Coriolis effect

The Coriolis effect is an apparent deflection of a moving object in a rotating frame of reference.

The Coriolis effect caused by the rotation of the Earth is responsible for the precession of a Foucault pendulum and for the direction of rotation of cyclones. Due to the effect, cyclones rotate counterclockwise in the Northern hemisphere and clockwise in the Southern hemisphere. However, contrary to popular opinion, the Coriolis effect is not a determining factor in the rotation of water in toilets or bathtubs (see the draining bathtubs/toilets section below).

The effect is named after Gaspard-Gustave Coriolis, a French scientist, who described it in 1835, though the mathematics appeared in the tidal equations of Laplace in 1778.

Formula

The formula for the Coriolis acceleration is

\mathbf{a}=-2\boldsymbol\omega\times\mathbf{v}

where (here and below) $\mathbf{v}$ is the velocity of the particle in the rotating system, and $\boldsymbol\omega$ is the angular velocity vector (which has magnitude equal to the rotation rate and points in the direction of the axis of rotation) of the rotating system. The equation may be multiplied by the mass of the relevant object to produce the Coriolis force. See Fictitious force for a derivation.

Note that this is vector multiplication. In non-vector terms: at a given rate of rotation of the observer, the magnitude of the Coriolis acceleration of the object will be proportional to the velocity of the object and also to the sine of the angle between the direction of movement of the object and the axis of rotation.

The Coriolis effect is the behavior added by the Coriolis acceleration. The formula implies that the Coriolis acceleration is perpendicular both to the direction of the velocity of the moving mass and to the rotation axis. So in particular:

if the velocity (as always, in the rotating system) is zero, the Coriolis acceleration is zero
if the velocity is parallel to the rotation axis, the Coriolis acceleration is zero
if the velocity is straight (perpendicularly) inward to the axis, the acceleration will follow the direction of rotation (i.e., be perpendicular to the rotation axis and the particle velocity)
if the velocity is following the rotation, the acceleration will be (perpendicularly) outward from the axis

In the formula above, the vectors are 3-d. If we are considering the simpler case of motion restricted to the surface of a rotating turntable the equation simplifies somewhat to:

\mathbf{a}=-2\omega \mathbf{k}\times (u,v)

where $\mathbf{k}$ is a unit local vertical and $(u,v) \$ is the velocity 2-d vector in the plane of the turntable.

When considering atmospheric dynamics, the Coriolis acceleration (strictly a 3-d vector in the first formula above) appears only in the horizontal equations, due to the neglect of products of small quantities and other approximations. The term that appears is then

- f \mathbf{k} \times (u,v)\,

where $f = 2 \omega \sin(\phi) \$ (where $\phi \,$ is the latitude) is called the Coriolis parameter and $(u,v) \$ are the horizontal components of the velocity.

What the Coriolis effect is not

The Coriolis effect does not depend on the curvature of the Earth, only on its rotation. However, the value of the Coriolis parameter, $f \$ , does vary with latitude, and that is due to the Earth's shape.
The Coriolis effect is not the fictitious Centrifugal force given by $\omega\times(\omega\times\mathbf{r})$ . However, the co-existence of Coriolis and centripetal forces makes simple explanations of the effect of Coriolis in isolation difficult.

Visualisation of the Coriolis effect

To demonstrate the Coriolis effect, a parabolic turntable can be used. On a flat turntable the centrifugal force, which always acts outwards from the rotation axis, would lead to objects being forced out off the edge. But if the surface of the turntable has the correct parabolic bowl shape, and is rotated at the correct rate, then the component of gravity tangential to the bowl surface will exactly balance the centrifugal force. This allows the Coriolis force to be displayed in isolation.

Discs cut from cylinders of dry ice can be used as pucks, moving around almost frictionlessly over the surface of the parabolic turntable, allowing effects of Coriolis on dynamic phenomena to show themselves. To get a view of the motions as seen from a rotating point of view, a video-camera is attached to the turntable in such a way that the camera is co-rotating with the turntable.

When the fluid is rotating on a flat turntable, the surface of the fluid naturally assumes the correct parabolic shape. This fact may be exploited in order to make a parabolic turntable, by using a fluid that sets after several hours, such as a synthetic resin.

Draining bathtubs/toilets

A popular misconception is that 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 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.

This is less of a puzzle once one remembers that the earth revolves once per day but that a bathtub takes only minutes to drain. When the water is being drawn towards the plughole, the radius with which it is spinning around it decreases, so its rate of rotation increases from the low background level to a noticeable spin in order to conserve its angular momentum (the same effect as ballet dancers bringing their arms in to cause them to spin faster).

