Coriolis effect

In physics, the Coriolis effect or Coriolis force is a manifestation of inertia first described in full by Gaspard-Gustave Coriolis, a French scientist, in 1835.

Formula

The formula for the Coriolis force is as follows.

<math>\vec{F}_C=2m(\vec{v} \times \vec{\omega})<math>

In this formula the arrow above the symbol indicates vector quantities, <math>\vec{F}_C<math> is the Coriolis force, m is mass, <math>\vec{v} <math> is the velocity, <math>\times<math> is the vector cross product, and <math>\vec{\omega}<math> is the angular velocity of the rotating system.

An image of the Coriolis effect

The animation shows a simplified representation of the Coriolis effect. An object is moving with respect to a rotating disk. There is no friction between the disk and the object. The object is moving in a straight line with constant velocity, the way objects move when no force is being exerted. Seen from the perspective of the rotating disk, the object's trajectory seems to be a curved trajectory.

Seen from above, the rotating Earth looks like a rotating disk. If, for example, a cannonball is fired, then from the moment of firing on, the cannonball is no longer co-rotating with objects that are in contact with the earth. Seen from a non-rotating perspective the cannonball follows a trajectory that is to be expected for a cannonball in Earth's gravity. But seen from the perspective of the rotating Earth, the cannonball appears to be deflected to one side compared to how it would move if the Earth were not rotating. (Deflected to the right if the cannonball is fired in the northern hemisphere, deflected to the left in the southern hemisphere.)

Note that the above demonstration is limited to the trajectory of objects that are free-moving. A cannonball that has been fired will from that moment on follow a parabolic trajectory (not counting air friction for now). A full discussion of the Coriolis effect must also cover non-free motion. For example, the water in the oceans is not free-moving, its motion is constrained, and how it will actually move is the result of both the constraints and the tendency to move in the direction it would move to if it would not be constrained.

The dynamics of the Coriolis effect

To present the full extent of the Coriolis effect the following setup is used: a rotating mercury mirror. This kind of device is in fact used in astronomy.¹ The surface of a rotating mercury pool is a perfect parabolic mirror. Every part of the rotating mercury mirror is in dynamic equilibrium. In this state of dynamic equilibrium the force towards the center is proportional to the distance to the center, hence the periodicity of circular motion is the same everywhere; all the mercury is co-rotating. An object floating in the mercury is in dynamic equilibrium everywhere, as long as it is co-rotating with the mercury. This provides the steady background for the full Coriolis effect to display itself.

First to consider is the situation when an object, say a tiny hovercraft, hovers above the surface of the mirror. (In this discussion any air resistance, which is small anyway, is not considered.) The hovercraft is not interacting with the mercury, so for now only the shape of the mirror matters, not its rotation. Seen from the perspective of the inertial frame, when the hovercraft is released from a standstill position close to the rim, with no propulsion on, from then on the hovercraft will oscillate back and forth across the surface. Because of the specific shape of the mirror, the oscillation will be a harmonic oscillation. The period of the oscillation will be the same as the period of rotation of the rotating mercury mirror.
It is also possible to have the hovercraft oscillate in two perpendicular directions. If the timing of the two oscillations is set up carefully, then the resultant motion of the two linear independent motions is in the shape of an ellipse, or in the case of full symmetry a circle.

Thinking of elliptical trajectories as linear combinations of harmonic oscillations helps to see the symmetries of the underlying physics, and it helps to visualize the non-constant velocity of the object following an elliptical trajectory.

Consider the situation where the hovercraft is moving along an elliptical trajectory. The period of this elliptical orbit is the same as the period of the rotating mercury mirror. Seen from the perspective of the rotating system, the hovercraft is following a small circular trajectory. For every revolution of the rotating system the hovercraft goes around the small circular trajectory twice. Mathematically, this small circular trajectory can be obtained by subtracting a concentric circular trajectory from the elliptical trajectory (with the subtracted circle carefully chosen to obtain the smallest possible difference).

Interaction between the two systems

The next thing to consider is what happens when there is some drag, some friction. The two systems involved are the inertial system and the rotating system. The direction of manifestation of inertia is determined by the direction of acceleration with respect to the inertial frame of reference, which is a non-rotating frame of reference. The rotating system is in this particular example the rotating mercury mirror that will drag along any object in contact with it. Usually manifestations of inertia and drag are pointing in the same direction, but when rotation is involved they aren't.

