Coriolis effect

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The Coriolis effect is an apparent deflection of a moving object in a rotating frame of reference. The effect is named after Gaspard-Gustave Coriolis, a French scientist, who discussed it in 1835, though the mathematics appeared in the tidal equations of Laplace in 1778.


Contents

Simple examples

The earth has a circumference at the Equator of 40,075 kilometers. Since it rotates once every 24 hours, objects at the Equator are travelling at 40,075 divided by 24 = 1670 kilometers per hour. However at the North Pole, objects do not have any significant speed of rotation. In between the Equator and North Pole, objects have intermediate speeds of rotation. The Earth's Coriolis effect arises because of the differences in the Earth's speed of rotation at different places.

In a non-rotating world an object could simply move from one place to another by accelerating in the direction required. This is our normal experience on the earth because the Earth's Coriolis effect is small on a human scale and because we are usually firmly attached to the Earth. However over larger distances for objects not attached to the Earth, the effect is significant.

Consider a missile fired from the North Pole towards a place on the Equator. Since the Earth is rotating, the missile must also pick up the rotational speed of its intended destination, otherwise its destination will rotate away from under it as it travels south. Without a correction, observers on the Earth will see the object apparently being deflected from its destination. However the force producing this apparent deflection is an illusion; the earth merely rotates as the missile is travelling. Nevertheless an allowance would have to be made for this illusion when aiming the missile.

The scale of the effect can be seen by taking the distance from the North Pole to the Equator, which is a quarter of the circumference, roughly 10,000 kilometers. The missile, or any other object, must gain 1670 km/h divided by 10,000 = 0.167 km/hr on average for each kilometer travelled south. This small difference in speed for each kilometer travelled is too small to be noticed by people attached to the surface of the earth. However weather systems occupy large areas and are not attached to the surface of the earth. Consequently the Coriolis effect is an important factor in meteorology.

Like the missile in the Northern Hemisphere, a mass of air travelling south has less speed than the air at its destination and so appears to be deflected west. Conversely air travelling north has excess speed and appears to be deflected east. As the wind blows from all sides to fill an area of low pressure, the Coriolis effect therefore creates rotation around the low pressure system. The winds around areas of low pressure circulate counter-clockwise in the Northern Hemisphere, while in the Southern Hemisphere thia circulation is clockwise. The rotation produces the characteristic swirls that can be seen on satellite photographs of weather systems, and of hurricanes in particular.

A common fallacy is that the Coriolis effect affects the rotation of water flowing through plug-holes. The water in a bath is less than one hundredth of a kilometer across. Consequently difference in the speed of the Earth's rotation from one side of the bath to the other is insignificant and cannot appreciably affect the rotation of the bath-water down the plug-hole. The rotation of the water is chiefly governed by the plumbing and random eddies in the water.

Although the rotation of the Earth provides the most obvious examples of the Coriolis effect, it arises in other rotating systems. For example, someone standing at the edge of a rotating carousel could throw a ball to someone standing at the center. If thrown without making allowance for the Coriolis effect, the ball would not reach its target.

Formula

The formula for the Coriolis acceleration is

<math>-2\boldsymbol\omega\times\mathbf{v}</math>

where (here and below) v is the velocity in the rotating system, <math>\omega</math> is the angular velocity (the rotation rate and orientation) 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 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
  • if the velocity is following the rotation, the acceleration will be (perpendicularly) outward from the axis

When considering atmospheric dynamics, the Coriolis acceleration (strictly a 3-d vector in the 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

<math>- f \mathbf{k} \times (u,v)</math>

where k is a unit local vertical, <math>f = 2 \omega sin(latitude)</math> is called the Coriolis parameter and <math>(u,v)</math> are the horizontal components of the velocity.

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. 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.

