was five years ago, and I didn’t really understand them then anyway, so I went and looked them up. I hadn’t remembered how cool they are. I felt the need to share that coolness with the internet, but the standard presentation of Rossby waves is pretty math heavy. So I’ve tried to explain them here without a single equation, heeding Einstein’s advice that “If you can’t explain something simply, you don’t understand it well enough.” Let me know how I did, and I’ll try to answer any questions in the comments. Ahem.

Oceanic Rossby waves are radically different from the surface waves we are all familiar with (i.e., the ones that enable fun activities like surfing and seasickness). They are huge, stretching for hundreds of kilometers horizontally. They travel only from east to west across the oceans–open-ocean Rossby waves are physically incapable of traveling in the opposite direction. They cause water to meander in wide loops north and south, but only move the surface of the ocean up or down a few centimeters. They routinely cross entire ocean basins, but take their sweet time doing it: a Rossby wave starting on the West Coast of the US might take ten years to reach Japan.

This video shows the height of the sea surface relative to its long-term average: reds are above normal, blues are below. The slow-moving, westward-propagating disturbances on either side of the equator in the Pacific are Rossby waves (they are labeled at 0:34).

Rossby waves are one of the trippy consequences of our existence on a rotating planet. Though you (probably) don’t feel as though you are right now, you are spinning around your vertical axis. Even if we remain absolutely still relative to our surroundings, the Earth’s rotation is rotating us too. If I were to fall asleep in my chair with my head lolled back for 12 hours, I would wake up with a stiff neck, facing, in absolute terms, a different direction than the one I was when my eyes closed. Of course, it wouldn’t look like it to me, since the chair, room, and everything else immediately around me would have rotated as well.

This example illustrates the difference between planetary and relative vorticity (“vorticity” is geophysicist for “spinniness”). Even sitting still in a chair, I am rotating with the background, or planetary, vorticity. If I get up and start spinning or running in circles, I will be adding my own personal vorticity. If I spin counterclockwise, my total vorticity will increase–since I will be rotating in the same direction as the planet [1]. If I go clockwise, I will decrease my total vorticity.

There is another way to change your vorticity, and that is to change your location on the planet. If I fall asleep for 12 hours facing north in Seattle, I will wake up facing in a direction about 90° from where I was looking when I fell asleep. If I do the same thing in a lawn chair on the pack ice a foot from the North Pole, I will wake up facing almost exactly 180° opposite where I was when my eyes closed. If, instead, I drift off gazing out on the ocean from the beach on the north shore of Kiritimati (1° 52′ N, a.k.a. Christmas Island), I will wake up facing almost exactly the same direction I was when the tropical drink slipped out of my hand. In other words, planetary vorticity increases from zero at the equator to 360° per day at the poles.

One more bit of physics is necessary to understand Rossby waves: the tendency of vorticity to be conserved. In other words, once something starts spinning–whether it is a bicycle wheel, figure skater, or 200-km-wide slab of water on the surface of the ocean–it will keep spinning at the same rate until something else interferes. So if, while I was asleep at the North Pole, I were teleported to a frictionless ice rink (bear with me here) on Kiritimati, I would wake up spinning slowly counterclockwise on the ice, since that was my rotation at the Pole. If the teleportation went the other way, I would wake up at the Pole spinning slowly clockwise relative to the ice. In this case, my personal vorticity would be zero, but the Earth would be spinning underneath me, giving an apparent rotation. The same phenomenon will take place, to a less extreme degree, with any change in latitude. As long as frictional effects are low enough, and the length scale long enough, changes in latitude produce noticeable changes in vorticity relative to the earth’s surface.

So picture a large parcel of water in the North Pacific. It is at rest relative to the earth’s surface, but still has positive (counterclockwise) absolute vorticity, equal to the planetary vorticity at that latitude. Now imagine it is pushed north, perhaps due to a burst of wind or some other perturbation. The water parcel’s vorticity is conserved, but the planetary vorticity is higher at the higher latitude, so the parcel now has an apparent clockwise rotation relative to the earth’s surface. On its west side, it thus pulls the leading edge of the new wave north. On the east side, water is displaced south, where it finds itself with a counterclockwise vorticity anomaly. And so on:

After a few minutes of examining this sketch, and, if you’re me, tilting your head back and forth while you twist your hands in front of you trying to act it out, this should start to make sense. In particular, it is apparent that any wave generated like this has to travel westward. Also, thinking about the differences in relative vorticity involved, the waves have to be pretty slow-moving. The maximum difference in planetary vorticity you can find anywhere is between the equator and the poles–a difference of one rotation per day. With a realistic displacement of a couple of degrees of latitude, the difference in relative vorticity between a parcel of water displaced to the north and one to the south will be quite small, meaning that the rotations pushing the wave along will be quite slow.

These waves were theorized to exist in the ocean by Carl-Gustaf Rossby in the 1930’s, but couldn’t be measured until the advent of satellites with radar altimiters on them. Using these instruments, we can observe lumps on the sea surface with resolution down to a few centimeters. Sure enough, we can see the small surface expressions of Rossby waves marching westward across the world’s oceans. Studies have shown their real velocities to be quite close to what Rossby’s theory predicted.

So who cares? Rossby waves, as it turns out, are important for a variety of processes on earth. In the ocean, they play a critical role in El Niño, interacting with several other forces to produce the sloshing of warm water back and forth across the tropical Pacific every two to seven years. Further, they are one of the major ways that this El Niño signal is transmitted across the oceans. In the atmosphere, Rossby waves generate the wide meanders of the Jet Stream that give us most of our weather in the mid-latitudes. But I just kinda like thinking about them–these huge, silent disturbances, propagating majestically across the oceans, guided by nothing but strange effects of the Earth’s shape and rotation. Who knew anything that weird existed here?

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[1] A side effect of doing this is illustrated here.