The Unswirled Truth of the Coriolis Effect
This blog is in response to an earlier entry about the direction in which the water drains when flushing a toilet in the southern hemisphere. The famous myth regards the Coriolis force, which arises from the Earth’s rotation and is responsible for tropical cyclones (hurricanes) spinning counterclockwise in the northern hemisphere and spinning clockwise in the southern hemisphere. I won’t explain why these spin directions occur, but this link does a good job of explaining the Coriolis force and why the tendency is counterclockwise in the northern hemisphere.
As evidence, the first picture (left) shows an image of Hurricane Katrina on August 28, 2005 as it approached New Orleans. Note the counterclockwise swirl. This second picture (right) shows Cyclone Larry (on March 20, 2006), which was the largest cyclone to hit the northeast coast of Australia since 1931, wiping out 90% of Australia’s banana crop. Note the clockwise swirl.
The Coriolis force is definitely real! As an aside, “tropical cyclone” is the general term for these swirling storms, which have different names in different parts of the world for historical and regional reasons. For example, the storms in the Atlantic (that impact North America) are called hurricanes and in the Northwest Pacific Ocean they are called typhoons. In the Indian Ocean and Southwest Pacific Ocean they’re called cyclones. Apparently, Australians have historically preferred the scientific term “willy-willy” (presumably because they make their heads spin willy-willy), but have adopted the term cyclone in modern times. (But I am told they still use willy-willy to describe “mini-dust devil type” phenomenon that sometimes occurs in the Australian desert).
Now, given that hurricanes are affected by the Coriolis force, I guess it seems logical that whatever occurs in the atmosphere must occur in my toilet. It reminds me of my favorite Simpson’s episode where Bart visits the American embassy in Australia and flushes the toilet which is equiped with a huge electro-magnetic-mechanical contraption that changes the direction of the water so that it flushes ‘the American way’. The short answer is that the myth is indeed a myth. The Coriolis effect is real, but in your toilet it is several orders of magnitude smaller than the local effects (such as the the exact geometry of the drain, or in particular, any microscopic residual motion in the water). It’s these random local effects that dominate and determine the final overall direction of spin. I have personally verified this in numerous toilets throughout Australia and Ecuador. That’s the short answer. For a more detailed answer… read on!
My friend, Piet Moeleker, and I decided to travel to Ecuador a few years ago to study this phenomenon in detail (okay… we were on vacation). Like every tourist, we visited the Equatorial monument (Mitad del Mundo) just outside Quito. The first thing to know is that the official Equatorial monument is in the wrong spot (GPS indicated 7 seconds North – about 300 ft North) – this fact is generally acknowledged by everyone. A small private museum has sprung up beside the official government tourist attraction with the claim that they have the ‘true’ equator as determined by GPS. They even have ‘science’ experiments to prove it. This experiment involved filling a basin with water, then pulling the drain plug and observing the direction of spin of the water. They had a line painted on the ground, which was the ‘Equator’. The guide moved the basin approximately 1 meter to the ‘south’ and did the experiment and observed the water spin CW. The guide then moved the basin 1 meter to the ‘north’ and watched the water spin CCW. The guide repeated the experiment a third time on the ‘Equator’ and observed it can go in either direction. Sure seemed convincing… except it defied our intuition of fluid mechanics. Even more puzzling was that their “Equatorial line” painted on the ground didn’t seem to be the true Equator (my handheld GPS was reading 3.5 seconds North – about 110 meters North). Hmmm… so what’s the science?
The governing equations for fluid mechanics can be written in a rotating reference frame such that all the important effects (including Coriolis effect) are represented by mathematical terms. Using some simplifying assumptions and rearranging the terms yields the Rossby number, a dimensionless number that represents the ratio of the inertial effects (acceleration) to the coriolis effect:
Where U is the fluid velocity, L is the characteristic length scale of the system, w is the rotation rate (=24 hours/day = 7.27×10^-5 rad/s on Earth) and theta is the latitude.
