One More Round with the Coriolis Force

Part 2. Challenging the Alleged Effects

Kevin T. Kilty

In Part 1, I listed a series of effects attributed to the Coriolis force. Now let's examine the alleged effects on my list one by one.

Water going down drains

This is small scale motion. The typical size of a household basin (L) is 0.2 m, the velocity (U) is 0.5 m/sec, and R_o = 25,000. The enormity of R_o shows that centripetal accelerations in any observed spiraling motion cannot possible come from the tiny Coriolis force from the Earth's rotation. Water does not go down the drain differently in Auckland than it does in Santa Fe. In order to demonstrate the Coriolis force in this experiment one must have a very shallow sink, and tiny drain, so that water drains over a time period comparable to a fair fraction of a day.

Bigger whorls

Extratropical cyclones are the big weather systems of the winter season. They have a scale (L) of about 2,500 km, a typical wind velocity of 10 m/sec, and the Rossby number is R_o = 0.04 or so. Obviously a Rossby number so small means that the Coriolis force is much greater than the observed centripetal acceleration, and some additional force is needed to maintain the observed curvature of motion. These systems are dominated by Coriolis force balancing mainly a pressure gradient.

Not too many people have seen a tornado. They are hypnotically fascinating. One form they can take is that of a long, slender funnel, which hangs from a cloud base while its rotating foot grinds along the Earth. It often roars. A typical scale size (L) is 1 km (or less); a typical velocity (U) is 100 m/sec. Therefore, R_o = 1000, and, the Coriolis force from Earth rotation is very small compared to the centripetal force needed for the tornado's steady flow. The tornado is mainly a pressure gradient providing centripetal acceleration.

What about the initiation and growth of a tornado? Does Coriolis force play no part in the initiation of a tornado? I think it must for a predominance of tornadoes rotate in the correct sense for each hemisphere (anticlockwise in the Northern hemisphere). However, it cannot be entirely the cause for not all tornadoes rotate as expected. About 4% rotate clockwise. Of the seven tornadoes that hit Grand Island, Nebraska, one day during the summer of 1980, three rotated clockwise and four rotated anticlockwise. There is an excess of cyclonic rotation in air masses in the Northern hemisphere, which undoubtedly results indirectly from the Earth's Coriolis force, but in some cases tornadoes don't care and rotate as they please. How tornadoes organize inflow and rotation is not understood, and I suspect that studying these oddballs of rotation sense would help illuminate this. Thus, when people say the Coriolis force causes the rotation of a tornado, one can only respond,``But not directly!''

Rails, wheel sets, and ordnance

No one I've consulted who works with trains or rails knows anything about such observations. Frictional forces on the rails, wheels, and against the air consume 20,000 horsepower at running speed, though, which imply forces that dwarf any possible Coriolis force.

A replica of a small caliber cavalry field gun, operated by a group of mechanical engineers at Casper College, shoots well to the right of its intended target at a range of 200 meters. Several people (engineers all) asked me if this could be the Earth's Coriolis force at work. But the time scale of the flight--2 seconds--is so short that the Earth rotates during this flight by an amount so small that no one could observe its effect. I am convinced that the shell is simply over-spun by rifling and develops lateral lift during its flight. The direction of lift is related to the direction of rifling.

Flushing away a myth

The argument regarding motion down a toilet is exactly that of draining a sink. However, an equally illuminating argument is the one I just mentioned in regard to the artillery shell. During the time a toilet flushes, and during the time a sink drains, the Earth completes very limited rotation. Thus, the rotation of the Earth is a negligible influence. This same analysis applies to the spin of a rattleback toy. The motion is too brief for the Coriolis effect to play a role.

Walking in circles

What can we conclude about blindfolded students, polar explorers, and penguins? These walk at velocities and over distances that are intermediate between tornadoes and water down the drain. Thus, we conclude that the Coriolis force plays no significant part in their stumbling around. In fact, if lost polar explorers do tend to walk in circles, perhaps a better explanation is that they also tend to follow a sun that circles around the horizon rather than travels across the sky from East to West.

Tectonic plates and crystals

The group of parameters making up the Rossby number imply that it is possible for a very slow velocity to offset a small scale size, so that even small scale phenomena might relate to the Coriolis force. The ship's gyroscope is an example. It maintains a pole-indicating orientation because of the Earth's Coriolis force. What, then, of crystal growth? Crystals of calcite in caves sometimes grow in the form of a helix, "helictites" they are called, and among the various explanations for them is one invoking the Coriolis force. Now this is not an entirely consistent effect. Many crystals within a single set of helictites curve in the wrong direction. According to one of our rules, however, the Earth's Coriolis force is consistent in each hemisphere; it doesn't sometimes cause curves in one sense and at other times cause curves in the opposite sense.

Some of these helical crystals grow 3 mm (L) in about 2 weeks (U = 2 x 10^(-9) m/sec). Thus, R_o = 0.006 or so. This indicates, misleadingly in this case, that Coriolis force overwhelms the needed centripetal force, and that some other force is needed to counterbalance the Coriolis force.

A similar calculation holds for tectonic plates. M. Kane of the USGS proposed a Coriolis force that tears apart continents and J. Sumner of the University of Arizona proposed that Coriolis force explains many consistent features of plate tectonics. Because of the very small velocity (U = 10^(-8) m/sec) and large extent (L = 10^(6)m) of tectonic plates, the Rossby number R_o is about 10^(-9).