Coriolis in Meteorology

Perhaps the most important instance of the Coriolis effect is in the large scale dynamics of the oceans and the atmosphere. In meteorology, Coriolis effects tend to dominate the centrifugal effects associated with, for example, the rotation of large cyclones, because the latter are usually balanced by an ambient pressure gradient (exactly analogously to the slope on a parabolic turntable).

Flow around a low-pressure area

If a low pressure area forms in the atmosphere, air will tend to flow in towards it, but will be deflected perpendicular to its velocity by the Coriolis acceleration. A system of equilibrium can then establish itself creating circular movement, or a cyclonic flow.

The force balance is largely between the pressure gradient force acting towards the low-pressure area and the Coriolis acceleration acting away from the center of the low pressure. Instead of flowing down the gradient, the air tends to flow perpendicular to the air-pressure gradient and forms a cyclonic flow. This is an example of a more general case of geostrophic flow in which air flows along isobars. On a non-rotating planet the air would flow along the straightest possible line, quickly leveling the air pressure. Note that the force balance is thus very different from the case of "inertial circles" (see below) which explains why mid-latitude cycles are larger by an order of magnitude than inertial circle flow would be.

This pattern of deflection, and the direction of movement, is called Buys-Ballot's law. The pattern of flow is called a cyclone. In the Northern Hemisphere the direction of movement around a low-pressure area is counterclockwise. In the Southern Hemisphere, the direction of movement is clockwise because the rotational dynamics is a mirror image there. However, at high altitudes, outward-spreading air rotates in the opposite direction ^* Cyclones cannot form on the equator, and they rarely travel towards the equator, because in the equatorial region the coriolis parameter is small, and exactly zero on the equator.

Inertial circles

An air or water mass moving with speed $v\,$ subject only to the Coriolis force travels in a circular trajectory called an 'inertial circle'. Since the force is directed at right angles to the motion of the particle, it will move with a constant speed, and perform a complete circle with frequency $f$ . The magnitude of the Coriolis force also determines the radius of this circle:

R=v/f\,

On the Earth, a typical mid-latitude value for $f$ is 10⁻⁴; hence for a typical atmospheric speed of 10 m/s the radius is 100 km, with a period of about 14 hours. In the ocean, where a typical speed is closer to 10 cm/s, the radius of an inertial circle is 1 km. These inertial circles are clockwise in the northern hemisphere (where trajectories are bent to the right) and anti-clockwise in the southern hemisphere.

If the rotating system is a parabolic turntable, then $f$ is constant and the trajectories are exact circles. On a rotating planet, $f$ varies with latitude and the paths of particles do not form exact circles. Since the parameter $f$ varies as the sine of the latitude, the oscillations associated with a given speed are smallest at the poles (latitude = $\pm 90^\circ$ ), and would increase indefinitely at the equator, except the dynamics ceases to apply close to the equator.

The dynamics of inertial circles are different from those of mid-latitude cyclones. In the latter case, the Coriolis force (directed outward) is in an approximate balance with the pressure gradient force (directed inward), a situation known as geostrophic balance. In particular, cyclones rotate in the opposite direction as inertial circles.

Length scales and the Rossby Number

The time, space and velocity scales are important in determining the importance of the Coriolis effect. Whether rotation is important in a system can be determined by its Rossby number, which is the ratio of the velocity of a system to the product of the Coriolis parameter, and the lengthscale of the motion:

Ro = \frac{U}{fL}

A small Rossby number signifies a system which is strongly affected by rotation, and a large Rossby number signifies a system in which rotation is unimportant. An atmospheric system moving at U = 10m/s occupying a spatial distance of L=1000km, has a Rossby number

Ro = \frac{10}{10^{-4}\times 1000\times10^3} = 0.1

A man playing catch may throw the ball at U=30m/s in a garden of length L=50m. The Rossby number in this case would be

Ro = \frac{30}{10^{-4}\times 50} = 6000

Needless to say, one does not worry about which hemisphere one is in when playing catch in the garden. However, an unguided missile obeys exactly the same physics as a baseball, but may travel far enough and be in the air long enough to notice the effect of Coriolis. Long range shells landed close to, but to the right of where they were aimed until this was noted (or left if they were fired in the southern hemisphere, though most were not).

The Rossby number can also tell us about the bathtub. If the lengthscale of the tub is about L=1m, and the water moves towards the drain at about 60cm/s, then the Rossby number is

Ro = \frac{0.6}{10^{-4}\times 1} = 6 000

Thus, the bathtub is, in terms of scales, much like a game of catch, and rotation is likely to be unimportant.