When some friction between the hovercraft and the mercury is added to the dynamics of the situation, then the elliptical orbit will gradually be rounded into a circular orbit.

Seen from the perspective of the rotating system, what was first a small circular trajectory is now spiraling inward. There is interaction between the two systems: the drag from the mercury is changing one dynamic equilibrium, elliptical orbit, into another dynamic equilibrium: circular orbit.

Eddies and Coriolis effect

Suppose that at a co-rotating device at some point sucks away some mercury, creating a low mercury level area. Mercury would start flowing towards that area from all directions.

The mercury that initially starts to flow in the radial direction is deflected as it flows. If the mercury mirror is rotating counterclockwise, then the mercury that initially flows in the radial direction is deflected to the right. The mercury that initially flows in the tangential direction is deflected too. The mercury that flows in the tangential direction is no longer in dynamic equilibrium; its velocity is no longer exactly the velocity that is necessary for dynamic equilibrium, so its distance to the center will change, a deflection. All mercury that is flowing down the mercury level gradient is deflected to the right with respect to its initial direction of motion. This models the effect that in meteorology led to the formulation of Buys-Ballot's law.

The mercury of the rotating mercury mirror will tend to spiral around the low mercury level area. If the rotating mercury mirror is rotating counterclockwise the eddy will rotate counterclockwise too. (In the free-motion example discussed above the inertial Coriolis motion corresponds to an apparent clockwise motion as seen in the rotating system.) The mercury level gradient force is deflecting the flowing mercury to the left. Without this mercury level gradient force, the mercury of the eddy would be deflected to the right.

If the eddy would contract, the center-directed force would do work, and the angular velocity of the eddy would increase. Friction tends to slow down the eddy, but as long as there is a center-directed force, contraction of the eddy has the effect of sustaining the eddy's velocity.

For stable circular motion with respect to an inertial system the magnitude of the centripetal force is given by: F=m<math>\omega<math>v (where <math>\omega<math> is the angular velocity of the rotating system, and v is the velocity of the mass with respect to the inertial system.)

In the case of the eddy on the rotating mercury mirror the magnitude of the manifestation of inertia is determined by the actual motion, the orbiting motion with respect to the inertial system. When this force is expressed in terms of the velocity with respect to the rotating system the magnitude of the force is given by: F=2m<math>\omega<math>v. (Where <math>\omega<math> is the angular velocity of the rotating system, and v is the velocity of the mass with respect to the rotating system.)

Defining Coriolis force

There are two ways in common usage, to categorize particular inertial forces as either 'centrifugal force' or 'Coriolis force' Some authors on the subject use the first, some authors use the second:

1. Start with the rotating system and define centrifugal force as any force that acts in radial direction with respect to a polar coordinates grid, and define Coriolis force as any force that acts in tangential direction with respect to a polar coordinates grid.
2. The way Coriolis force is implemented in meteorological calculations that take the rotation of Earth into account: the steady background is the centrifugal force associated with the rotation of Earth (modeled in this article by the rotation of the rotating mercury mirror). Coriolis force is then defined as the velocity-dependent inertial force that needs to be taken into account.

The second way to define the distinction between centrifugal force and Coriolis force acting in a system as a whole is mathematically much more elegant.

The Coriolis effect in the atmosphere

Hurricane Isabel east of the Bahamas on 2003-09-15. Photograph courtesy NASA.

Like in the example of the rotating mercury mirror, the atmosphere of the Earth would be in steady, co-rotating dynamic equilibrium if nothing would stir it.

Air masses are being heated at the equator. The decrease in density increases their buoyancy and they rise, and they are replaced by air moving over the Earth's surface towards the equator. Since there is not enough friction between the surface of the earth and the air, the masses of air are not brought up to the velocity necessary to remain in co-rotation with the rotation of the Earth. Because of that there are relatively steady winds at certain latitudal regions. These are known as the trade winds. In the northern hemisphere the trade winds blow from the North-East, and in the southern hemisphere they blow from the South-East. The air that has risen at the equator does not move all the way to the poles. It is prevented from doing so because the Coriolis effect tends to turn moving air full circle.