Do these tiny Rossby numbers in the case of helictites and tectonic plates mean necessarily that the Coriolis force plays a dominant role in crystal growth and plate tectonics? Of course not. R_o measures the relative contribution of Coriolis force to the required centripetal force in a curved motion. It does not, alone, make a sufficient analysis for complicated dynamical problems. Other forces often play some part in many observations on our list, helictites and tectonic plates in particular, and we require more complex analyses.

Change with time

For instance, we might wonder if friction or viscosity exerts an influence. Consider a mass such as a tectonic plate. If it is at rest with respect to the surface of the Earth, and then moves toward the Earth's axis of rotation at a speed U, for example by drifting North over the surface, then this mass will develop apparent spin at a rate related to omega U. This is a characteristic rate of generating vorticity. Viscosity disperses this motion to the surrounding material, per unit mass, at a characteristic rate of nu U/L^2, where nu is the kinematic viscosity of the fluid.

Note that this quantity is the gradient of a shear stress due to viscosity, and is therefore a frictional force per unit mass. The ratio of these two rates (or forces) is a new dimensionless group called the Stokes number, S_k = omega L^2/nu, and, if I apply a kinematic viscosity typical of the soft portion of the Earth's mantle (10^(-9) in SI units), then S_k = 10^(-4). The smallness of the Rossby number indicates that large, slowly moving tectonic plates should develop rotation from the Coriolis force, but the smallness of the Stokes number shows that viscosity disperses this rotation throughout the body of the Earth as rapidly as it develops. The Coriolis force plays no role in plate tectonics.

Baer's Law

Rivers of the Northern Hemisphere tend to erode chiefly on the right bank; those of the Southern hemisphere chiefly on the left bank. Does this reflect an influence from the Coriolis force? According to Albert Einstein, it is the velocity gradient that causes river bank erosion so that, "We must then concentrate our attention on the circumstances which affect the steepness of the velocity gradient at the wall."

Let's do this. Suppose there is a perfectly straight section of a river. The flow is down slope. The river develops a flow profile where water velocity is greatest in the center of the stream at its surface, and friction with the river bed and banks brings water velocity near to zero at the boundaries. Such a flow is symmetric and would not tend to cut more aggressively into either its right or left bank.

However, the situation is not perfectly symmetric. The Coriolis force pushes water slightly to the right in the Northern hemisphere, so that the water surface actually slopes from one bank to the other. The resulting pressure change causes river flow to spiral downstream, and this will cause the right bank to erode more quickly. At this point the river's flow line becomes slightly curved, and I expect that the centripetal accelerations of such a slightly curved path are much larger than the wimpy Coriolis force can deliver.

In a highly symmetrical situation, and in the absence of random influences, a tiny effect like the Coriolis force might offer a way to initiate Baer's observations. However, once the river begins to follow a curved bed, then other factors become more important. The river meanders back and forth, in conformance with Baer's law and against it as well. It would be extremely interesting to learn what rivers Baer analyzed in order to form his law, and decide what other influences were at work.

The Gulf and Jet streams

Despite one of these pertaining to an atmospheric current and the other to an oceanic current, they both have a common origin as boundary currents. They occur at the boundary between cold fluid to the North and warm fluid to the South. Take the Polar Jet Stream for example. It has a characteristic speed of U=80 m/Sec and maximum extent of L=2 times 10^(7)m. For it R_o is about 0.028. The Gulf Stream, on the other hand, has a maximum speed of U=1 m/Sec and extent of L=2 times 10^(6)m. R_o is about 0.004 suggesting an even greater Coriolis influence for it. Yet, the Coriolis force, being directed toward the right of the flow in the Northern Hemisphere, points toward the center of curved flow in the case of the Gulf Stream, but away from the center of curvature in the Polar Jet Stream. Obviously the Coriolis force cannot provide the centripetal force needed to keep the Polar Jet Stream in its path, and other forces are obviously involved. Both of these flows also show a tendency to deform into loops that are 20 times smaller than the extents I used to calculate R_o, which break off from the main flow and spin for long periods of time on their own. These topics are too complex to make an adequate analysis using the simple tools I've offered here, even though the Coriolis force must play an important role.

The Rattleback

A Rattleback is a children's toy. The Edmund Scientific Company sold these at one time under the trade name"Space Pets." These toys have a shape without an axis of symmetry. Their peculiar shape couples the various modes of motion of the toy to one another. For example, it is impossible to give the toy a rocking motion without it eventually developing a spin in a preferred direction. Spinning it against the preferred direction, no matter how carefully done, will cause it to rock. Yet, our interest here lies with what influence the Earth's spin might have on the motion of the toy. Any reasonable spin imparted to the toy results in a rate of rotation that is measured in cycles of a second or two at the longest. The ratio of this rate to the Earth's rate of rotation is R_o for the toy. R_o has a value in the thousands in this case. Thus, the toy completes its motion in such a brief period of time that the rotation of Earth has no effect. Obviously the preferred direction of spin will not be reversed by changing hemispheres.

This concludes this 2-part series.

References

Henry M. Stommel and Dennis W. Moore, An introduction to the Coriolis Force, Columbia University Press, 1989.
Isaac Asimov, What is the Coriolis Effect? Science Digest 69, 82-83, 1971.
I. Amato, The Curling Crystal Club, Science News 135, 124-125, 1989.
Anon., Coriolis force for continents, Science News 101, 215, 1972.
Anon., Do slabs rotate as the earth turns? Science News 125, 358, 1984.
Albert Einstein, Die Naturwissenschaften, 14, 1926.