Master of trades and gyres

The Coriolis effect plays a critical role in a number of important oceanographic and meteorological phenomena. The effect describes how the Earth’s rotation influences moving objects and deflects them to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. The Coriolis effect has a critical importance in all large-scale motions on the Earth, especially in the ocean and the atmosphere. Without the Coriolis effect there would be no trade winds or westerlies, no hurricanes, no high and low pressure systems, no large ocean gyres, and no major ocean currents like the Gulf Stream, to name only a few examples.

The analogy often used to help explain the Coriolis effect is the example of a merry-go-round that is rotating counterclockwise, with two boys sitting on opposite sides (one with a ball to be thrown to the other), and a person standing in the playground watching the merry-go-round. When one boy tries to throw the ball across to the boy on the opposite side of the merry-go-round, the ball appears (to the boys) to curve sharply to the right and miss the target. The person on the ground, however, sees what really happens. The boy does indeed throw the ball straight, but by the time the ball gets across to the other side the other boy is no longer there, having been rotated around by the merry-go-round to another position. The ball appears to curve only to the boys on the merry-go-round, not to the observer on the ground.

Physics equations can be formulated, relative to the rotating merry-go-round, that describe the motion of the thrown ball, but these equations must include a “force” acting perpendicular to the motion of the ball that “pushes” the ball to the right. This force, called the Coriolis force, is a fictitious force, since it comes about because we are observing motion from within a rotating reference frame. Although fictitious, the Coriolis “force” feels like a real force to someone on the rotating merry-go-round; in that reference frame, it acts on mass like a real force.

This is not the only fictitious force that the boys on the merry-go-round notice. They also feel a force trying to push them outward and off the merry-go-round. This is called the centrifugal force and is also fictitious. An object set in motion tends to stay in motion and to travel in a straight line, unless acted on by another force. This is called Newton’s First Law of Motion, and the tendency to keep moving (unless stopped by some force) is called inertia. If you attach a rock to a string and swing it around in a circle, but then suddenly cut the string, the rock will travel off in a straight line (that is tangent to the circle it had been tracing). Before it was cut, the string exerted a real force (called centripetal force), pulling on the rock to keep it from flying off. The centrifugal force that the boys think they feel is really their inertia; i.e., their bodies trying to maintain their inertial straight-line motion relative to the playground, but their seats keep holding them on the merry-go-round and pulling them into the circular motion of the merry-go-round. Similarly, the thrown ball is maintaining its inertial straight-line motion (relative to the playground), and the boys observe a fictitious Coriolis force causing the ball to curve (relative to the rotating merry-go-round).

Now we change the example by replacing the counterclockwise-rotating merry-go-round with the rotating Earth. The person in the playground reference frame is replaced with an observer in a reference frame among the stars. The Earth rotates once every 24 hours from west to east, which is counterclockwise when looking down from the North Pole. We replace the thrown ball with a rocket launched from one location and aimed to hit a target location. In this Earth example the same deflection to the right occurs (in the Northern Hemisphere, where the rotation is counterclockwise), since by the time the rocket goes the distance required to hit the second location, that location has been moved out of harm’s way by the rotating Earth.

We have to make a slight modification because now we are dealing with a rotating sphere instead of a flat merry-go-round. The “inertial straight line” on the merry-go-round is replaced by an inertial “great circle” that the rocket would follow around the Earth if it continued traveling for a long distance (essentially like being in orbit).

Deflected rockets

On the rotating Earth the Coriolis effect is easiest to visualize if the rocket is launched from the North Pole and aimed at a second location directly south. Then, by the time the rocket travels the necessary distance, the second location has rotated eastward out of the rocket’s path and thus appears to deflect to the right (westward in this case) when viewed from the Earth. If we launched the rocket from a location some distance south of the North Pole, that location will also be rotating eastward, but at a slower speed than the speed of rotation at the target location farther south. So, along with the rocket’s large southward speed, the rocket will also have some eastward speed (due to the Earth), but not enough to keep up with the eastward motion at the target’s latitude, so it will still deflect to the right when viewed from the Earth. Launching the rocket northward from the equator is an opposite situation, but still easy to visualize. The rocket leaves the equator with a certain amount of eastward motion (due to the rotating Earth), and travels northward over parts of the Earth that have less eastward motion, so it will again deflect to the right (eastward in this case) when viewed from the Earth.