The Coriolis effect plays a strong role in weather patterns, where it affects prevailing winds and the rotation of storms, as well as in the direction of ocean currents due to the Ekman spiral. Above the atmospheric boundary layer, friction plays a relatively minor role, as air parcels move mostly parallel to each other. Here, an approximate balance between pressure gradient force and Coriolis force exists, causing the geostrophic wind, which is the wind effected by these two forces only, to blow along isobars (along lines of constant geopotential height, to be precise). Thus a northern hemispheric low pressure system, or cyclone, rotates in a counterclockwise direction, while northern hemispheric high pressure systems, or cyclones in the southern hemisphere, rotate in a clockwise manner, as described by Buys-Ballot's law.

Coriolis flow meter

A practical application of the Coriolis force is the mass flow meter, an instrument that measures the mass flow rate of a fluid through a tube. The operating principle was introduced in 1977 by Micro Motion Inc. Simple flow meters measure volume flow rate, which is proportional to mass flow rate only when the density of the fluid is constant. If the fluid has varying density, or contains bubbles, then the volume flow rate multiplied by the density is not an accurate measure of the mass flow rate. The Coriolis mass flow meter works by inducing a vibration of the tube through which the fluid passes, and subsequently monitoring and analysing the inertial effects that occur in response to the combination of the induced vibration and the mass flow.

More occurrences of the Coriolis effect

See Taylor-Proudman theorem for a startling consequence of the Coriolis effect: in a rotating reference frame, if the flow has low Rossby number but high Reynolds number, all steady solutions to the Navier-Stokes equations have the property that the fluid velocity is uniform along any line parallel to the rotation axis. In oceanic flow, it is possible to ignore the non-vertical components of the Earth's rotation, so if the conditions of the theorem apply (<math>Re >\!\!> 1<math> is universal but using <math>0.1{\rm m/s}<math> as a typical flow speed and using 4km as a depth, <math>f=10^{-4}{\rm s}^{-1}<math> gives <math>Ro\simeq 0.25<math> which is marginal), the fluid velocity is identical at all points along any single vertical line (known as a Taylor column). The Taylor-Proudman theorem is widely used when considering limnological flows, astrophysical flows (such as solar and jovian dynamics) and some industrial problems such as turbine design.

People often ask whether the Coriolis effect determines the direction in which bathtubs or toilets drain, and whether water always drains in one direction in the Northern Hemisphere, and in the other direction in the Southern Hemisphere. The answer is almost always no. The Coriolis effect is a few orders of magnitude smaller than other random influences on drain direction, such as the geometry of the sink, toilet, or tub; whether it is flat or tilted; and the direction in which water was initially added to it. If one takes great care to create a flat circular pool of water with a small, smooth drain; to wait for eddies caused by filling it to die down; and to remove the drain from below (or otherwise remove it without introducing new eddies into the water) — then it is possible to observe the influence of the Coriolis effect in the direction of the resulting vortex. There is a good deal of misunderstanding on this point, as most people (including many scientists) do not realize how small the Coriolis effect is on small systems.² On the other hand, the Coriolis effect can have a visible effect over large amounts of time, and has been observed to cause uneven wear on railroad tracks and to cause rivers to dig their beds deeper on one side.

Effects due to the Coriolis force also appear in atomic physics. In polyatomic molecules, the molecule motion can be described by a rigid body rotation and internal vibration of atoms about their equilibrium position. As a result of the vibrations of the atoms, the atoms are in motion relative to the rotating coordinate system of the molecule. A Coriolis force is therefore present and will cause the atoms to move in a direction perpendicular to the original oscillations. This leads to a mixing in molecular spectra between the rotational and vibrational levels.

Insects of the group Diptera are thought to use two small vibrating structures at the side of their bodies to detect the effects of the Coriolis force. These so called Halteres play an important role in these insects' ability to perform aerobatics.

External links

• Note 1: Large Zenith Telescope (http://www.astro.ubc.ca/LMT/lzt/index.html) A 6 meter diameter, 5 revolutions per minute rotating mercury mirror.

• Note 2: Coriolis effect misconceptions (http://www.ems.psu.edu/~fraser/Bad/BadCoriolis.html)

• Note 3: Usenet Physica FAQ, bathtub (http://math.ucr.edu/home/baez/physics/General/bathtub.html)cs:Coriolisova síla

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