Draining bathtubs/toilets

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.1 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. The increase in rotational speed around the plug hole is because water is being drawn towards the plughole and so its radius of its mass to the point it is spinning around 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 bringing one's arms in on a swivel chair making it spin faster)

The time and space scales are important in determining the importance of the Coriolis effect. Weather systems are large enough to feel the curvature of the earth and generally rotate less than once a day so a similar timescale to the earth's rotation so the Coriolis effect is dominant. An unguided missile, if fired far enough, will travel far enough and be in the air long enough to notice the effect but the dominant effect is the direction it was fired in. 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 weren't). You don't worry about which hemisphere you're in when playing catch in the garden though this is exactly the same physics at a smaller scale. A bathtub is best approximated (in terms of scale) by a game of catch.

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.

Ballistics

In firing projectiles over a significant distance, the rotation of the Earth must be taken into account. During its flight, the projectile moves in a straight line (not counting gravitation and air resistance for now). The target, co-rotating with the Earth, is a moving target, so the gun must be aimed not directly at the target, but at a point where the projectile and the target will arrive simultaneously.

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.

Visualisation of the Coriolis effect


To demonstrate the Coriolis effect, a turntable can be used. If the turntable is flat, then the centrifugal force, which always acts outwards from the rotation axis, would lead to objects being forced out off the edge. If the turntable has a bowl shape, then the component of gravity tangential to the bowl surface will tend to counteract the centrifugal force. If the bowl is parabolic, and spun at the appropriate rate, then gravity exactly counteracts the centrifugal force and the only net force (bar friction, which can be minimised) acting is then the Coriolis force. If the turntable is a dish with a rim and filled with liquid, then when the liquid is rotating it naturally assumes a parabolic shape (for the same reasons). If a liquid that sets after several hours is used, such as a synthetic resin, a permanent shape is obtained.

Disks cut from cylinders of dry ice can be used as pucks, moving around almost frictionlessly over the surface of the parabolic turntable, allowing dynamic phenomena to show themselves. To also 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. This type of setup, with a parabolic turntable, at the center about a centimeter deeper than the rim, is used at Massachusetts Institute of Technology (MIT) for teaching purposes. 2

Inertial circles


If an object moves subject only to the Coriolis force, it will move in a circular trajectory called an 'inertial circle'.

In an inertial circle, the force balance is sometimes most easily understood as being between two fictitious forces, the centrifugal force (directed outwards) and the Coriolis force (directed inwards). The dynamics is thus quite different to mid-latitude cyclones or hurricanes, in which cases the force balance is between the pressure gradient force (directed inwards) and the Coriolis force (directed out). In particular, this means that the direction of orbit is opposite to that of mid-latitude cyclones.

The frequency of these oscillations is given by f, the coriolis parameter; and their radius by [1]:

<math>v/f</math>,

where v is the velocity of the air mass. 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. For a turntable rotating about once every 6 seconds, f is one, hence the radius of the circles, in cm, is numerically the same as the speed, in cm/s.

The centrifugal force is <math>v^2/r</math> and the Coriolis force <math>vf</math>, hence the forces balance when <math>v^2/r = vf</math>, i.e. <math>v/f = r</math>, giving the expression above for the radius of the circles.

If the rotating system is a turntable, then <math>f</math> is constant and the trajectories are exact circles. On a rotating planet, <math>f</math> varies with latitude and the circles do not exactly close.

Closer to the equator the component of the velocity towards or away from the Earth's axis is smaller, this component varies as <math>sin(latitude)</math>, and this is taken into account in the parameter f. For a given velocity the oscillations are smallest at the poles as shown by the picture and would increase indefinitely at the equator, except the dynamics ceases to apply close to the equator. On a rotating planet the oscillations are only approximately circular [2] and do not form closed loops as indicated in the simplified picture.

References

Physics and meteorology references

  • Gill, AE 'Atmospher-Ocean dynamics, Academic Press, 1982.
  • Durran, D. R., and S. K. Domonkos, 1996: An apparatus for demonstrating the inertial oscillation. Bulletin of the American Meteorological Society, 77, 557–559.
    1300 KB PDF-file of the above article
  • Marion, Jerry B. 1970, Classical Dynamics of Particles and Systems, Academic Press.
  • Symon, Keith. 1971, Mechanics, Addison-Wesley

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.


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