In countries near the Earth’s equator, tourists are often dazzled by a demonstration of a mysterious physical phenomenon. A presenter will position three buckets of water – one in the Northern Hemisphere, one in the Southern Hemisphere, and one directly on the equator – and let the water drain out.
Tourists are shown that, as the water drains, the water in the northern bucket rotates in one direction, the water in the southern bucket rotates in the other direction, and the water at the equator doesn’t rotate at all.
The demonstrator might claim that this strange phenomenon is governed by physics, that it’s an example of the Coriolis effect.
The Coriolis effect is based on the idea that the spinning of the Earth introduces a physical force, known as the Coriolis force, which affects the way objects appear to move to us Earthbound observers. The Coriolis force causes objects on the Earth’s surface to be deflected in different directions depending on whether they are above or below the equator. The effect is strongest near the poles and weakest at the equator.
The Coriolis effect is legitimately responsible for the behavior of some natural phenomena, like hurricanes, that meterologists and physical oceanographers like the two of us study. But in domestic settings, the spinning of the Earth actually has very little effect on how water behaves. Math can explain how this works – or doesn’t work – in a kitchen sink.
The math behind the phenomenon
Geophysicists use certain mathematical equations, known as the Navier-Stokes equations, to describe the behavior of fluids. Roughly, the Navier-Stokes equations relate the change of fluid velocity – how the fluid moves – to the forces acting on the fluid, subject to a few physical constraints. For example, the equations assume that the overall amount of fluid in the system doesn’t change over time.
But just because physicists and mathematicians can write down these equations, it doesn’t mean we can solve them. In fact, these equations are so difficult to solve that you would win a Millennium Prize and US$1 million if you could do it.
Although there is no known complete solution to Navier-Stokes equations, meteorologists and physical oceanographers can still obtain useful partial solutions. One way to obtain these partial solutions is to compare various terms in the Navier-Stokes equations to determine which ones are most important.
These comparisons are often recorded as ratios and have no associated physical unit, thereby earning them the name “dimensionless numbers.”
What happens in your sink?
In the context of the Coriolis effect, perhaps the most important dimensionless number is the Rossby number, named for the early 20th-century meteorologist Carl-Gustav Rossby. The Rossby number compares the dynamics of the fluid with the Earth’s rotation rate, taking into account how big the system is and how fast it’s moving.
A small Rossby number indicates that the Coriolis force has a strong effect on the system, while a large Rossby number signifies that the Coriolis force has a negligible effect. For example, the Rossby number for an average hurricane is of the order of 1, indicating that the dynamics of the fluid and the Earth’s rotation rate are of similar relevance. It is true that hurricanes tend to rotate clockwise in the Southern Hemisphere and counterclockwise in the Northern Hemisphere.
The same math that applies to large-scale phenomena like hurricanes also applies to the water in your bathroom sink. In this setting, the system is relatively small, and so the Rossby number will be much larger than 1 – more than 10,000 times larger. This observation indicates that the Coriolis force is negligible on water draining in a bathroom sink.
In fact, the Rossby number predicts that the water would need to move at an almost imperceptible speed for the Coriolis force to become significant. So even though the way water swirls down the drain may be consistent, that isn’t due to the Coriolis effect.
So what did the tourists see?
The same logic applies to the equatorial attractions. Given the size of the system, physical oceanographers can comfortably conclude that the Coriolis force is not responsible for what the tourists see in those buckets or bowls.
This conclusion is also supported by examining the same kind of presentation in different countries.
The water in the Northern Hemisphere rotates counterclockwise in one video but clockwise in another video. If the rotation were due to the Coriolis effect, the result would be the same in both videos.
Although physical oceanographers can’t deny what the tourists see, we know that the magic trick isn’t due to the Coriolis effect at such a small scale.