We can see that if the numerator is large, then inertial (or acceleration) effects dominate, and if the denominator is large, then Coriolis force dominates. A general rule of thumb is Ro < 1 means Coriolis force is important. Lets examine our two cases, but first we need some numbers. Using a highly calibrated tape measure, I measured my toilet bowl to be 12 in (0.3 m). Let’s assume that the water approaches the drain at around 1 m/s (which is really fast… I guess I have a ‘turbo toilet’). Most of the hurricanes that viciously attack the US actually start their life as perturbations in the Easterly trade winds just off the coast of Africa. These low pressure atmospheric systems (tropical depressions that eventually form into hurricanes) have wind speeds of about 10 m/s with length scales of the order of 1000 km. Hurricane formation is a complicated process (subject of another blog?), but it turns out that the rotation of the hurricane is determined at these early stages of formation. We’ll assume a latitude of 20 degrees.
|Toilet||Tropical Depression (Hurricane)|
|U||1 m/s||10 m/s|
|L||0.3 m||1000 km|
|Latitude||20 deg||20 deg|
Well… that’s pretty clear. The tropical depression is definitely influenced by the Coriolis force, whereas the toilet is not!! It turns out that the local forces in our toilets are much more powerful than a hurricane!
Seriously, what’s happening is that, on the small length scales and fast rotation rates (compared to the Earth’s rotation) of water draining in the sink, the effect of the Coriolis force is relatively weak. Other effects, in particular, any microscopic residual motion of the water (prior to pulling the drain plug or flushing the toilet) dominates the final direction of spin. In principle, if you did a very carefully controlled experiment and waited long enough (days) such that all residual motion in the sink had ceased, you would then see the effect of Coriolis force and the water would spin in appropriate direction. In practice (ie: everyday use) – this does not happen.
A fascinating discussion arises from Figure below which shows the Rossby number as a function of latitude for both the toilet and tropical depression. Remember, Rossby number less than one means Coriolis force dominates, so we see that the Coriolis force is smallest at the Equator and largest at the poles. This is in agreement with the basic definition of Coriolis force which predicts that it is greatest at the poles and identically zero at the Equator. At all latitudes, the toilet does not feel the effects of the Coriolis force since the Rossby number is huge. Large scale atmospheric phenomenon (like hurricanes), however are affected, provided the latitude is greater than 5 degrees (where it crosses the Rossby = 1 line).
From the point of view of Coriolis force alone, the strongest ‘cyclonic storms’ are likely to occur at high latitudes (ie: ‘nor’easters in the US) since the Rossby number becomes much smaller than 1. Polar lows are “hurricane-like” storms (with the characteristic swirl) that form near the arctic – but no one really lives there, so we tend to hear less about them. Luckily, neither of these can form into hurricanes because they lack the warm ocean water which is the required energy source for hurricane formation. So… interesting, hurricanes are highly unlikely to form within 5 degrees of the Equator because the Coriolis force is not strong enough so close to the Equator. As it turns out, the warm waters required for hurricanes are typically found at latitudes less than 20 degrees…. hence tropical cyclones are almost always formed between the latitudes of 5 and 20 degrees!
Now… back to our Ecuadorian equatorial paradox. What was going on? Piet and I proceeded to perform our own experiments with the “magic” water basin using small leaves to visualize the flowfield.
We were able to conclusively demonstrate that we could indeed get the water to swirl in any arbitrary direction. The museum guide was apparently a “sham artist” and deliberately left a little bit of ‘spin’ (no pun intended) in the water basin – that’s what set the spin direction of the draining water. What about the location of the Equator? The museum claimed to use GPS to identify the location of the Equator. Well, it turns out they probably did. However, until the late 1990′s the US deliberately degraded the GPS signal for civilian users to approximately 100 meter accuracy – and this “equator” was just about 100 meters off. Piet and I set off to find the true equator using my handheld GPS which had 30 meters accuracy. Sure enough we found it – on some kind of highway with dump trucks and buses passing by at high speed. We proceeded to build our own lasting monument to the TRUE equator using garbage, Sprite bottles and an old bucket to document our historic find. And that, ladies and gentlemen, is the unswirled truth!