On Hurricanes and Bathtub Drains

We've all heard the stories about cruise ship passengers who sail south of the equator and report that after crossing the equator the water started going down the bathtub drain in the opposite direction from the northern hemisphere. There have even been TV stories on this phenomena, with (I must say gullible) reporters led down the garden path by "pseudo scientists." Well, after reading this article, you'll get to the bottom of this curiosity, and discover that the questions it raises are more profound than you might have thought.

Like almost everything else in physics, it all starts with our friend Sir Issac Newton. His famous three laws of motion should contain a little footnote, that says that they are only valid in an "inertial frame." What's an inertial frame? What are Newton's laws of motion? Just briefly, Newtons first law basically says that things either stay put or move in a straight line with constant velocity. Believe it or not, this was a brilliant insight, and was not all that obvious at the time. People were used to things staying put, but not very many things in ordinary experience move in a straight line with constant velocity, for here on Earth we have this thing called gravity to contend with. But if you've ever seen those fun loving astronauts goofing around in the space shuttle, you've seen objects which when left to themselves, move in straight lines at constant velocity. So what is an "inertial frame?" The simple answer is that it is a place in the universe that "relative to the fixed stars" (they're not really fixed, but for all practical purposes they are) is standing still or moving with a constant velocity. These kind of places are hard to come by, but they exist in theory. Now then Newton's idea of using "inertial frames" in order to describe the physics of moving bodies turned out to be a stroke of genius, for it made the equations that describe motion really simple. The equation that really says it all is F=ma, or Force = mass x acceleration. But this equation is only valid in "inertial frames."

Well, what does all that have to do with water draining down the bathtub? Well, even though the surface of the Earth isn't an "inertial frame" for a lot of problems it is close enough, and we can use Newtons laws to describe bodies in motion. If we can afford to neglect messy forces like friction and gravity, the surface of the Earth is a pretty good approximation to an inertial frame. There is one remaining problem though, the Earth is spinning. How fast? Well about once every 24 hours, or at a blindingly fast 1/(24*60) = .0006944 rpm. That might seem rather slow, but since the Earth is rather large, the actual speed that the equator is moving through space (the horizontal velocity) because of this spin is the radius of the Earth x the rpms x 2PI = about 4000 miles x 1/24 revolutions per hour = 1046 miles per hour. That's a lot faster than I drive. Notice also, that how fast the ground under your feet is moving depends on your latitude. Namely at the north or south poles, your "horizontal velocity" is zero.

Okay, the stage is set. Imagine now that you fire a cannonball from the north pole, and aim it directly at Mazatlan. Once the ball leaves the barrel, and ignoring gravity and friction, it suddenly finds itself in an inertial frame. Thus, relative to the "fixed stars" it is going to travel in a straight line. The problem is that while it is in the air, Mazatlan is moving to the east at about 1000 miles per hour (a little slower than the equator, because we are north of it.) Thus if it takes a minute to reach us, we will have traveled 1000 / 60 = 16.6 miles to the east, and it is going to miss by that amount. (Of course if it were a 100 megaton warhead instead of a cannonball that would still be close enough.) From the perspective of the guy at the north pole, the cannonball is being "deflected" to the right, as though some mysterious force is acting on it and making it move away from its intended target. The same thing happens if we shoot back at the north pole. Our cannonball is also going to be deflected to the right by the same amount. If you now imagine yourself in the southern hemisphere, you'll see that from the perspective of someone shooting from the south pole, his cannonball will be deflected to the left.