Although less easy to visualize, there is also a Coriolis effect when the rocket is aimed east (or west). This is because the rocket must travel along an inertial great circle, which means that it will not continue traveling in the initial east or west direction, since a latitudinal circle is not a great circle (except at the equator). The one situation in which a rocket aimed east (or west) will not be deflected is if it is launched from the equator and aimed at a target location also on the equator. The equator is a great circle, and so the rocket’s path will stay along the equator, and the second location will not be rotated out of its way. Thus, there is no Coriolis effect right at the equator. In this case, the rocket’s orbit will look the same whether viewed from the Earth or from outer space. Anywhere else on the Earth this will not be true. An observer fixed to the Earth’s surface will rotate around the Earth’s axis of rotation along a latitudinal circle, but a moving object will “orbit” around the Earth in a great circle that is different from the observer’s motion. This difference increases with latitude, being most pronounced at the North Pole (from where the rocket must always head south). Thus, the Coriolis effect increases from zero at the equator to maximum at the North Pole.

The strength of the Coriolis effect depends on the speed of the moving object (the ball or the rocket) compared with the speed of the rotating reference frame (the merry-go-round or the Earth). Thus, the more slowly the boy on the merry-go-round throws the ball, the more time there will be for the second boy to rotate away, and the more the ball will appear to curve away.

The faster an object moves on the Earth the less Coriolis effect there will be. However, the Coriolis effect can still be important for fast-moving objects, if they travel far enough. The first serious consideration of the Coriolis effect was for firing artillery at distant targets. There is one well-known naval engagement between the British and Germans in World War I near the Falkland Islands, where the Coriolis effect played an important role. The British gunners had been taught about the Coriolis effect on the shells fired long distances from their cannons, and they made what they determined to be the necessary adjustments. Yet they consistently hit approximately 100 yards to the left of the German ships. The one thing that they had apparently not considered was that in the Southern Hemisphere the deflection will be to the left and not to the right. They had done their Coriolis adjustment for 50o N, not for 50o S (the latitude of the Falkland Islands), so their shells hit a distance from the ships that was twice the distance caused by the Coriolis deflection.

Slow-moving water and air

Parcels of water in the ocean and parcels of air in the atmosphere move much more slowly than cannon shells and rockets, and the Coriolis effect is thus much more important. The greater the distance the water moves along the Earth’s surface the more pronounced the effect.

We mentioned above that the Coriolis effect increases with latitude. The speed at which the surface of the Earth moves around the rotational axis of the Earth is different at different latitudes because the Earth is a sphere. Although the Earth rotates with the same “angular” velocity everywhere (one cycle per day), the “linear” speed at the surface will be largest at the equator, where the radius of rotation around the Earth’s axis is largest. It is smaller at higher latitudes, because the surface is a shorter distance from the axis of rotation. The linear speed decreases more and more quickly as one approaches the North Pole, finally reaching zero. For a rocket launched northward from the equator, the Coriolis force keeps increasing as the rocket moves northward, because its eastward motion (gained by being launched from the equator) gets larger and larger compared with the eastward motion of the Earth’s surface under it.

The fact that the Coriolis effect is zero at the equator is the reason why hurricanes never form right at the equator, even though the warmest water temperatures are there (the heat being needed to drive the hurricane). Most hurricanes are generated between 5o and 20o north or south of the equator, where there is enough Coriolis effect to start the air turning.

Hurricanes might seem to turn in the wrong direction (i.e., counterclockwise in the Northern Hemisphere) when wind turning to the right would seem to imply that they should turn clockwise. However, a hurricane is a low pressure area with higher pressure air masses on all sides. The air masses flow in from the north, south, east, and west, each air mass being pushed toward the right by the Coriolis effect. These multiple pushes, however, drive the rotation around the low-pressure center of the hurricane in a counterclockwise direction.

One can see from the importance of Coriolis in forming low and high pressure systems, hurricanes, the trade winds, westerlies, and easterlies that, without its rotation and resulting Coriolis force, the entire Earth would have weather that does not change much (as is the case in the tropics where the Coriolis effect is very small or zero). This is, in fact, the case on Venus, which rotates very slowly (one rotation every 243 days). Jupiter, on the other hand, rotates much faster than the Earth and thus has a very dynamic atmosphere, including the giant red spot (which is actually a high pressure system rather than a low pressure system like in a hurricane). The sun also rotates on its axis, and the Coriolis effect is a controlling factor in the directions of rotation of sun spots.