However, if the experiment is very carefully controlled to remove all other forces from the system, rotation can play a role in bathtub dynamics. An article in the British "Journal of Fluid Mechanics" in the 1930's describes this. The key is to put a few drops of ink into the bathtub water, and observing when the ink stops swirling, meaning the viscosity of the water has dissipated its initial vorticity (or curl; i.e. $\nabla \times U = 0$ ) then, if the plug is extracted ever so slowly so as not to introduce any additional vorticity, then the tub will empty with a counterclockwise swirl in England.

Terrestrial effects summarized

A summary of Coriolis effects on the Earth's surface. Note that some of these assume that we are considering a "2-d" velocity, in the plane tangential to the planet's surface (if this restriction is removed, the latitude dependence of the strength of the Coriolis effect disappears).

the magnitude of the Coriolis effect changes with the latitude and the speed of the air (and water).
the Coriolis effect is greatest in polar regions where the surface of the Earth is at right angles to the axis of rotation.
the Coriolis effect decreases nearer the equator because the surface of the Earth is parallel to the axis of rotation.
the Coriolis effect causes air (and water) masses to turn right in the northern hemisphere and causes air (and water) masses to turn left in the southern hemisphere.
the Coriolis effect gives rise to geostrophic winds and currents.
a geostropic motion is one in which a pressure gradient force is balanced by an equal and opposite Coriolis force.
the effect works in opposite directions in the two hemispheres.

The Coriolis effect strongly affects the large-scale atmospheric circulation, leading to the Hadley, Ferrel, and Polar cells. In the oceans, Coriolis is responsible for the propagation of Kelvin waves and the establishment of the Sverdrup balance.

Coriolis Elsewhere

Coriolis flow meter

A practical application of the Coriolis effect 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 operating principle essentially involves rotation, though not through a full circle. It 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.

Molecular 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. Coriolis effects will therefore be 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.

Ballistics

The Coriolis effects became important in external ballistics for calculating the trajectories of very long-range artillery shells. The most famous historical example was the Paris gun, used by the Germans during World War I to bombard Paris from a range of about 120 km.

References

Physics and meteorology references

Gill, AE 'Atmospher-Ocean dynamics'', Academic Press, 1982.

Durran, D. R., 1993: Is the Coriolis force really responsible for the inertial oscillation?, Bull. Amer. Meteor. Soc., 74, 2179–2184; Corrigenda. Bulletin of the American Meteorological Society, 75, 261

Durran, D. R., and S. K. Domonkos, 1996: An apparatus for demonstrating the inertial oscillation, Bulletin of the American Meteorological Society, 77, 557–559.

Marion, Jerry B. 1970, Classical Dynamics of Particles and Systems, Academic Press.

Persson, A., 1998 How do we Understand the Coriolis Force? Bulletin of the American Meteorological Society 79, 1373-1385.

Symon, Keith. 1971, Mechanics, Addison-Wesley

Norman Ph. A., 2000 An Explication of the Coriolis Effect, Bulletin of the American Meteorological Society: Vol. 81, No. 2, pp. 299–303.

Historical references

Grattan-Guinness, I., Ed., 1994: Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences. Vols. I and II. Routledge, 1840 pp.
1997: The Fontana History of the Mathematical Sciences. Fontana, 817 pp. 710 pp.

Khrgian, A., 1970: Meteorology—A Historical Survey. Vol. 1. Keter Press, 387 pp.

Kuhn, T. S., 1977: Energy conservation as an example of simultaneous discovery. The Essential Tension, Selected Studies in Scientific Tradition and Change, University of Chicago Press, 66–104.

Kutzbach, G., 1979: The Thermal Theory of Cyclones. A History of Meteorological Thought in the Nineteenth Century. Amer. Meteor. Soc., 254 pp.

Footnotes

External links

The definition of the Coriolis effect from the Glossary of Meteorology

The Coriolis Effect PDF-file. 17 pages. A general discussion by Anders Persson of various aspects of the coriolis effect, including Foucault's Pendulum and Taylor columns.
Anders Persson The Coriolis Effect: Four centuries of conflict between common sense and mathematics, Part I: A history to 1885 History of Meteorology 2 (2005)
Coriolis Force - from ScienceWorld
The Coriolis Effect: An Introduction. Details of the causes of prevailing wind patterns. Targeted towards ages 5 to 18.
Coriolis Effect and Drains An article from the NEWTON web site hosted by the Argonne National Laboratory.
Do bathtubs drain counterclockwise in the Northern Hemisphere? by Cecil Adams.
Bad Coriolis. An article uncovering misinformation about the Coriolis effect. By Alistair B. Fraser, Emeritus Professor of Meteorology at Pennsylvania State University

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