Okay, so we here on Earth discover this mysterious force that is deflecting our cannonballs. This was known about since the mid 1800s and was described by a man named Gustave Coriolis. What we want to do next is get some idea of how big this force is, and what it depends on. Since this force is "caused" by the fact that we are on a spinning body, and the fact that the body is spinning relative to the inertial frame of the fixed stars, the force is going to depend on how fast we are spinning, which in turn depends on our latitude. Also thinking back on our cannonball, how much it has been "deflected" depends on how much time it spent in the air, with the Earth moving beneath it. Now we need to think a minute about forces, velocities, and deflections. If something is moving with a constant velocity, say 100 miles per hour, then in one hour it will have moved 100 miles. See it doesn't take a genius to do physics. In two hours it will have moved 200 miles, etc. Similarly, if something is accelerating at a constant rate, the velocity will increase proportionately. Think about it, if your car is accelerating at 10 miles per hour per second, then if you were standing still, after one second you'll be going 10 miles per hour. After 2 seconds you'll be going 20 mph, and after 10 seconds you'll be getting a ticket (going 100 mph.) Now fire the cannonball from the north pole, at watch it as it heads south. Initially its horizontal velocity is zero, and it is traveling due south at say V miles per hour. Lets say the Earth is rotating at w revolutions per hour. Then after one second, the cannonball is R = V / (60*60) miles south of us. At this point, the Earth is moving beneath it with a apparent velocity of 2 PI x w * R miles per hour. Thus from our perspective it is as though the cannonball has accelerated by 2 PI * w * V miles per hour per second. Thus we see that the acceleration, and hence the force on the cannonball is proportional the speed of rotation of the Earth, and the speed of the object that is moving relative to the Earth. Physicists write this as acceleration = k (a constant) x w (angular rotation of the Earth) x V (velocity of the object relative to the Earth) or k * w * V for short. (By the way, the k constant is on the order of 2, not 20 billion.) Now lets apply this formula to bathtub drains. w is 2 * 3.14 / (24*60*60) = .000072 revolutions (actually radians) per second, and water drains at around 1 foot per second, so the acceleration of the water is about .000072 feet per second per second. At that rate it would take almost a minute (47 seconds) for the water to be deflected one inch. In comparison to the other forces at work in a draining bathtub, this is almost insignificant, and no experiment that you could do on a cruise ship or in your bathtub could detect a force of this magnitude. The water near the drain is already gone in a matter of a second or less, so there just isn't any time for a force of this magnitude to do any real work on the water. This effect has been demonstrated under laboratory conditions, but the amount of care taken to eliminate outside disturbances is very great. The water must be allowed to "settle down" for a period of weeks, and the drain has to be opened very, very carefully. Under these conditions, you can demonstrate the effect of the Coriolis force on water flowing down a drain.

The situation is quite different when it comes to hurricanes however. Here we have winds blowing a many feet per second over periods of hours rather than seconds. Thus this small force can cause substantial deflections over time, and is responsible for the counter clockwise rotation of hurricanes in the northern hemisphere. Why counter clockwise? As the air rushes towards a region of low pressure, the Coriolis force deflects it to the right, generating a counter clockwise rotating airflow.

One final note. The existence of this Coriolis force was the first real "evidence" that the Earth does indeed rotate. You may have seen those giant pendulums swinging back and forth in some science museums slowly knocking over little pegs on the ground. These ingenious devices were first constructed by J. Foucault in 1851, and proved conclusively that the Earth does indeed rotate. Foucault took a very large and heave iron ball, and attached it to a 200 foot steel wire and strung it to the ceiling of the Pantheon in Paris. Using a candle to burn through a thin thread that he used to pull the pendulum to one side, he was very careful not to give the ball any initial sideways motion. The audience watched an waited, and after an hour all could see that the plane of the pendulums motion had shifted by 11 degrees, exactly the amount predicted by Foucault based on his calculations of the Coriolis force. For the first time in history, the fact of the Earth's rotation had be demonstrated experimentally.

The easiest way to visualize this is to imagine the pendulum swinging at the north pole. Every 24 hours the Earth will make a complete revolution beneath it, and the pendulum will knock over all of the pegs that it passes over. From the perspective of the guy watching the pendulum, our old friend the Coriolis force is once again at working causing the pendulum to slowing be deflected to the right. This happens, albeit more slowly, at lower latitudes, and can still be observed.