A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2010 Nov 23, 21:02 -0800
Gary, you wrote:
"That is because Coriolis varies with the sine of the latitude which is zero at the equator so no Coriolis acceleration (or force) to make the water swirl."
Of course that wouldn't explain it all by itself. The projection of the Coriolis acceleration onto a horizontal plane is indeed proportional to the sine of latitude but that's not enough information to counter the demo shown to gullible tourists. After all, the sine of latitude is positive ten feet north of the equator and negative ten feet south of the equator. So (says gullible tourist) the factor of sine(latitude) is consistent with what we see, right?
The Coriolis acceleration is a vector given by 2wXv where w is a vector parallel to the Earth's axis with a magnitude equal to 2pi/T (T is the length of the day in seconds for mks units), v is the vector velocity of the object or fluid element, and the "X" represents the vector cross product. It is the small magnitude of w that makes the Coriolis effect completely insignificant in common sinks, toilets, etc. For example, if we have a fluid element in a basin moving at 1 meter per second, the magnitude of the acceleration is around 0.00007 m/s^2 or roughly 0.000007 g's. Note that the cross product yields a factor proportional to sine(theta) where theta is the angle between w and v (in other words, the tilt of v relative to the Earth's axis). Many people who have learned about the Coriolis acceleration from meteorology textbooks are surprised to learn that there's no factor for latitude in this. The Coriolis acceleration is constant all over the surface of the Earth for any given angle of theta. That's why the Coriolis acceleration can affect a freely moving (3d) gyroscope the same way at all latitudes. But for meteorology, flight, and anything involving the atmosphere or the ocean (and also for any situation where the motion is constrained to a horizontal plane, like a rolling ball or the bob of a Foucault pendulum), we always look at the projection of the Coriolis acceleration onto the horizontal plane. This is done because the atmosphere and the oceans are always in near perfect hydrostatic equilibrium --you won't find a low pressure system in altitude with higher pressure above it (but if you did you could have horizontal hurricanes! wouldn't that be fun). Because there's hydrostatic equilibrium in altitude, the Coriolis acceleration doesn't do much of anything interesting in the vertical direction and can usually be ignored.
Now if we take that very tiny value for the Coriolis acceleration, 0.00007 m/s^2, and multiply by the sine of the latitude with latitude set to, let's say, one nautical mile north or south of the equator, it is reduced by a factor of about 1/3438 (the number of minutes of arc in one radian). That leaves a really tiny number, 0.00000002 m/s^2 which clearly, over the course of a minute or so (the time it takes a sink to drain) could have no impact on the motion of the water. But what's still surprising to many people, since it's such a popular science urban legend, is that it also doesn't matter for ordinary basins of water at middle latitudes. At 35 degrees latitude, the acceleration is still a very tiny 0.00004 m/s^2 for a fluid element moving at 1 m/s. Toilets drain clockwise or counter-clockwise based on design and the residual vorticity from the "filling process". Coriolis effect only becomes significant in a practical sense when the velocities are large, like in airplanes, or when the time and distances over which the acceleration has an opportunity to build up its effect are larger, like in weather systems.
Sorry for the physics diversion.
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