Tornadoes are sometimes mentioned as being caused by the Coriolis effect, but they are too small, and their wind speeds too great, for Coriolis to have any effect. Likewise, the direction of rotation of water going down the drain in a sink is not affected by the Coriolis effect, its size being much too small.

The Coriolis effect is very small, but the long distances that water travels in an ocean current provide plenty of time for the Coriolis effect to accumulate. In special situations motions over limited distances can demonstrate a cumulative effect if observed over long time periods. The classic example is the Foucault pendulum (named after French physicist Jean Bernard Foucault), which is a pendulum with a heavy weight hung on a very long wire (several stories high) from an approximately frictionless pivot. These are often seen in science museums. The back-and-forth motion of the weight appears to stay in the same vertical plane, but if one waits long enough one will notice that the weight is not coming back to exactly the same spot at the full extent of each swing. Typically, small wooden blocks are set up in a large circle around the pendulum at just the right distance to be hit by the weight. Over a day or more each block is eventually knocked over by the oscillating weight. The plane of oscillation of the pendulum is thus seen to be slowly rotating (clockwise in the Northern Hemisphere) around a vertical axis perpendicular to the floor (the Earth’s surface). This rotation is caused by the Coriolis effect.

Pendulum period tied to latitude

The amount of time it takes for the oscillating weight of the Foucault pendulum to come back to the first block it knocked down depends on the latitude where the pendulum is located. At the North Pole it takes 24 hours. Here it is easy to visualize the Earth actually turning under the oscillating pendulum, which itself is really staying in the exact same oscillating plane relative to the stars. If the pendulum is somewhere south of the North Pole (but not on the equator) its plane of oscillation still rotates, but it takes longer for the pendulum to come back to where it started. This is more difficult to visualize, because now the whole pendulum is traveling around with the rotating earth along a latitudinal circle. As it does so, its oscillations still stay in the same direction relative to the stars, so that the plane of oscillation rotates relative to the Earth’s surface. On the equator, the pendulum will stay in the same plane relative to the Earth, since its plane of oscillation is perpendicular to the Earth’s axis of rotation.

There are also special types of very long waves that are affected by the Coriolis effect, or even caused by it. For example, when the tide propagates southward as a very long wave along an east coast in the Northern Hemisphere, the Coriolis effect on southerly flowing flood currents causes a raised water level at the coast. The Coriolis effect on northerly flowing ebb currents causes a lowered water level at the coast. The result is a greater tide range at the coast than offshore. This long tide wave is called a coastal Kelvin wave. The restoring force to the vertical oscillation of the water surface in this wave is gravity, but it is the Coriolis effect that causes the slope in water surface toward the coast. Kelvin waves can also propagate eastward along the equator, where there obviously is no coast, but where the fact that the Coriolis force is zero acts like a boundary.

The Coriolis force can also be a restoring force in a wave, in this case causing horizontal oscillations. The Gulf Stream and other strong currents on the western sides of the oceans are caused by the change in Coriolis force with latitude. This change in Coriolis force with latitude can also be the restoring force in a wave called a Rossby wave. In such a wave, which propagates westward across an ocean, parcels of water oscillate north and south about a latitude line. The current is approximately in geostrophic balance, but when it moves a little northward it is forced back southward by the change in Coriolis force, and vice versa.

Both eastward-propagating equatorial Kelvin waves and westward-propagating Rossby waves play key roles in the phenomena of El Niño. When the westward trade winds collapse and the warm water in the western Pacific moves eastward to the South American coast, it is in the form of equatorial Kelvin waves. At the coast these waves split, heading north toward California and south toward Peru in the form of coastal Kelvin waves. Some of the energy is also reflected back westward in the form of Rossby waves. This all plays some (as yet not fully understood) role in the timing of El Niño cycles.

Rossby waves are also found in the atmosphere at high elevations above the Earth. Around the North Pole (where the change in Coriolis force with latitude is large) there are typically between four and six very long horizontal waves, with a wavelength greater than the width of the U.S.

In Paris, in 1835, when Gustave Gaspard de Coriolis published the paper that first explained the effect that is now named after him, he probably did not realize how important that effect would be in explaining motions in the atmosphere and ocean. His other major (and much longer) publication that same year was probably viewed with more interesta 176-page book explaining the mathematical theory of billiards.

Bruce Parker is the chief of the Coast Survey Development Laboratory, National Ocean Service, NOAA.

By Ocean Navigator