The Coriolis Force, explained and debunked

Now that we're firmly into hurricane season, with two hurricanes already having made landfall in the United States, it's an appropriate time to discuss the Coriolis force, which drives the circulation of hurricanes and all other major weather systems.

The Coriolis force (named for G.G. Coriolis, who discovered the force in 1835), like centrifugal force, is something of a pseudo-force in that it has no source. Gravity and electricity, for example, have mass and charge sources. However, anybody who's weathered a hurricane can tell that this pseudo-force often packs a very non-pseudo punch.

The Coriolis force and centrifugal force are both caused by one simple phenomenon, a rotating reference frame. An observer is in a rotating reference frame if while trying to stay at rest he is rotating around an axis, much as we are on the earth.

Centrifugal force arises from the fact that as you rotate around your rotation axis you're actually changing the direction of your motion constantly. For instance, imagine you're sitting at the end of the second hand on a clock, facing outward. At the 12 o'clock position, your momentum is to the right, but by the time you get to the six's position, your momentum is completely toward the opposite, to the left. The force that actually changed your direction of motion was the friction between your pants and the top of the second hand, but all the while your momentum (inertia) was trying to keep you moving in a straight line. You felt this changing of your momentum as the centrifugal force outward.

To understand the Coriolis force, it's easiest to think of yourself as being on the interior ring of a merry-go-round (moving counter-clockwise, say) while trying to throw a ball to your friend on the outermost ring. If you throw the ball right at your friend, you'll see that it sails wide right, behind his back. It will look to you as if the ball curved by several feet. Why? Because the outer ring is moving much faster than the inner ring, so your ball, traveling on a straight line relative to the ground, finds your friend zipping by to the left much faster than you were moving (relative to the ground) on the inside of the ride.

We can use our merry-go-round example to see how Coriolis forces work on the earth. A low-pressure system such as a hurricane pulls air inward. So, it pulls air down from the north and up from the south. But in the north, the earth's rotation has a slower velocity because the distance from the rotation axis is smaller. Think about the North Pole, for instance. There, the velocity due to rotation is zero because the distance from the axis is zero. So, the air being sucked in from the north is much like the ball thrown from the inside of the merry-go-round. It lags behind the storm and misses to the west. South of the storm the rotation velocity is greater, so air from the south is moving faster than air at the center. Thus, when it gets sucked northward it zooms ahead of the storm and misses the center to the east. This causes a counter-clockwise rotation of low-pressure cells, which can be easily seen in film loops of hurricanes.

A quick sidebar: A common misconception is that the Coriolis force has anything to do with the direction water spirals down the bathtub. While the notion is qualitatively justified, the numbers just don't add up. Experts at vector calculus can show that the acceleration due to the Coriolis force has a magnitude of 4 * pi * f * v, where f is the frequency of rotation of the earth (once per day), and v is the speed of the object in motion relative to the earth (or how fast you threw the ball on the merry-go-round, relative to where you were riding). One can quickly compute that the Coriolis acceleration has a magnitude of (0.000145) * v. That means if the water on one side of the tub is moving by a factor of just 14 hundred-thousandths faster than it is on the other side, the Coriolis force is washed out. Furthermore, if we take a meter per second as the speed of the water down the drain, then we have an acceleration of 0.000145 meters per second squared. Compare that with the gravitational acceleration, which is 9.8 meters per second squared, and we discover that the Coriolis force (at the North Pole, where it is greatest) has a magnitude which is 15 millionths that of gravity. That means that it only takes 15 millionths more water streaming down one side of the tub, compared to the other to beat the Coriolis force. Given that the water is about a centimeter deep, this means that if just 0.14 microns more water is flowing down one side of the tub, the gravitational acceleration on that tiny difference will be greater than the Coriolis force. If you're still skeptical, just move your foot around near the drain when you take a shower, and you'll see the water spiral down one way, then another.

Why, then, is the Coriolis force enough to drive hurricanes? Because hurricanes are so large that the local random velocities that would disrupt the flow of water in your tub get washed out over the 500-mile journey to the eye of the storm. The Coriolis force, however, always acts in the same direction over the entire journey, and thus accumulates. If we go back to the numbers, we find that the acceleration (4*pi*f*v), where v is now 20 miles per hour, is 0.0013 meters per second squared. Adding this up over one day, we find that the velocity due to the Coriolis force alone is over 200 miles per hour by day's end.

The reason bathtubs and hurricanes are so different is that a hurricane is much larger than a bathtub, and it takes a lot longer for water to move across the hurricane than across the bathtub.